research article generalized lorentz spaces and...

Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 302941 14 pageshttpdxdoiorg1011552013302941

Research ArticleGeneralized Lorentz Spaces and Applications

Hatem Mejjaoli

Department of Mathematics College of Sciences Taibah University PO B 30002 Al Madinah Al Munawarah Saudi Arabia

Correspondence should be addressed to HatemMejjaoli hatemmejjaoliipestrnutn

Received 21 March 2013 Accepted 6 June 2013

Academic Editor Jose Luis Sanchez

Copyright copy 2013 Hatem Mejjaoli This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We define and study the Lorentz spaces associated with the Dunkl operators onR119889 Furthermore we obtain the Strichartz estimatesfor the Dunkl-Schrodinger equations under the generalized Lorentz norms The Sobolev inequalities between the homogeneousDunkl-Besov spaces and generalized Lorentz spaces are also considered

ldquoDedicated to Khalifa Trimecherdquo

1 Introduction

Dunkl operators 119879119895 (119895 = 1 119889) introduced by Dunklin [1] are parameterized differential-difference operators onR119889 that are related to finite reflection groups Over thelast years much attention has been paid to these operatorsin various mathematical (and even physical) directionsIn this prospect Dunkl operators are naturally connectedwith certain Schrodinger operators for Calogero-Sutherland-type quantum many-body systems [2ndash4] Moreover Dunkloperators allow generalizations of several analytic structuressuch as Laplace operator Fourier transform heat semigroupwave equations and Schrodinger equations [5ndash11]

In the present paper we intend to continue our studyof generalized spaces of type Sobolev associated with Dunkloperators started in [12 13] In this paper we study thegeneralized Lorentz spaces andwe establish Sobolev inequal-ities between the homogeneous Dunkl-Besov spaces andmany spaces as the homogeneous Dunkl-Riesz spaces andgeneralized Lorentz spaces

As an application we consider the Dunkl-Schrodingerequation

119894120597119905119906 + 119896119906 = 119891 (119905 119909) (119905 119909) isin (0infin) timesR119889

119906|119905=0 = 119892

(1)

where119896 = sum119889

119895=11198792

119895is the Dunkl Laplace operator We study

the previous equation focusing on the following problems

(1) Establish the Strichartz estimate under the general-ized Lorentz norms

(2) Illustrate applications to well posedness

The contents of the paper are as follows In Section 2we recall some basic results about the harmonic analysisassociated with the Dunkl operators In Section 3 we intro-duce the homogeneous Dunkl-Besov spaces the homoge-neous Dunkl-Triebel-Lizorkin spaces and the homogeneousDunkl-Riesz potential spaces and we prove new embeddingSobolev theorem In Section 4 we recall some facts about areal interpolation method Next we define the generalizedLorentz spaces and will pay special attention to the interpo-lation definition of these spaces Section 5 is devoted to givea complete picture of the Sobolev type inequalities for thefractional Dunkl-Laplace operators In Section 6 Strichartzestimates for the solution of the Dunkl-Schrodinger evolu-tion equation are considered on a mixed normed space withgeneralized Lorentz norm with respect to the time variableFinally we establish Sobolev inequalities between the homo-geneous Dunkl-Besov spaces and generalized Lorentz spacesand we give many applications

2 Journal of Function Spaces and Applications

2 Preliminaries

In order to confirm the basic and standard notations webriefly overview the theory of Dunkl operators and relatedharmonic analysis Main references are [1 5 6 11 14ndash17]

21 Root System Reflection Group and Multiplicity FunctionLetR119889 be the Euclidean space equipped with a scalar product⟨ ⟩ and let ||119909|| = radic⟨119909 119909⟩ For 120572 in R119889

0 120590120572 denotes thereflection in the hyperplane 119867120572 sub R119889 perpendicular to 120572that is for 119909 isin R119889 120590120572(119909) = 119909 minus 2120572

minus2⟨120572 119909⟩120572 A finite set

119877 sub R119889 0 is called a root system if 119877 cap R120572 = plusmn120572 and

120590120572119877 = 119877 for all 120572 isin 119877 We normalize each 120572 isin 119877 as ⟨120572 120572⟩ = 2We fix a 120573 isin R119889

cup120572isin119877119867120572 and define a positive root system119877+

of 119877 as 119877+ = 120572 isin 119877 | ⟨120572 120573⟩ gt 0 The reflections 120590120572 120572 isin 119877generate a finite group 119866 sub 119874(119889) called the reflection groupA function 119896 119877 rarr C on 119877 is called a multiplicity functionif it is invariant under the action of119866 We introduce the index120574 as

120574 = 120574 (119896) = sum

120572isin119877+

119896 (120572) (2)

Throughout this paper we will assume that 119896(120572) ge 0 for all120572 isin 119877 We denote by 120596119896 the weight function on R119889 given by

120596119896 (119909) = prod

120572isin119877+

|⟨120572 119909⟩|2119896(120572)

(3)

which is invariant and homogeneous of degree 2120574 In the casethat the reflection group 119866 is the group Z119889

2of sign changes

the weight function 120596119896 is a product function of the formprod

119889

119895=1|119909119895|

119896119895 119896119895 ge 0 We denote by 119888119896 the Mehta-type constant

defined by

119888119896 = intR119889

119890minus||119909||22120596119896 (119909) 119889119909 (4)

We note that Etingof (cf [18]) has given a derivation of theMehta-type constant valid for all finite reflection groups

In the following we denote by

119862(R119889) the space of continuous functions on R119889

119862119901(R119889

) the space of functions of class 119862119901 on R119889119862119901

119887(R119889

) the space of bounded functions of class 119862119901

E(R119889) the space of 119862infin functions on R119889

S(R119889) the Schwartz space of rapidly decreasing functions onR119889

119863(R119889) the space of119862infin functions onR119889which are of compactsupport

S1015840(R119889

) the space of temperate distributions on R119889

22 The Dunkl Operators Let 119896 119877 rarr C be a multiplicityfunction on 119877 and 119877+ a fixed positive root system of 119877 Thenthe Dunkl operators 119879119895 1 le 119895 le 119889 are defined on 119862

1(R119889

) by

119879119895119891 (119909) =120597

120597119909119895

119891 (119909) + sum

120572isin119877+

119896 (120572) 120572119895

119891 (119909) minus 119891 (120590120572 (119909))

⟨120572 119909⟩ (5)

where 120572 = (1205721 1205722 120572119889) Similarly as ordinary derivativeseach119879119895 satisfies for all119891 119892 in1198621

(R119889) and at least one of them

is 119866-invariant

119879119895 (119891119892) = (119879119895119891) 119892 + 119891 (119879119895119892) (6)

and for all 119891 in 1198621

119887(R119889

) and 119892 in S(R119889)

intR119889

119879119895119891 (119909) 119892 (119909) 120596119896 (119909) 119889119909 = minusintR119889

119891 (119909) 119879119895119892 (119909) 120596119896 (119909) 119889119909

(7)

Furthermore according to [1 14] the Dunkl operators119879119895 1 le 119895 le 119889 commute and there exists the so-called Dunklrsquosintertwining operator 119881119896 such that 119879119895119881119896 = 119881119896(120597120597119909119895) for1 le 119895 le 119889 and 119881119896(1) = 1 We define the Dunkl-Laplaceoperator119896 on R119889 by

119896119891 (119909) =

119889

sum

119895=1

1198792

119895119891 (119909)

= 119891 (119909) + 2 sum

120572isin119877+

119896 (120572) (⟨nabla119891 (119909) 120572⟩

⟨120572 119909⟩

minus119891 (119909) minus 119891 (120590120572 (119909))

⟨120572 119909⟩2

)

(8)

where and nabla are the usual Euclidean Laplacian and nablaoperators on R119889 respectively Since the Dunkl operatorscommute their joint eigenvalue problem is significant andfor each 119910 isin R119889 the system

119879119895119906 (119909 119910) = 119910119895119906 (119909 119910) 119895 = 1 119889 119906 (0 119910) = 1 (9)

admits a unique analytic solution 119870(119909 119910) 119909 isin R119889 called theDunkl kernel which has a holomorphic extension toC119889

timesC119889For 119909 119910 isin C119889 the kernel satisfies

(a) 119870(119909 119910) = 119870(119910 119909)(b) 119870(120582119909 119910) = 119870(119909 120582119910) for 120582 isin C(c) 119870(119892119909 119892119910) = 119870(119909 119910) for 119892 isin 119866

23 The Dunkl Transform For functions 119891 on R119889 we define119871119901-norms of 119891 with respect to 120596119896(119909)119889119909 as

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

= (intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596119896 (119909) 119889119909)

1119901

(10)

if 1 le 119901 lt infin and 119891119871infin119896(R119889) = ess sup

119909isinR119889 |119891(119909)| We denoteby 119871119901

119896(R119889

) the space of all measurable functions119891 onR119889 withfinite 119871119901

119896-norm

The Dunkl transformF119863 on 1198711

119896(R119889

) is given by

F119863 (119891) (119910) =1

119888119896

intR119889

119891 (119909)119870 (119909 minus119894119910) 120596119896 (119909) 119889119909 (11)

Journal of Function Spaces and Applications 3

Some basic properties are the following (cf [5 6]) For all119891 isin 119871

1

119896(R119889

)

(a) F119863(119891)119871infin119896(R119889) le 119888

minus1

119896 1198911198711

119896(R119889)

(b) F119863(119891(sdot120582))(119910) = 1205822120574+119889F119863(119891)(120582119910) for 120582 gt 0

(c) ifF119863(119891) belongs to 1198711

119896(R119889

) then

119891 (119910) =1

119888119896

intR119889

F119863 (119891) (119909)119870 (119894119909 119910) 120596119896 (119909) 119889119909 ae (12)

and moreover for all 119891 isin S(R119889)

(d) F119863(119879119895119891)(119910) = 119894119910119895F119863(119891)(119910)

(e) if we defineF119863(119891)(119910) = F119863(119891)(minus119910) then

F119863F119863 = F119863F119863 = 119868119889 (13)

Proposition 1 The Dunkl transform F119863 is a topologicalisomorphism from S(R119889

) onto itself and for all f in S(R119889)

intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

2120596119896 (119909) 119889119909 = int

R119889

1003816100381610038161003816F119863 (119891) (120585)1003816100381610038161003816

2120596119896 (120585) 119889120585 (14)

In particular the Dunkl transform 119891 rarr F119863(119891) can beuniquely extended to an isometric isomorphism on 119871

2

119896(R119889

)

We define the tempered distribution T119891 associated with119891 isin 119871

119901

119896(R119889

) by

⟨T119891 120601⟩ = intR119889

119891 (119909) 120601 (119909) 120596119896 (119909) 119889119909 (15)

for 120601 isin S(R119889) and denote by ⟨119891 120601⟩119896 the integral in the right

hand side

Definition 2 The Dunkl transform F119863(120591) of a distribution120591 isin S1015840

(R119889) is defined by

⟨F119863 (120591) 120601⟩ = ⟨120591F119863 (120601)⟩ (16)

for 120601 isin S(R119889)

In particular for 119891 isin 119871119901

119896(R119889

) it follows that for120601 isin S(R119889

)

⟨F119863 (119891) 120601⟩ = ⟨F119863 (T119891) 120601⟩ = ⟨T119891F119863 (120601)⟩

= ⟨119891F119863(120601)⟩119896

(17)

Proposition 3 The Dunkl transform F119863 is a topologicalisomorphism from S1015840

(R119889) onto itself

24 The Dunkl Convolution By using the Dunkl kernel inSection 22 we introduce a generalized translation and aconvolution structure in our Dunkl setting For a function119891 isin S(R119889

) and 119910 isin R119889 the Dunkl translation 120591119910119891 is definedby

120591119910119891 (119909) =1

119888119896

intR119889

F119863 (119891) (119911)119870 (119894119909 119911)119870 (119894119910 119911) 120596119896 (119911) 119889119911

(18)

Clearly 120591119910119891(119909) = 120591119909119891(119910) and by using the Dunklrsquos inter-twining operator 119881119896 120591119910119891 is related to the usual translationas 120591119910119891(119909) = (119881119896)119909(119881119896)119910((119881119896)

minus1(119891)(119909+119910)) (cf [11 17]) Hence

120591119910 can also be defined for 119891 isin E(R119889) We define the Dunkl

convolution product 119891lowast119863 119892 of functions 119891 119892 isin S(R119889) as

follows

119891lowast119863 119892 (119909) = intR119889

120591119909119891 (minus119910) 119892 (119910) 120596119896 (119910) 119889119910 (19)

This convolution is commutative and associative (cf [17])Since F119863(120591119910119891)(119909) = 119870(119894119909 119910)F119863(119891)(119909) by the previous

definition of 120591119910119891 it follows that

(a) for all 119891 119892 isin 119863(R119889) (resp S(R119889

)) 119891lowast119863 119892 belongs to119863(R119889

) (resp S(R119889)) and

F119863 (119891lowast119863 119892) (119910) = F119863 (119891) (119910)F119863 (119892) (119910) (20)

Moreover as pointed in [16] and Sections 4 and 7 theoperator 119891 rarr 119891lowast119863 119892 is bounded on 119871

119901

119896(R119889

) 1 le 119901 le infinprovided that 119892 is a radial function in 119871

1

119896(R119889

) or an arbitraryfunction in 119871

1

119896(R119889

) for 119866 = Z119889

2 Hence the standard

argument yields the following Youngrsquos inequality

(b) Let 1 le 119901 119902 119903 le infin such that 1119901 + 1119902 minus 1119903 = 1Assume that 119891 isin 119871

119901

119896(R119889

) and 119892 isin 119871119902

119896(R119889

) If120591119909119892119871

119902

119896(R119889) le 119862119892

119871119902

119896(R119889) for all 119909 isin R119889 then119891lowast119863 119892 isin

119871119903

119896(R119889

) and

1003817100381710038171003817119891lowast119863 1198921003817100381710038171003817119871119903119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119902

119896(R119889)

(21)

Definition 4 The Dunkl convolution product of a distribu-tion 119878 in S1015840

(R119889) and a function 120601 in S(R119889

) is the function119878lowast119863 120601 defined by

119878lowast119863 120601 (119909) = ⟨119878119910 120591minus119910120601 (119909)⟩ (22)

Proposition 5 Let 119891 be in 119871119901

119896(R119889

) 1 le 119901 le infin and 120601 inS(R119889

) Then the distributionT119891lowast119863 120601 is given by the function119891lowast119863 120601 If one assumes that 120601 is arbitrary for 119889 = 1 and radialfor 119889 ge 2 then T119891lowast119863 120601 belongs to 119871

119901

119896(R119889

) Moreover for all120595 isin S(R119889

)

⟨T119891lowast119863120601 120595⟩ = ⟨120601lowast119863⟩119896 (23)

where (119909) = 120595(minus119909) and

F119863 (T119891lowast119863 120601) = F119863 (T119891)F119863 (120601) (24)

For each 119906 isin S1015840(R119889

) we define the distributions119879119895119906 1 le 119895 le 119889 by

⟨119879119895119906 120595⟩ = minus⟨119906 119879119895120595⟩ (25)


for all 120595 isin S(R119889) Then ⟨119896119906 120595⟩ = ⟨119906119896120595⟩ and these

distributions satisfy the following properties (see Section 23(d))

F119863 (119879119895119906) = 119894119910119895F119863 (119906)

F119863 (119896119906) = minus10038171003817100381710038171199101003817100381710038171003817

2F119863 (119906)

(26)

In the following we denote T119891 given by (15) by 119891 forsimplicity

3 B119904119896

119901119902 F119904119896

119901119902(R119889

) and H119904

119901119896Spaces and

Basic Properties

One of the main tools in this paper is the homogeneousLittlewood-Paley decompositions of distributions associatedwith the Dunkl operators into dyadic blocs of frequencies

Lemma 6 Let one define by C the ring of center 0 of smallradius 12 and great radius 2 There exist two radial functions120595 and 120593 the values of which are in the interval [0 1] belongingto119863(R119889

) such that

supp120595 sub 119861 (0 1) supp120593 sub C

forall120585 isin R119889 120595 (120585) + sum

119895ge0

120593 (2minus119895120585) = 1

forall120585 isin C sum

119895isinZ

120593 (2minus119895120585) = 1

|119899 minus 119898| ge 2 997904rArr supp120593 (2minus119899sdot) cap supp120593 (2minus119898

sdot) = 0

119895 ge 1 997904rArr supp120595 cap supp120593 (2minus119895sdot) = 0

(27)

Notations We denote by

Δ 119895119891 = Fminus1

119863(120593(

120585

2119895)F119863 (119891)) 119878119895119891 = sum

119899le119895minus1

Δ 119899119891

forall119895 isin Z

(28)

The distribution Δ 119895119891 is called the jth dyadic block of thehomogeneous Littlewood-Paley decomposition of 119891 associ-ated with the Dunkl operators

Throughout this paper we define 120601 and 120594 by 120601 = Fminus1

119863(120593)

and 120594 = Fminus1

119863(120595)

When dealing with the Littlewood-Paley decompositionit is convenient to introduce the functions and 120593 belongingto119863(R119889

) such that equiv 1 on supp 120595 and 120593 equiv 1 on supp 120593

Remark 7 We remark that

F119863 (119878119895119891) (120585) = (120585

2119895)F119863 (119878119895119891) (120585)

F119863 (Δ 119895119891) (120585) = 120593(120585

2119895)F119863 (Δ 119895119891) (120585)

(29)

We put

120601 = Fminus1

119863(120593) 120594 = F

minus1

119863() (30)

Definition 8 Let one denote by S1015840

ℎ119896(R119889) the space of

tempered distribution such that

lim119895rarrminusinfin

119878119895119906 = 0 in S1015840(R

119889) (31)

On the follow we define analogues of the homogeneousBesov Triebel-Lizorkin and Riesz potential spaces associatedwith the Dunkl operators on R119889 and obtain their basicproperties

From now we make the convention that for all non-negative sequence 119886119902119902isinZ the notation (sum

119902119886119903

119902)1119903 stands for

sup119902119886119902 in the case 119903 = infin

Definition 9 Let 119904 isin R and 119901 119902 isin [1infin] The homogeneousDunkl-Besov spaces B119904119896

119901119902(R119889

) are the space of distribution inS1015840

ℎ119896(R119889) such that

10038171003817100381710038171198911003817100381710038171003817B119904119896119901119902

(R119889)= (sum

119895isinZ

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

)

119902

)

1119902

lt infin (32)

Definition 10 Let 119904 isin R and 1 le 119901 119902 le infin thehomogeneous Dunkl-Triebel-Lizorkin space F119904119896

119901119902(R119889

) is thespace of distribution in S1015840

ℎ119896(R119889) such that

10038171003817100381710038171198911003817100381710038171003817F119904119896119901119902

(R119889)=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

211990411989511990210038161003816100381610038161003816

Δ 11989511989110038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

119896(R119889)

lt infin (33)

Let us recall that the operators (minus119896)1199042 and (119868 minus 119896)

1199042

have been defined respectively by (cf [19])

(minus119896)1199042119891 = F

minus1

119863(sdot

119904F119863119891)

(119868 minus 119896)1199042119891 = F

minus1

119863((1 + sdot

2)1199042

F119863119891)

(34)

The operators (119868 minus 119896)minus1199042 for 119904 gt 0 are called Dunkl-Bessel

potential operators and they are given by Dunkl convolutionwith the Dunkl-Bessel potential

(119868 minus 119896)minus1199042

119891 = 119891lowast119863 119861119896119904(35)

where

119861119896119904 (119910) =1

Γ (1199042)int

infin

0

119890minus119905119890minus11991024119905119905(119904minus119889minus2120574)2 119889119905

119905 (36)

We note that 119861119896119904(119910) ge 0 for all 119910 isin R119889 119861119896119904 isin 1198711

119896(R119889

) and

119861119896119904 (119910) le 11986210038171003817100381710038171199101003817100381710038171003817

119904minus119889minus2120574119890minus11991022

10038171003817100381710038171199101003817100381710038171003817 gt 0 (37)

Definition 11 For 119904 isin R and 1 le 119901 le infin the Dunkl-Bessel potential space 119867

119904

119901119896(R119889

) is defined as the space(119868 minus 119896)

1199042(119871

119901

119896(R119889

)) equipped with the norm 119891119867119904119901119896

(R119889) =

(119868 minus 119896)1199042119891

119871119901

119896(R119889)


Furthermore 119901 = 2119867119904

2119896(R119889

) = 119867119904

119896(R119889

)

Definition 12 The operators (minus119896)minus1199042

119896 0 lt 119904 lt 119889 + 2120574 are

called Dunkl-Riesz potentials operators and one has

(minus119896)minus1199042

119891 = 119877119896119904lowast119863119891(38)

where 119877119896119904 is the Dunkl-Riesz potential given by

119877119896119904 (119910) = 119862 (119896 119904 119889)10038171003817100381710038171199101003817100381710038171003817

119904minus119889minus2120574

where 119862 (119896 119904 119889) =Γ ((119889 + 2120574 minus 119904) 2)

2(119889+2120574minus119904)2Γ (1199042)

(39)

Definition 13 For 119904 isin R and 1 le 119901 le infin the homogeneousDunkl-Riesz potential space H119904

119901119896(R119889

) is defined as the space(minus119896)

1199042(119871

119901

119896(R119889

)) equipped with the norm 119891H119904119901119896

(R119889) =

(minus119896)1199042119891

119871119901

119896(R119889)

Proposition 14 Let 119902 isin (1infin) and let 119904 isin R such that0 lt 119904 lt (119889 + 2120574)119902 then one has

B119904119896

119902119902(R

119889)

= F119904119896

119902119902(R

119889) 997893rarr F

119904119896

119902infin(R

119889) 997893rarr F

119904minus(119889+2120574)119902119896

infininfin(R

119889)

(40)

H119904

119902119896(R

119889)

= F119904119896

1199022(R

119889) 997893rarr F

119904119896

119902infin(R

119889) 997893rarr F

119904minus(119889+2120574)119902119896

infininfin(R

119889)

(41)

Proof We obtain these results by similar ideas used in thenonhomogeneous case (cf [12])

Theorem 15 Let 119886 119887 gt 0 and let 1199021 1199022 isin [1infin] Let120579 = 119886(119886 + 119887) isin (0 1) and let 1119901 = (1 minus 120579)1199021 +

1205791199022 Then there exists a constant 119862 such that for every119891 isin F119886119896

1199021infin(R119889

) cap Fminus119887119896

1199022infin(R119889

) then one has

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

120579

(42)

In particular one gets

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

F1198861198961199021infin(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus1198871198961199022infin(R119889) (43)

Proof Let 119891 be a Schwartz class we have1003816100381610038161003816119891 (119909)

1003816100381610038161003816 le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ

min(2minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816))

(44)

We define119873(119909) as the largest index such that

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816) le 2

minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816) (45)

and we write

1003816100381610038161003816119891 (119909)1003816100381610038161003816

le sum

119895le119873(119909)

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895isinZ

(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

le 119862(sup119895isinZ

2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

119887(119886+119887)

(sup119895isinZ

2minus119887119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

119886(119886+119887)

(46)

Thus (42) is proved In order to obtain (43) it is enough toapply Holderrsquos inequality in the expression previous since wehave 120579 = 119886(119886 + 119887) isin (0 1) and 1119901 = (1 minus 120579)1199021 + 1205791199022

Corollary 16 Let 119902 isin (1infin) and let 119904 isin R such that0 lt 119904 lt (119889 + 2120574)119902 then one has

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus119902119901

Bminus((2120574+119889)119902minus119904)119896

infininfin (R119889)

10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904119896119902119902(R119889) (47)

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus119902119901

Bminus((2120574+119889)119902minus119904)119896119896


10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902119896(R119889)

(48)

where 119901 = 119902(2120574 + 119889)(2120574 + 119889 minus 119902119904)

Proof We take 119886 = 119904 gt 0 minus119887 = 119904 minus (119889 + 2120574)119902 lt 0 1199021 = 119902and 1199022 = infin and we deduce the inequality (47) from therelations (43) and (40) In the same way we deduce (48) fromthe relations (43) and (41)

Theorem 17 (see [13]) (1) Let 119904 gt 0 and 119901 119903 isin [1infin] ThenB119904119896

119901119903(R119889

) cap 119871infin

119896(R119889

) is an algebra and there exists a positiveconstant 119862 such that

119906VB119904119896119901119903(R119889)

le 119862 [119906119871infin119896(R119889)VB119904119896

119901119903(R119889) + V119871infin

119896(R119889)119906B119904119896

119901119903(R119889)]

(49)

(2) Moreover for any (1199041 1199042) any 1199012 and any 1199032 such that1199041 + 1199042 gt (119889 + 2120574)1199011 and 1199041 lt (119889 + 2120574)1199011 one has

119906VB11990411989611990121199032(R119889)

le 119862[119906B1199041119896

1199011infin(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

1199011infin(R119889)

]

(50)

where 119904 = 1199041 + 1199042 minus (119889 + 2120574)1199011


(3) Moreover for any (1199041 1199042) any 1199012 and any (1199031 1199032) suchthat 1199041 + 1199042 gt (119889 + 2120574)1199011 1199041 lt (119889 + 2120574)1199011 11199031 + 11199032 = 1one has119906VB119904119896

119901infin(R119889)

le 119862[119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

11990111199031(R119889)

]

(51)

(4) Moreover for any (1199041 1199042) any (1199011 1199012 119901) and any(1199031 1199032) such that 119904119895 lt (119889 + 2120574)119901119895 1199041 + 1199042 gt (119889 + 2120574)(11199011 +

11199012 minus 1119901) and 119901 ge max(1199011 1199012) one has

119906VB11990412119896

119901119903(R119889)

le 119862119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

(52)

with 11990412 = 1199041 + 1199042 minus (119889 + 2120574)(11199011 + 11199012 minus 1119901) and119903 = max(1199031 1199032)

4 A Primer to Real Interpolation Theory andGeneralized Lorentz Spaces

Fromnowwe denote by 119897119902(Z) the set of sequence (119886119895)119895isinZ suchthat

(sum

119895isinZ

10038161003816100381610038161003816119886119895

10038161003816100381610038161003816

119902

)

1119902

lt infin (53)

stands for sup119895|119886119895| in the case 119902 = infin

The theory of interpolation spaces was introduced in theearly sixties by J Lions and J Peetre for the real method andby Calderon for the complex method (cf [20])

In this section we present the real method There aremany equivalent ways to define the method we will presentthe discrete J-method and the K-method which are thesimplest ones

We consider two Banach spaces 1198600 and 1198601 which arecontinuously imbedded into a common topological vectorspace 119881 and 119905 gt 0

The J-method and the K-method consist to consider theJ-functional and the K-functional defined on 1198600 ⋂1198601 by

119869 (119905 119886 1198600 1198601) = max (1198861198600

1199051198861198601

)

119870 (119905 119886 1198600 1198601) = min (1003817100381710038171003817119886010038171003817100381710038171198600

+ 11990510038171003817100381710038171198861

10038171003817100381710038171198601

119886 = 1198860 + 1198861)

(54)

Definition 18 (J-method of interpolation) For 0 lt 120579 lt 1 and1 le 119902 le infin the interpolation space [1198600 1198601]120579119902119869 is defined asfollows 119886 isin [1198600 1198601]120579119902119869 if and only if 119886 can be written as asum 119886 = sum

119895isinZ 119886119895 where the series converge in 1198600 + 1198601 each119886119895 belongs to 1198600 ⋂1198601 and (2

minus119895120579119869(2

119895 119886119895 1198600 1198601))119895isinZ isin 119897

119902(Z)

The norm of [1198600 1198601]120579119902119869 is defined by

119886[11986001198601]120579119902119869

= inf119886=sum119895isinZ 119886119895

(sum

119895isinZ

2minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

1119902

+ (sum

119895isinZ

2119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

1119902

(55)

Definition 19 (K-method of interpolation) For 0 lt 120579 lt 1

and 1 le 119902 le infin the space [1198600 1198601]120579119902119870 is defined by119886 isin [1198600 1198601]120579119902119870 if and only if 119886 isin 1198600 + 1198601 and(2

minus119895120579119870(2

119895 119886 1198600 1198601))119895isinZ isin 119897

119902(Z)

The norm of [1198600 1198601]120579119902119870 is defined as follows

119886[11986001198601]120579119902119870

= (sum

119895isinZ

2minus119895120579119902

119870(2119895 119886 1198600 1198601)

119902

)

1119902

(56)

Proposition 20 (Equivalence theorem) For 0 lt 120579 lt 1 and1 le 119902 le infin one has [1198600 1198601]120579119902119870 = [1198600 1198601]120579119902119869

Remark 21 In the following we will denote this space by[1198600 1198601]120579119902

Lemma 22 For 119886 = sum119895isinZ 119886119895 and 984858 gt 0 with 984858 = 1 one has

119886[11986001198601]120579119902

le 119862 (119902 120579 984858)(sum

119895isinZ

984858minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

(1minus120579)119902

times (sum

119895isinZ

984858119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

120579119902

(57)

Proposition 23 (i) For 1205790 = 1205791 one has

[[1198600 1198601]12057901199020

[1198600 1198601]12057911199021

]120579119902

= [1198600 1198601](1minus120579)1205790+1205791205791119902 (58)

(ii) For 1205790 = 1205791 (58) is still valid if 1119902 = (1 minus 120579)1199020 + 1205791199021

Proposition 24 (Duality theorem for the real method) Oneconsiders the dual spaces 1198601015840

0 119860

1015840

1and [1198600 1198601]

1015840

120579119902for 0 lt 120579 lt 1

and 1 le 119902 lt infin of the spaces 1198600 1198601 and [1198600 1198601]120579119902 If1198600 ⋂1198601 is dense in 1198600 and in 1198601 one has [1198600 1198601]

1015840

120579119902=

[1198601015840

0 119860

1015840

1]1205791199021015840

where 1199021015840 is the conjugate component of 119902

For any measurable function 119891 on R119889 we define itsdistribution and rearrangement functions

119889119891119896 (120582) = int119909isinR119889 |119891(119909)|ge120582

120596119896 (119909) 119889119909

119891lowast

119896(119904) = inf 120582 119889119891119896 (120582) le 119904

(59)

For 1 le 119901 le infin and 1 le 119902 le infin define1003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

=

(int

infin

0

(1199041119901

119891lowast

119896(119904))

119902 119889119904

119904)

1119902

if 1 le 119901 119902 lt infin

sup119904gt0

1199041119901

119891lowast

119896(119904) if 1 le 119901 le infin 119902 = infin

(60)

The generalized Lorentz spaces 119871119901119902119896(R119889

) is defined as the setof all measurable functions 119891 such that ||119891||

119871119901119902

119896(R119889) lt infin


Proposition 25 (i) For 1 lt 119901 lt infin 1 le 119902 le infin

119871119901119902

119896(R

119889) = [119871

1

119896(R

119889) 119871

infin

119896(R

119889)]

120579119902 (61)

with 1119901 = 1 minus 120579(ii) For 1199010 = 1199011 one has

[11987111990101199020

119896(R

119889) 119871

11990111199021

119896(R

119889)]

120579119902= [119871

1199010

119896(R

119889) 119871

1199011

119896(R

119889)]

120579119902

= 119871119901119902

119896(R

119889)

(62)

with 1119901 = (1 minus 120579)1199010 + 1205791199011(iii) In the case 1199010 = 1199011 = 119901 one has

[1198711199011199020

119896(R

119889) 119871

1199011199021

119896(R

119889)]

120579119902= 119871

119901119902

119896(R

119889) (63)

with 1119902 = (1 minus 120579)1199020 + 1205791199021(iv) If 1 le 119901 le infin and 1 le 1199021 lt 1199022 le infin then

1198711199011199021

119896(R

119889) 997893rarr 119871

1199011199022

119896(R

119889) (64)

Proof We obtain these results by similar ideas used in theEuclidean case

Proposition 26 (i) Let 1 lt 119901 lt infin 1 le 119902 le infin Thenthere exists a constant 119862 such that every 119891 isin 119871

119901119902

119896(R119889

) can bedecomposed as 119891 = sum

119895isinZ 119891119895 where

1003817100381710038171003817100381710038171003817(2

minus119895(119901minus1)11990110038171003817100381710038171003817119891119895

100381710038171003817100381710038171198711119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903+

1003817100381710038171003817100381710038171003817(2

11989511990110038171003817100381710038171003817119891119895

10038171003817100381710038171003817119871infin119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(65)

the 119891119895 have disjoint supports if 119895 = 119899 119891119895119891119899 = 0(ii) Let 1 lt 119901 lt infin 1 le 119902 le infin Then there exists

a constant 119862 such that every 119891 isin 119871119901119902

119896(R119889

) and every 119892 isin

119871119901(119901minus1)119902(119902minus1)

119896(R119889

) one has 119891119892 isin 1198711

119896(R119889

) and

1003816100381610038161003816100381610038161003816intR119889

119891 (119909) 119892 (119909) 120596119896 (119909) 119889119909

1003816100381610038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901(119901minus1)119902(119902minus1)

119896(R119889)

(66)


5 Inequalities for the FractionalDunkl-Laplace Operators

Lemma 27 Let 119904 be a real number such that 0 lt 119904 lt 119889 + 2120574and let 1 lt 119901 lt 119902 lt infin satisfy

1

119901minus1

119902=

119904

2120574 + 119889 (67)

For 119891 isin 119871119901

119896(R119889

) one has100381710038171003817100381710038171003817(119868 minus 119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(68)

Proof We obtain this result by similar ideas used for theDunkl-Riesz potential (cf [21])

Proposition 28 Let 119904 lt (119889 + 2120574)2 and 119902 = (2119889 + 4120574)(119889 +

2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889)

(69)

Proof Let us first observe that since 119863(R119889) is dense in

119867119904

119896(R119889

) it is enough to prove (69) for 119891 isin 119863(R119889) Let

119891 119892 isin 119863(R119889) Then we have

⟨119891 119892⟩1198712119896(R119889)

= ⟨F119863(119891)F119863 (119892)⟩1198712119896(R119889)

= intR119889

(1 +10038171003817100381710038171205851003817100381710038171003817

2)1199042

F119863 (119891) (120585)F119863 (119892) (120585)(1 +10038171003817100381710038171205851003817100381710038171003817

2)minus1199042

times 120596119896 (120585) 119889120585

= ⟨(119868 minus 119896)1199042119891 (119868 minus 119896)

minus1199042119892⟩

1198712119896(R119889)

(70)

Hence100381610038161003816100381610038161003816⟨119891 119892⟩

1198712119896(R119889)

100381610038161003816100381610038161003816le100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

100381710038171003817100381710038171003817(119868 minus 119896)

minus11990421198921003817100381710038171003817100381710038171198712119896(R119889)

(71)

Now by the previous lemma we obtain100381610038161003816100381610038161003816⟨119891 119892⟩

1198712119896(R119889)

100381610038161003816100381610038161003816le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901

119896(R119889)

(72)

where 119901 = (2119889 + 4120574)(119889 + 2120574 + 2119904) Now let us take 119892 = 119891119902minus1

with 1119901 + 1119902 = 1 that is 119902 = (2119889 + 4120574)(119889 + 2120574 minus 2119904) Thenthe relation (72) gives that

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus1

119871119902

119896(R119889)

(73)

Thus we obtain (69)

Proposition 29 Let 1 le 119901 1199012 lt infin 0 lt 120579 lt 119901 lt infin0 lt 119904 lt 119889 + 2120574 and 1 lt 1199011 lt (119889 + 2120574)119904 Then one hasthe inequality

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(74)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (75)

Proof Holderrsquos inequality yields

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

1198711199010

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119901minus120579

1198711199012

119896(R119889)

(76)


where

1

1199010

=1

120579(1 minus

1

119901)

1

1199012

=1

(119901 minus 120579) 119901 (77)

Applying Lemma 27 with 1199011 = ((119889 + 2120574)1199010)(119889 + 2120574 + 1199041199010)we obtain the result

Theorem 30 Let 1 lt 119902 lt infin 0 lt 119904 lt 119889 + 2120574 and1 lt 1199011 lt (119889 + 2120574)119904 Then the inequality

exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(78)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (79)

Proof Using the convexity of the function 119892(ℎ) =

ℎ ln(intR119889

|119891(119909)|1ℎ

120596119896(119909)119889119909) and the logarithmic Holderrsquosinequality proved by Merker [22] we obtain

intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(80)

for 0 lt 119902 lt 119901 le infin We can choose 119901 = ((119889 + 2120574)119902)(119889 +

2120574 minus 119902119904) isin (119902infin) for 1199012 = 119902 and 120579 satisfying the condition ofProposition 29 and we get

intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

(119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

)

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

le119902120579

119901 minus 119902ln(

119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(81)

By a simple calculation we obtain the result

Corollary 31 Let 0 lt 119904 lt 119889 + 2120574 and 1 lt 119902 lt (119889 + 2120574)119904119891 isin 119867

119904

119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(82)

Proof It suffices to apply the previous theorem for 119902 = 1199011

Lemma 32 (see [23]) One assumes that 119866 = Z119889

2 If

119891 isin 11987111990111199021

119896(R119889

) 119892 isin 11987111990121199022

119896(R119889

) and 11199011 + 11199012 gt 1 then119891lowast119863119892 isin 119871

11990131199023

119896(R119889

) where 11199013 = 11199011 + 11199012 minus 1 and 1199023 ge 1

is any number such that 11199023 le 11199021 + 11199022 Moreover1003817100381710038171003817119891lowast119863119892

100381710038171003817100381711987111990131199023

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(83)

Remark 33 The analogues of this lemma for the generalreflection group119866 together with other additional results willappear in a forthcoming paper

Theorem 34 One assumes that 119866 = Z119889

2 Let 1 le 119901 lt infin

1 le 1199012 119902 1199021 1199022 lt infin 0 lt 120579 lt 119902 0 lt 119904 lt 119889 + 2120574 and1 lt 1199011 lt (119889 + 2120574)119904 Then the inequality

10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(84)

holds for

120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(85)

Proof Applying the Holder inequality and simple computa-tion yields

10038171003817100381710038171198911003817100381710038171003817

119902

119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

11987111990131199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus120579

11987111990121199022

119896(R119889)

(86)

where1

1199013

=1

119901minus1

119902+

1

1199021

1

1199012

=1

119901minus1

119902+

1

1199022

(87)

Note that

119891 (119909) = (119868 minus 119896)1199042119891lowast119863119861119896119904 (119909)

(88)

where 119861119896119904 is the Dunkl-Bessel kernel defined by rela-tion (36) From the relation (37) we see that 119861119896119904 isin

119871(119889+2120574)(119889+2120574minus119904)infin

119896(R119889

) Using now Lemma 32 we deduce that

1003817100381710038171003817119891100381710038171003817100381711987111990131199021

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

119904211989110038171003817100381710038171003817100381711987111990111199021

119896(R119889)

(89)


for

1

1199013

=1

1199011

minus119904

119889 + 2120574 0 lt 119904 lt

119889 + 2120574

1199011

(90)

The result then follows

Now we state the results for the Dunkl-Riesz potentialoperators The proofs are essentially as for the Dunkl-Besselpotential operators We will not repeat them


2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889) (91)

Proposition 36 Let 1 le 119901 1199012 lt infin 0 lt 120579 lt 119901 lt infin0 lt 119904 lt 119889 + 2120574 and 1 lt 1199011 lt (119889 + 2120574)119904 Then the inequality

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(92)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (93)


exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(94)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (95)

Corollary 38 Let 0 lt 119904 lt 119889 + 2120574 and 1 lt 119902 lt (119889 + 2120574)119904119891 isin H119904

119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(96)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(97)

holds for120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(98)

Remark 40 (i) We assume that G = Z119889

2 It follows from the

special case 1199011 = 1199021 and 1199012 = 1199022 of (97) that the inequality

10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

1198711199012

119896(R119889)

(99)

with 119902 = 119901(1 minus 120579119904(119889 + 2120574)) Equation (99) can be thought ofa refinement of (92) from (64)

(ii) We assume that 119866 = Z119889

2 It follows from the special

case 1199011 = 119902 = 120579 that (99) becomes1003817100381710038171003817119891

1003817100381710038171003817119871119902(119889+2120574)(119889+2120574minus119902119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(100)

which can also be thought of as a refinement of the Hardy-Littlewood-Sobolev fractional integration theorem in Dunklsetting (cf [21])

100381710038171003817100381710038171003817(minus119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(101)

(iii) We note that the results of Dunkl-Riesz potential ofthis section are in sprit of the classical case (cf [24])


2 Let 1 lt 119901 lt infin

0 lt 119904 lt (119889 + 2120574)119901 and 1 le 119902 le infin There exists a positiveconstant 119862 such that one has

10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(102)

For proof of this result we need the following lemmawhich we prove as the Euclidean case

Lemma 42 Let 1 le 1199011 1199012 1199021 1199022 le infin If 119891 isin 11987111990111199021

119896(R119889

) and119892 isin 119871

11990121199022

119896(R119889

) then1003817100381710038171003817119891119892

1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(103)

where 1119901 = 11199011 + 11199012 and 1119902 = 11199021 + 11199022

Proof of Theorem 41 Let 1 lt 119901 lt infin and 119904 isin (0 (119889 + 2120574)119901)We take 119892(119909) = 1119909

119904 and apply (103) in the specific form1003817100381710038171003817119891119892

1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199011119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (104)

where 119903 = (119889 + 2120574)119904 and 1199011 = (119902(119889 + 2120574))(119889 + 2120574 minus 119902119904) As119892 isin 119871

119903infin

119896(R119889

) we have10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

(105)

with 1 le 119902 le infin On the other hand from [23] Theorem 12we have

10038171003817100381710038171198911003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

119904119904119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(106)

for any 119891 isin 119871119901119902

119896(R119889

) with 1 le 119902 le infin 1 lt 119901 lt infin and0 lt 119904 lt (119889 + 2120574)119901 Thus we obtain (102)


6 Dispersion Phenomena

Notations Wedenote byI119896(119905) theDunkl-Schrodinger semi-group on 119871

2

119896(R119889

) defined by

I119896 (119905) V =1

119888119896|119905|120574+1198892

119890minus119894(119889+2120574)(1205874) sgn 119905

119890119894(sdot24119905)

times [F119863 (119890119894(sdot24119905)V)] (

sdot

2119905)

(107)

1198821119903

119896(R119889

) (1 le 119903 le infin) Banach space of (classes of)measurable functions 119906 R119889

rarr C such that 119879120583119906 isin 119871

119903

119896(119877

119889)

in the sense of distributions for every multi-index 120583 with|120583| le 11198821119903

119896(R119889

) is equipped with the norm

1199061198821119903

119896(R119889) = sum

|120583|le1

10038171003817100381710038171198791205831199061003817100381710038171003817119871119903119896(R119889)

(108)

1198821119903

119896119866(R119889

) (1 le 119903 le infin) the subspace of1198821119903

119896(R119889

) which theseelements are 119866-invariant

Definition 43 One says that the exponent pair (119902 119903) is(119889 + 2120574)2-admissible if 119902 119903 ge 2 (119902 119903 (119889 + 2120574)2) = (2infin 1)and

1

119902+119889 + 2120574

2119903le119889 + 2120574

4 (109)

If equality holds in (109) one says that (119902 119903) is sharp (119889+2120574)2-admissible otherwise one says that (119902 119903) is nonsharp (119889 +

2120574)2-admissible Note in particular that when 119889 + 2120574 gt 2the endpoint

119875 = (22119889 + 4120574

119889 + 2120574 minus 2) (110)

is sharp (119889 + 2120574)2-admissible

Lemma 44 (see [25]) Let 119864 and 119865 be Banach spaces and letL 119871119901119903(0infin 119864) rarr 119871

119902119904(0infin 119865) be an integral operator for

some 119901 119903 119902 119904 with a kernel 119896(119905 120591) such that

L119891 (119905) = int

infin

0

119896 (119905 120591) 119891 (120591) 119889120591 (111)

If 1 le 119901 le 119903 lt 119904 le 119902 lt infin then one has10038171003817100381710038171003817L119891

10038171003817100381710038171003817119871119902119904(0infin119865)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119903(0infin119864)

(112)

where L is the low diagonal operator defined by

L119891 (119905) = int

119905

0

119896 (119905 120591) 119891 (120591) 119889120591 (113)

Lemma 45 For any (119889 + 2120574)2-admissible pair (119902 119903) with119902 gt 2

1003817100381710038171003817I119896 (119905) 11989110038171003817100381710038171198711199022(0infin119871119903

119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198712119896(R119889)

(114)

10038171003817100381710038171003817100381710038171003817

int

119905

0

I119896 (119905 minus 120591) 119892 (120591) 119889120591

100381710038171003817100381710038171003817100381710038171198711199022(0infin119871119903119896(R119889))cap119871infin(0infin1198712

119896(R119889))

le 1198621003817100381710038171003817119892100381710038171003817100381711987111990210158402(0infin119871119903

1015840

119896(R119889))

(115)

Proof From the dispersion ofI119896(119905) such that1003817100381710038171003817I119896 (119905) 119892

1003817100381710038171003817119871119903119896(R119889)

le 119862119905minus(119889+2120574)(12minus1119903)1003817100381710038171003817119892

10038171003817100381710038171198711199031015840

119896(R119889)

(116)

for any 119903 isin [2infin] (cf [8]) and the fact that

119905minus(119889+2120574)(12minus1119903)

isin 1198712119903(119889+2120574)(119903minus2)infin

for any 119903 isin [22 (119889 + 2120574)

119889 + 2120574 minus 2]

(117)

one can easily prove the result

Theorem 46 Suppose that 119889 ge 1 (119902 119903) and (1199021 1199031) are(119889 + 2120574)2-admissible pairs and 2 lt 119886 le 119902 If 119906 is a solution tothe problem

119894120597119905119906 (119905 119909) + 119896119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060

(118)

for some data 1199060 119891 then

119906119871119902119886(R119871119903119896(R119889)) + 119906119862(R1198712

119896(R119889))

le 119862(10038171003817100381710038171199060

10038171003817100381710038171198712119896(R119889)

+1003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(R1198711199031015840

1

119896(R119889))⋂1198712(R119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

)

(119)

Proof Let 119906 be a solution of (118) We write 119906 as

119906 (119905 119909) = I119896 (119905) 1199060 (119909) + int

119905

0

I119896 (119905 minus 120591) 119891 (120591 119909) 119889120591

(119905 119909) isin R timesR119889

(120)

Let 119896(119905 120591) = I119896(119905 minus 120591) 119864 = 1198711199031015840

1

119896(R119889

) or 119871(2119889+4120574)(119889+2120574+2)2119896

(R119889)

119865 = 119871119903

119896(R119889

) and L119891(119905) = intinfin

0119896(119905 120591)119891(120591)119889120591 Then since

1199021015840

1le 2 lt 119904 le 119902 in view of Lemma 44 we only have to show

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817119871119902119904(0infin119871119903119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(0infin119871

1199031015840

1

119896(R119889))cap1198712(0infin119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

(121)

To show this observe from (114) and 119871119902119904

sub 1198711199022 for all 119904 ge 2

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817

2

119871119902119904(0infin119871119903119896(R119889))

le 119862intint

infin

0

⟨I119896 (minus120591) 119891 (120591) I119896 (minus119910) 119891 (119910)⟩ 119889120591119889119910

(122)

Then from the endpoint result of Keel andTao [26] the right-hand side of (122) is bounded by 1198912

1198712(0infin119871(2119889+4120574)(119889+2120574+2)2

119896(R119889))

The remaining part of theorem can be obtained by the dualityof Lorentz space (119871119902119904)1015840 = 119871

11990210158401199041015840

and the second part of (115)


As an application of the previous theorem we can deriveStrichartz estimates of the solution to the following nonlinearproblem

119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)

Theorem 47 If the initial data is sufficiently small and119866-invariant then there exists a unique solution 119906 isin

119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889

) for every sharp (119889 + 2120574)2-admissible pair(119902 119903) with 119902 gt 2 and 2 lt 119904 le 119902

Proof The existence of a unique1198671

119896119866(R119889

)-solution is provedin [9] it suffices to prove that 119906 isin 119871

119902119904(0infin119882

1119903

119896119866(R119889

)) FromDuhamelrsquos principle we deduce that

119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)

Using (114) and (119) we have

119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)

We can always find an admissible pair (1199020 1199030)with 1199030 lt 119889+2120574

and 2 lt 1199040 lt 1199020 and (1199021 1199031) and 1 lt 1199041 lt 2 such that

1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast

= ((119889 + 2120574)1199030)(119889 + 2120574 minus 1199030) Thus from theLeibnitz rule Holderrsquos inequality on Lorentz space andSobolev embedding we deduce that

11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)

Since ||1199060||1198671119896(R119889) is small we have

11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)

Finally since we can choose (1199021 1199031) arbitrarily to be (119889+2120574)2-admissible for any (119889 + 2120574)2-admissible pair (119902 119903) and 119904

with 119902 gt 2 and 2 lt 119904 le 119902 we have

119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)

In a similar way we can also derive from the smallness of||1199060||1198671

119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)

7 Embedding Sobolev Theoremsand Applications

Theorem 48 Let 119904 119905 gt 0 1199021 1199022 isin [1infin] with 1199021 = 1199022 Let120579 = 119904(119904 + 119905) isin (0 1) 1119901 = (1 minus 120579)1199021 + 1205791199022 and 119903 isin [1infin]If 119891 isin B119904119896

1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)

Proof We start picking 1199011 1199012 such that 1 le 1199021 lt 1199011 lt

119901 lt 1199012 lt 1199022 le infin with 2119901 = 11199011 + 11199012 We have then1119901119894 = (1 minus 119886119894)1199021 + 1198861198941199022 with 119886119894 isin (0 1) and 119894 = 1 2 Wewrite

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)

Using Holderrsquos inequality and by simple calculations weobtain

sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)

where 984858 = 2minus2(119904(1minus119886

1)minus1199051198861)

gt 0 From this and applyingProposition 25 we deduce that if 119891 isin B119904119896

1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889

) Furthermoreusing (57) we finally have

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)

Corollary 49 Let 119904 be a real number in the interval(0 (119889 + 2120574)119902) and let 119902 be a real number in [1infin] Thereis a constant 119862 such that for any function 119891 isin B119904119896

119902119902(R119889

) thefollowing inequality holds

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)


Proof Let 119901 isin (1infin) and 119904 isin (0 (119889 + 2120574)119902) with1119901 = 1119902 minus 119904(119889 + 2120574) We take 119892(119909) = 1||119909||

119904 and apply(103) in the specific form

10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)

Combining this with (131) we obtain (135)

Theorem 50 Let 0 lt 119904 lt (119889 + 2120574)2 be given There existsa positive constant 119862 such that for all function 119906 isin H119904

2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)

For proof of this theorem we need the following lemmawhich we obtain by simple calculations

Lemma 51 Let 119904 be a real number in the interval (0 120574 + 1198892)Then the function 119909 997891rarr ||119909||

minus2119904 belongs to the Dunkl-Besovspace B119889+2120574minus2119904119896

1infin(R119889

)

Proof of Theorem 50 Let us define

119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)

Using homogeneous Littlewood-Paley decomposition andthe fact that 1199062 belongs to S1015840

ℎ119896(R119889) we can write

119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)

times Δ 119899 (sdotminus2119904

) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)

Lemma 51 claims that sdot minus2119904 belongs to B(119889+2120574)2minus2119904119896

2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)

The following results of this section are in sprit of theclassical case (cf [27])

Theorem 52 Let 119904 119905 gt 0 120579 = 119904(119904 + 119905) and let 1199021 1199022 1199031 1199032 isin

[1infin] 119901 1199030 isin [1infin) with 1119901 = (1 minus 120579)1199021 + 120579119902211199030 = (1 minus 120579)1199031 + 1205791199032

(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)

(ii) Moreover this inequality is valid for 119903 = 1199030 in thefollowing cases

(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901

(iii) Finally the condition 119903 ge 1199030 is sharp

Proof (i) Case 119903 gt 1199030 With no loss of generality we mayassume that 1199021 lt 1199022 and we fix 120576 gt 0 such that

1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)

The proof follows essentially the same ideas used in theprevious theorem Indeed we have for119872119895 = 2

119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)

As 1199031 = 1199032 we can only say that (1198721minus120579+120576120576119894

119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894

where 1984858119894 = (1minus120579+120576120576119894)1199031+(120579minus120576120576119894)1199032Wemay use (57) butwe get only that 119891 isin 119871

119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889

)]12984858 with984858 = max(9848581 9848582) and that satisfies (143) with 119903 = 984858 Howeverwe may choose 120576 as small as we want and thus 984858 as close to 1199030as we want thus 119891 satisfies (143) for every 119903 gt 1199030

(ii) Case 119903 = 1199030

(a) If 119903 = 1199031 = 1199032 this case was treated in Theorem 48(b) If 1199031 = 1199021 and 1199032 = 1199022 this is a direct consequence of

(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)

and get by Holderrsquos inequality

10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)

We then use the embedding B0119896

119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)

which is valid for 119901 le 2

Theorem 53 Let 119904 119905 gt 0 let 1199021 1199022 isin [1infin] with 1199021 lt 1199022 Let120579 = 119904(119904 + 119905) isin (0 1) and let 1119901 = (1 minus 120579)1199021 + 1205791199022

(i) If 1199021 le 1199031 le 1199022 and let 1119903 = (1 minus 120579)1199031 + 1205791199022 Thenone has

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)

(ii) If 1199021 le 1199032 le 1199022 and let 1119903 = (1 minus 120579)1199021 + 1205791199032 Thenone has

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)

Proof We only prove the first inequality as the proof for thesecond one is similar Since 119891 isin B119904119896

11990211199031

(R119889) noting that

120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)

we have (120582119895)119895isinZ isin 1198971199031 Thus using

Proposition 26 (i) for the interpolation

1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)

with 11199031 = (1 minus 119886)1199021 + 1198861199022 we see that we have a partitionZ = sum

119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)

Moreover since 119891 isin Bminus119905119896

11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)

Let us note that 119872119899 = (sum119895isin119885119899

2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899

Δ 119895119891 We apply now (147) andTheorem 48 to obtain

10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579

1198992minus119899(1minus119886)(1minus120579)

(155)

Since we have 119891 = sum119899isinZ 119891119899 with these two inequalities at

hand and using (57) we find that 119891 isin [119871119901

119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903with 1119903 = (1minus119886)119901+1198861199022 but since 11199031 = (1minus119886)1199021+1198861199022

and 1119901 = (1minus120579)1199021+1205791199022 we obtain [119871119901

119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022

Theorem 54 Let 119904 119905 gt 0 and let 1199021 1199022 isin [1infin]with 1199021 lt 1199022Let 120579 = 119904(119904 + 119905) isin (0 1) and let 1119901 = (1 minus 120579)1199021 + 1205791199022 Let1199021 le 1199031 le 1199032 le 1199022 and let 1119903 = (1 minus 120579)1199031 + 1205791199032 Then onehas

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)

Proof Once the previous theorem is proved it is enoughto reapply similar arguments to obtain Theorem 54 As1199021 lt 1199031 lt 1199032 lt 1199022 we start using

1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)

instead of (152) and we obtain a partition Z = sum119895isinZ 119885119895 such

that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)

with 11199031 = (1 minus 119886)1199021 + 1198861199032 and where 120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)

belongs to 1198971199031 since 119891 isin B119904119896

11990211199031

(R119889) Moreover since 119891 isin

Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899

Δ 119895119891 We apply now (151) and Theorem 48 instead of(155) to obtain

10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)

Finally we have via (57) that119891 isin [119871119901119887

119896(R119889

) 1198711199011199032

119896(R119889

)]119886119903 with1119903 = (1minus119886)119887+1198861199032 To conclude we use the fact that 1119887 =(1minus120579)1199021+1205791199032 and 11199031 = (1minus119886)1199021+1198861199032 in order to obtainthat 119891 isin 119871

119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032

Conjecture 55 Theorems 34 39 and 41 are true for thegeneral reflection group 119866

Acknowledgments

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah University on materialand moral support in the financing of this research ProjectNo 4001 The author is deeply indebted to the refereesfor providing constructive comments and for helping inimproving the contents of this paper

References

[1] C F Dunkl ldquoDifferential-difference operators associated toreflection groupsrdquo Transactions of the American MathematicalSociety vol 311 no 1 pp 167ndash183 1989

[2] T H Baker and P J Forrester ldquoNon symmetric Jack polynomi-als and integral kernelsrdquoDukeMathematical Journal vol 95 no1 pp 1ndash50 1998

[3] J F van Diejen and L Vinet Calogero-Sutherland-Moser Mod-els CRM Series in Mathematical Physics Springer New YorkNY USA 2000

[4] K Hikami ldquoDunkl operator formalism for quantum many-body problems associated with classical root systemsrdquo Journalof the Physical Society of Japan vol 65 no 2 pp 394ndash401 1996

[5] M F E de Jeu ldquoThe dunkl transformrdquo Inventiones Mathemati-cae vol 113 no 1 pp 147ndash162 1993

[6] C F Dunkl ldquoHankel transforms associated to finite reflectiongroupsrdquo Contemporary Mathematics vol 138 pp 123ndash138 1992

[7] H Mejjaoli ldquoStrichartz estimates for the Dunkl wave equationand applicationrdquo Journal of Mathematical Analysis and Applica-tions vol 346 no 1 pp 41ndash54 2008

[8] H Mejjaoli ldquoDispersion phenomena in Dunkl-Schrodingerequation and applicationsrdquo Serdica Mathematical Journal vol35 pp 25ndash60 2009

[9] H Mejjaoli ldquoGlobal well-posedness and scattering for a class ofnonlinear Dunkl-Schrodinger equationsrdquo Nonlinear AnalysisTheory Methods and Applications vol 72 no 3-4 pp 1121ndash11392010

[10] H Mejjaoli ldquoDunkl-heat semigroup and applicationsrdquoApplica-ble Analysis 2012

[11] M Rosler ldquoGeneralized Hermite polynomials and the heatequation for Dunkl operatorsrdquo Communications in Mathemati-cal Physics vol 192 no 3 pp 519ndash542 1998

[12] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[13] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[14] C F Dunkl ldquoIntegral kernels with re ection group invariantrdquoCanadian Journal of Mathematics vol 43 pp 1213ndash1227 1991

[15] M Rosler ldquoA positive radial product formula for the Dunklkernelrdquo Transactions of the AmericanMathematical Society vol355 no 6 pp 2413ndash2438 2003

[16] S Thangavelu and Y Xu ldquoConvolution operator and maximalfunction for the Dunkl transformrdquo Journal drsquoAnalyse Mathema-tique vol 97 pp 25ndash55 2005

[17] K Trimeche ldquoPaley-Wiener theorems for the Dunkl transformand Dunkl translation operatorsrdquo Integral Transforms andSpecial Functions vol 13 no 1 pp 17ndash38 2002

[18] P Etingof ldquoA uniform proof of the macdonald-Mehta-Opdamidentity for finite coxeter groupsrdquo Mathematical Research Let-ters vol 17 no 2 pp 277ndash282 2010

[19] SThangavelyu and Y Xu ldquoRiesz transform and Riesz potentialsfor Dunkl transformrdquo Journal of Computational and AppliedMathematics vol 199 no 1 pp 181ndash195 2007

[20] J Bergh and J Lofstrom Interpolation Spaces An IntroductionSpringer New York NY USA 1976

[21] S Hassani S Mustapha and M Sifi ldquoRiesz potentials andfractional maximal function for the dunkl transformrdquo Journalof Lie Theory vol 19 no 4 pp 725ndash734 2009

[22] JMerker ldquoRegularity of solutions to doubly nonlinear diffusionequationsrdquo Electronic Journal of Differential Equations vol 17pp 185ndash195 2009

[23] M G Hajibayov ldquoBoundedness of the Dunkl convolutionoperatorsrdquo in Analele Universitatii de Vest vol 49 of TimisoaraSeria Matematica Informatica pp 49ndash67 2011

[24] H Hajaiej X Yu and Z Zhai ldquoFractional Gagliardo-Nirenbergand Hardy inequalities under Lorentz normsrdquo Journal of Math-ematical Analysis and Applications vol 396 no 2 pp 569ndash5772012

[25] C Ahn and Y Cho ldquoLorentz space extension of Strichartzestimatesrdquo Proceedings of the American Mathematical Societyvol 133 no 12 pp 3497ndash3503 2005

[26] M Keel and T Tao ldquoEndpoint Strichartz estimatesrdquo AmericanJournal of Mathematics vol 120 no 5 pp 955ndash980 1998

[27] D Chamorro and P G Lemarie-Rieusset ldquoReal Interpola-tion methodLorentz spaces and refined Sobolev inequalitiesrdquohttparxivorgabs12113320

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 Preliminaries

In order to confirm the basic and standard notations webriefly overview the theory of Dunkl operators and relatedharmonic analysis Main references are [1 5 6 11 14ndash17]

21 Root System Reflection Group and Multiplicity FunctionLetR119889 be the Euclidean space equipped with a scalar product⟨ ⟩ and let ||119909|| = radic⟨119909 119909⟩ For 120572 in R119889

0 120590120572 denotes thereflection in the hyperplane 119867120572 sub R119889 perpendicular to 120572that is for 119909 isin R119889 120590120572(119909) = 119909 minus 2120572

minus2⟨120572 119909⟩120572 A finite set

119877 sub R119889 0 is called a root system if 119877 cap R120572 = plusmn120572 and

120590120572119877 = 119877 for all 120572 isin 119877 We normalize each 120572 isin 119877 as ⟨120572 120572⟩ = 2We fix a 120573 isin R119889

cup120572isin119877119867120572 and define a positive root system119877+

of 119877 as 119877+ = 120572 isin 119877 | ⟨120572 120573⟩ gt 0 The reflections 120590120572 120572 isin 119877generate a finite group 119866 sub 119874(119889) called the reflection groupA function 119896 119877 rarr C on 119877 is called a multiplicity functionif it is invariant under the action of119866 We introduce the index120574 as

120574 = 120574 (119896) = sum

120572isin119877+

119896 (120572) (2)

Throughout this paper we will assume that 119896(120572) ge 0 for all120572 isin 119877 We denote by 120596119896 the weight function on R119889 given by

120596119896 (119909) = prod

120572isin119877+

|⟨120572 119909⟩|2119896(120572)

(3)

which is invariant and homogeneous of degree 2120574 In the casethat the reflection group 119866 is the group Z119889

2of sign changes

the weight function 120596119896 is a product function of the formprod

119889

119895=1|119909119895|

119896119895 119896119895 ge 0 We denote by 119888119896 the Mehta-type constant

defined by

119888119896 = intR119889

119890minus||119909||22120596119896 (119909) 119889119909 (4)

We note that Etingof (cf [18]) has given a derivation of theMehta-type constant valid for all finite reflection groups

In the following we denote by

119862(R119889) the space of continuous functions on R119889

119862119901(R119889

) the space of functions of class 119862119901 on R119889119862119901

119887(R119889

) the space of bounded functions of class 119862119901

E(R119889) the space of 119862infin functions on R119889

S(R119889) the Schwartz space of rapidly decreasing functions onR119889

119863(R119889) the space of119862infin functions onR119889which are of compactsupport

S1015840(R119889

) the space of temperate distributions on R119889

22 The Dunkl Operators Let 119896 119877 rarr C be a multiplicityfunction on 119877 and 119877+ a fixed positive root system of 119877 Thenthe Dunkl operators 119879119895 1 le 119895 le 119889 are defined on 119862

1(R119889

) by

119879119895119891 (119909) =120597

120597119909119895

119891 (119909) + sum

120572isin119877+

119896 (120572) 120572119895

119891 (119909) minus 119891 (120590120572 (119909))

⟨120572 119909⟩ (5)

where 120572 = (1205721 1205722 120572119889) Similarly as ordinary derivativeseach119879119895 satisfies for all119891 119892 in1198621

(R119889) and at least one of them

is 119866-invariant

119879119895 (119891119892) = (119879119895119891) 119892 + 119891 (119879119895119892) (6)

and for all 119891 in 1198621

119887(R119889

) and 119892 in S(R119889)

intR119889

119879119895119891 (119909) 119892 (119909) 120596119896 (119909) 119889119909 = minusintR119889

119891 (119909) 119879119895119892 (119909) 120596119896 (119909) 119889119909

(7)

Furthermore according to [1 14] the Dunkl operators119879119895 1 le 119895 le 119889 commute and there exists the so-called Dunklrsquosintertwining operator 119881119896 such that 119879119895119881119896 = 119881119896(120597120597119909119895) for1 le 119895 le 119889 and 119881119896(1) = 1 We define the Dunkl-Laplaceoperator119896 on R119889 by

119896119891 (119909) =

119889

sum

119895=1

1198792

119895119891 (119909)

= 119891 (119909) + 2 sum

120572isin119877+

119896 (120572) (⟨nabla119891 (119909) 120572⟩

⟨120572 119909⟩

minus119891 (119909) minus 119891 (120590120572 (119909))

⟨120572 119909⟩2

)

(8)

where and nabla are the usual Euclidean Laplacian and nablaoperators on R119889 respectively Since the Dunkl operatorscommute their joint eigenvalue problem is significant andfor each 119910 isin R119889 the system

119879119895119906 (119909 119910) = 119910119895119906 (119909 119910) 119895 = 1 119889 119906 (0 119910) = 1 (9)

admits a unique analytic solution 119870(119909 119910) 119909 isin R119889 called theDunkl kernel which has a holomorphic extension toC119889

timesC119889For 119909 119910 isin C119889 the kernel satisfies

(a) 119870(119909 119910) = 119870(119910 119909)(b) 119870(120582119909 119910) = 119870(119909 120582119910) for 120582 isin C(c) 119870(119892119909 119892119910) = 119870(119909 119910) for 119892 isin 119866

23 The Dunkl Transform For functions 119891 on R119889 we define119871119901-norms of 119891 with respect to 120596119896(119909)119889119909 as

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

= (intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596119896 (119909) 119889119909)

1119901

(10)

if 1 le 119901 lt infin and 119891119871infin119896(R119889) = ess sup

119909isinR119889 |119891(119909)| We denoteby 119871119901

119896(R119889

) the space of all measurable functions119891 onR119889 withfinite 119871119901

119896-norm

The Dunkl transformF119863 on 1198711

119896(R119889

) is given by

F119863 (119891) (119910) =1

119888119896

intR119889

119891 (119909)119870 (119909 minus119894119910) 120596119896 (119909) 119889119909 (11)



1

119896(R119889

)

(a) F119863(119891)119871infin119896(R119889) le 119888

minus1

119896 1198911198711

119896(R119889)

(b) F119863(119891(sdot120582))(119910) = 1205822120574+119889F119863(119891)(120582119910) for 120582 gt 0

(c) ifF119863(119891) belongs to 1198711

119896(R119889

) then

119891 (119910) =1

119888119896

intR119889

F119863 (119891) (119909)119870 (119894119909 119910) 120596119896 (119909) 119889119909 ae (12)


(d) F119863(119879119895119891)(119910) = 119894119910119895F119863(119891)(119910)


F119863F119863 = F119863F119863 = 119868119889 (13)



intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

2120596119896 (119909) 119889119909 = int

R119889

1003816100381610038161003816F119863 (119891) (120585)1003816100381610038161003816

2120596119896 (120585) 119889120585 (14)


2

119896(R119889

)


119901

119896(R119889

) by

⟨T119891 120601⟩ = intR119889

119891 (119909) 120601 (119909) 120596119896 (119909) 119889119909 (15)


hand side



⟨F119863 (120591) 120601⟩ = ⟨120591F119863 (120601)⟩ (16)

for 120601 isin S(R119889)


119896(R119889


)

⟨F119863 (119891) 120601⟩ = ⟨F119863 (T119891) 120601⟩ = ⟨T119891F119863 (120601)⟩

= ⟨119891F119863(120601)⟩119896

(17)





120591119910119891 (119909) =1

119888119896

intR119889

F119863 (119891) (119911)119870 (119894119909 119911)119870 (119894119910 119911) 120596119896 (119911) 119889119911

(18)


minus1(119891)(119909+119910)) (cf [11 17]) Hence



follows

119891lowast119863 119892 (119909) = intR119889

120591119909119891 (minus119910) 119892 (119910) 120596119896 (119910) 119889119910 (19)





) (resp S(R119889)) and

F119863 (119891lowast119863 119892) (119910) = F119863 (119891) (119910)F119863 (119892) (119910) (20)


119901

119896(R119889


1

119896(R119889


1

119896(R119889

) for 119866 = Z119889




119901

119896(R119889

) and 119892 isin 119871119902

119896(R119889

) If120591119909119892119871

119902

119896(R119889) le 119862119892

119871119902


119871119903

119896(R119889

) and

1003817100381710038171003817119891lowast119863 1198921003817100381710038171003817119871119903119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119902

119896(R119889)

(21)




119878lowast119863 120601 (119909) = ⟨119878119910 120591minus119910120601 (119909)⟩ (22)


119896(R119889



119901

119896(R119889


)

⟨T119891lowast119863120601 120595⟩ = ⟨120601lowast119863⟩119896 (23)

where (119909) = 120595(minus119909) and

F119863 (T119891lowast119863 120601) = F119863 (T119891)F119863 (120601) (24)

For each 119906 isin S1015840(R119889


⟨119879119895119906 120595⟩ = minus⟨119906 119879119895120595⟩ (25)




F119863 (119879119895119906) = 119894119910119895F119863 (119906)

F119863 (119896119906) = minus10038171003817100381710038171199101003817100381710038171003817

2F119863 (119906)

(26)


3 B119904119896

119901119902 F119904119896

119901119902(R119889

) and H119904


Basic Properties



) such that


forall120585 isin R119889 120595 (120585) + sum

119895ge0

120593 (2minus119895120585) = 1


119895isinZ

120593 (2minus119895120585) = 1


sdot) = 0


(27)


Δ 119895119891 = Fminus1

119863(120593(

120585

2119895)F119863 (119891)) 119878119895119891 = sum

119899le119895minus1

Δ 119899119891

forall119895 isin Z

(28)



119863(120593)


119863(120595)




F119863 (119878119895119891) (120585) = (120585

2119895)F119863 (119878119895119891) (120585)

F119863 (Δ 119895119891) (120585) = 120593(120585

2119895)F119863 (Δ 119895119891) (120585)

(29)

We put

120601 = Fminus1

119863(120593) 120594 = F

minus1

119863() (30)





119878119895119906 = 0 in S1015840(R

119889) (31)



119902119886119903

119902)1119903 stands for



119901119902(R119889


ℎ119896(R119889) such that

10038171003817100381710038171198911003817100381710038171003817B119904119896119901119902

(R119889)= (sum

119895isinZ

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

)

119902

)

1119902

lt infin (32)


119901119902(R119889


ℎ119896(R119889) such that

10038171003817100381710038171198911003817100381710038171003817F119904119896119901119902

(R119889)=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

211990411989511990210038161003816100381610038161003816

Δ 11989511989110038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

119896(R119889)

lt infin (33)


1199042


(minus119896)1199042119891 = F

minus1

119863(sdot

119904F119863119891)

(119868 minus 119896)1199042119891 = F

minus1

119863((1 + sdot

2)1199042

F119863119891)

(34)



(119868 minus 119896)minus1199042

119891 = 119891lowast119863 119861119896119904(35)

where

119861119896119904 (119910) =1

Γ (1199042)int

infin

0


119905 (36)


119896(R119889

) and

119861119896119904 (119910) le 11986210038171003817100381710038171199101003817100381710038171003817


10038171003817100381710038171199101003817100381710038171003817 gt 0 (37)


119904

119901119896(R119889


1199042(119871

119901

119896(R119889


(R119889) =

(119868 minus 119896)1199042119891

119871119901

119896(R119889)


Furthermore 119901 = 2119867119904

2119896(R119889

) = 119867119904

119896(R119889

)


119896 0 lt 119904 lt 119889 + 2120574 are



119891 = 119877119896119904lowast119863119891(38)


119877119896119904 (119910) = 119862 (119896 119904 119889)10038171003817100381710038171199101003817100381710038171003817

119904minus119889minus2120574

where 119862 (119896 119904 119889) =Γ ((119889 + 2120574 minus 119904) 2)

2(119889+2120574minus119904)2Γ (1199042)

(39)


119901119896(R119889


1199042(119871

119901

119896(R119889


(R119889) =

(minus119896)1199042119891

119871119901

119896(R119889)


B119904119896

119902119902(R

119889)

= F119904119896

119902119902(R

119889) 997893rarr F

119904119896

119902infin(R

119889) 997893rarr F

119904minus(119889+2120574)119902119896

infininfin(R

119889)

(40)

H119904

119902119896(R

119889)

= F119904119896

1199022(R

119889) 997893rarr F

119904119896

119902infin(R

119889) 997893rarr F

119904minus(119889+2120574)119902119896

infininfin(R

119889)

(41)




1199021infin(R119889

) cap Fminus119887119896

1199022infin(R119889

) then one has

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

120579

(42)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

F1198861198961199021infin(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus1198871198961199022infin(R119889) (43)


1003816100381610038161003816 le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ


(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816))

(44)


2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816) le 2


(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816) (45)

and we write

1003816100381610038161003816119891 (119909)1003816100381610038161003816

le sum

119895le119873(119909)

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895isinZ

(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)


2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

119887(119886+119887)

(sup119895isinZ

2minus119887119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

119886(119886+119887)

(46)



10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus119902119901

Bminus((2120574+119889)119902minus119904)119896


10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904119896119902119902(R119889) (47)

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus119902119901

Bminus((2120574+119889)119902minus119904)119896119896


10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902119896(R119889)

(48)

where 119901 = 119902(2120574 + 119889)(2120574 + 119889 minus 119902119904)



119901119903(R119889

) cap 119871infin

119896(R119889


119906VB119904119896119901119903(R119889)

le 119862 [119906119871infin119896(R119889)VB119904119896

119901119903(R119889) + V119871infin

119896(R119889)119906B119904119896

119901119903(R119889)]

(49)


119906VB11990411989611990121199032(R119889)

le 119862[119906B1199041119896

1199011infin(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

1199011infin(R119889)

]

(50)

where 119904 = 1199041 + 1199042 minus (119889 + 2120574)1199011



119901infin(R119889)

le 119862[119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

11990111199031(R119889)

]

(51)



119906VB11990412119896

119901119903(R119889)

le 119862119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

(52)




(sum

119895isinZ

10038161003816100381610038161003816119886119895

10038161003816100381610038161003816

119902

)

1119902

lt infin (53)






119869 (119905 119886 1198600 1198601) = max (1198861198600

1199051198861198601

)

119870 (119905 119886 1198600 1198601) = min (1003817100381710038171003817119886010038171003817100381710038171198600

+ 11990510038171003817100381710038171198861

10038171003817100381710038171198601

119886 = 1198860 + 1198861)

(54)



minus119895120579119869(2

119895 119886119895 1198600 1198601))119895isinZ isin 119897

119902(Z)


119886[11986001198601]120579119902119869

= inf119886=sum119895isinZ 119886119895

(sum

119895isinZ

2minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

1119902

+ (sum

119895isinZ

2119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

1119902

(55)



minus119895120579119870(2

119895 119886 1198600 1198601))119895isinZ isin 119897

119902(Z)


119886[11986001198601]120579119902119870

= (sum

119895isinZ

2minus119895120579119902

119870(2119895 119886 1198600 1198601)

119902

)

1119902

(56)




119886[11986001198601]120579119902

le 119862 (119902 120579 984858)(sum

119895isinZ

984858minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

(1minus120579)119902

times (sum

119895isinZ

984858119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

120579119902

(57)


[[1198600 1198601]12057901199020

[1198600 1198601]12057911199021

]120579119902

= [1198600 1198601](1minus120579)1205790+1205791205791119902 (58)



0 119860

1015840

1and [1198600 1198601]

1015840

120579119902for 0 lt 120579 lt 1


1015840

120579119902=

[1198601015840

0 119860

1015840

1]1205791199021015840



119889119891119896 (120582) = int119909isinR119889 |119891(119909)|ge120582

120596119896 (119909) 119889119909

119891lowast

119896(119904) = inf 120582 119889119891119896 (120582) le 119904

(59)


1003817100381710038171003817119871119901119902

119896(R119889)

=

(int

infin

0

(1199041119901

119891lowast

119896(119904))

119902 119889119904

119904)

1119902

if 1 le 119901 119902 lt infin

sup119904gt0

1199041119901

119891lowast


(60)



119871119901119902

119896(R119889) lt infin



119871119901119902

119896(R

119889) = [119871

1

119896(R

119889) 119871

infin

119896(R

119889)]

120579119902 (61)


[11987111990101199020

119896(R

119889) 119871

11990111199021

119896(R

119889)]

120579119902= [119871

1199010

119896(R

119889) 119871

1199011

119896(R

119889)]

120579119902

= 119871119901119902

119896(R

119889)

(62)


[1198711199011199020

119896(R

119889) 119871

1199011199021

119896(R

119889)]

120579119902= 119871

119901119902

119896(R

119889) (63)


1198711199011199021

119896(R

119889) 997893rarr 119871

1199011199022

119896(R

119889) (64)



119901119902

119896(R119889


119895isinZ 119891119895 where

1003817100381710038171003817100381710038171003817(2

minus119895(119901minus1)11990110038171003817100381710038171003817119891119895

100381710038171003817100381710038171198711119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903+

1003817100381710038171003817100381710038171003817(2

11989511990110038171003817100381710038171003817119891119895

10038171003817100381710038171003817119871infin119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(65)



119896(R119889


119871119901(119901minus1)119902(119902minus1)

119896(R119889

) one has 119891119892 isin 1198711

119896(R119889

) and

1003816100381610038161003816100381610038161003816intR119889

119891 (119909) 119892 (119909) 120596119896 (119909) 119889119909

1003816100381610038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901(119901minus1)119902(119902minus1)

119896(R119889)

(66)




1

119901minus1

119902=

119904

2120574 + 119889 (67)

For 119891 isin 119871119901

119896(R119889

) one has100381710038171003817100381710038171003817(119868 minus 119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(68)



2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889)

(69)


119867119904

119896(R119889



⟨119891 119892⟩1198712119896(R119889)

= ⟨F119863(119891)F119863 (119892)⟩1198712119896(R119889)

= intR119889

(1 +10038171003817100381710038171205851003817100381710038171003817

2)1199042

F119863 (119891) (120585)F119863 (119892) (120585)(1 +10038171003817100381710038171205851003817100381710038171003817

2)minus1199042

times 120596119896 (120585) 119889120585

= ⟨(119868 minus 119896)1199042119891 (119868 minus 119896)

minus1199042119892⟩

1198712119896(R119889)

(70)

Hence100381610038161003816100381610038161003816⟨119891 119892⟩

1198712119896(R119889)

100381610038161003816100381610038161003816le100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

100381710038171003817100381710038171003817(119868 minus 119896)

minus11990421198921003817100381710038171003817100381710038171198712119896(R119889)

(71)


1198712119896(R119889)

100381610038161003816100381610038161003816le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901

119896(R119889)

(72)



10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus1

119871119902

119896(R119889)

(73)

Thus we obtain (69)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(74)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (75)


10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

1198711199010

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119901minus120579

1198711199012

119896(R119889)

(76)


where

1

1199010

=1

120579(1 minus

1

119901)

1

1199012

=1

(119901 minus 120579) 119901 (77)



exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(78)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (79)


ℎ ln(intR119889

|119891(119909)|1ℎ


intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(80)



intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

(119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

)

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

le119902120579

119901 minus 119902ln(

119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(81)



119904

119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(82)



2 If

119891 isin 11987111990111199021

119896(R119889

) 119892 isin 11987111990121199022

119896(R119889


11990131199023

119896(R119889



100381710038171003817100381711987111990131199023

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(83)





10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(84)

holds for

120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(85)


10038171003817100381710038171198911003817100381710038171003817

119902

119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

11987111990131199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus120579

11987111990121199022

119896(R119889)

(86)

where1

1199013

=1

119901minus1

119902+

1

1199021

1

1199012

=1

119901minus1

119902+

1

1199022

(87)

Note that

119891 (119909) = (119868 minus 119896)1199042119891lowast119863119861119896119904 (119909)

(88)


119871(119889+2120574)(119889+2120574minus119904)infin

119896(R119889


1003817100381710038171003817119891100381710038171003817100381711987111990131199021

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

119904211989110038171003817100381710038171003817100381711987111990111199021

119896(R119889)

(89)


for

1

1199013

=1

1199011

minus119904

119889 + 2120574 0 lt 119904 lt

119889 + 2120574

1199011

(90)




2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889) (91)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(92)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (93)


exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(94)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (95)


119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(96)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(97)

holds for120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(98)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

1198711199012

119896(R119889)

(99)





1003817100381710038171003817119871119902(119889+2120574)(119889+2120574minus119902119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(100)


100381710038171003817100381710038171003817(minus119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(101)





10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(102)



119896(R119889

) and119892 isin 119871

11990121199022

119896(R119889

) then1003817100381710038171003817119891119892

1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(103)

where 1119901 = 11199011 + 11199012 and 1119902 = 11199021 + 11199022



1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199011119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (104)


119903infin

119896(R119889

) we have10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

(105)


10038171003817100381710038171198911003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

119904119904119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(106)

for any 119891 isin 119871119901119902

119896(R119889





2

119896(R119889

) defined by

I119896 (119905) V =1

119888119896|119905|120574+1198892

119890minus119894(119889+2120574)(1205874) sgn 119905

119890119894(sdot24119905)

times [F119863 (119890119894(sdot24119905)V)] (

sdot

2119905)

(107)

1198821119903

119896(R119889



119903

119896(119877

119889)


119896(R119889


1199061198821119903

119896(R119889) = sum

|120583|le1

10038171003817100381710038171198791205831199061003817100381710038171003817119871119903119896(R119889)

(108)

1198821119903

119896119866(R119889


119896(R119889



1

119902+119889 + 2120574

2119903le119889 + 2120574

4 (109)



119875 = (22119889 + 4120574

119889 + 2120574 minus 2) (110)





L119891 (119905) = int

infin

0

119896 (119905 120591) 119891 (120591) 119889120591 (111)


10038171003817100381710038171003817119871119902119904(0infin119865)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119903(0infin119864)

(112)


L119891 (119905) = int

119905

0

119896 (119905 120591) 119891 (120591) 119889120591 (113)


1003817100381710038171003817I119896 (119905) 11989110038171003817100381710038171198711199022(0infin119871119903

119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198712119896(R119889)

(114)

10038171003817100381710038171003817100381710038171003817

int

119905

0

I119896 (119905 minus 120591) 119892 (120591) 119889120591


119896(R119889))

le 1198621003817100381710038171003817119892100381710038171003817100381711987111990210158402(0infin119871119903

1015840

119896(R119889))

(115)


1003817100381710038171003817119871119903119896(R119889)

le 119862119905minus(119889+2120574)(12minus1119903)1003817100381710038171003817119892

10038171003817100381710038171198711199031015840

119896(R119889)

(116)


119905minus(119889+2120574)(12minus1119903)

isin 1198712119903(119889+2120574)(119903minus2)infin

for any 119903 isin [22 (119889 + 2120574)

119889 + 2120574 minus 2]

(117)



119894120597119905119906 (119905 119909) + 119896119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060

(118)


119906119871119902119886(R119871119903119896(R119889)) + 119906119862(R1198712

119896(R119889))

le 119862(10038171003817100381710038171199060

10038171003817100381710038171198712119896(R119889)

+1003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(R1198711199031015840

1

119896(R119889))⋂1198712(R119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

)

(119)


119906 (119905 119909) = I119896 (119905) 1199060 (119909) + int

119905

0

I119896 (119905 minus 120591) 119891 (120591 119909) 119889120591

(119905 119909) isin R timesR119889

(120)

Let 119896(119905 120591) = I119896(119905 minus 120591) 119864 = 1198711199031015840

1

119896(R119889

) or 119871(2119889+4120574)(119889+2120574+2)2119896

(R119889)

119865 = 119871119903

119896(R119889

) and L119891(119905) = intinfin

0119896(119905 120591)119891(120591)119889120591 Then since

1199021015840


that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817119871119902119904(0infin119871119903119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(0infin119871

1199031015840

1

119896(R119889))cap1198712(0infin119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

(121)


sub 1198711199022 for all 119904 ge 2

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817

2

119871119902119904(0infin119871119903119896(R119889))

le 119862intint

infin

0

⟨I119896 (minus120591) 119891 (120591) I119896 (minus119910) 119891 (119910)⟩ 119889120591119889119910

(122)


1198712(0infin119871(2119889+4120574)(119889+2120574+2)2

119896(R119889))


11990210158401199041015840




119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)


119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889



119896119866(R119889


119902119904(0infin119882

1119903

119896119866(R119889


119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)


119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)



1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast


11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)


11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)



119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)


119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)



1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)



10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)


sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)


1)minus1199051198861)


1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)


119902119902(R119889


(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)




10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1

119896(R119889

)

(a) F119863(119891)119871infin119896(R119889) le 119888

minus1

119896 1198911198711

119896(R119889)

(b) F119863(119891(sdot120582))(119910) = 1205822120574+119889F119863(119891)(120582119910) for 120582 gt 0

(c) ifF119863(119891) belongs to 1198711

119896(R119889

) then

119891 (119910) =1

119888119896

intR119889

F119863 (119891) (119909)119870 (119894119909 119910) 120596119896 (119909) 119889119909 ae (12)


(d) F119863(119879119895119891)(119910) = 119894119910119895F119863(119891)(119910)


F119863F119863 = F119863F119863 = 119868119889 (13)



intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

2120596119896 (119909) 119889119909 = int

R119889

1003816100381610038161003816F119863 (119891) (120585)1003816100381610038161003816

2120596119896 (120585) 119889120585 (14)


2

119896(R119889

)


119901

119896(R119889

) by

⟨T119891 120601⟩ = intR119889

119891 (119909) 120601 (119909) 120596119896 (119909) 119889119909 (15)


hand side



⟨F119863 (120591) 120601⟩ = ⟨120591F119863 (120601)⟩ (16)

for 120601 isin S(R119889)


119896(R119889


)

⟨F119863 (119891) 120601⟩ = ⟨F119863 (T119891) 120601⟩ = ⟨T119891F119863 (120601)⟩

= ⟨119891F119863(120601)⟩119896

(17)





120591119910119891 (119909) =1

119888119896

intR119889

F119863 (119891) (119911)119870 (119894119909 119911)119870 (119894119910 119911) 120596119896 (119911) 119889119911

(18)


minus1(119891)(119909+119910)) (cf [11 17]) Hence



follows

119891lowast119863 119892 (119909) = intR119889

120591119909119891 (minus119910) 119892 (119910) 120596119896 (119910) 119889119910 (19)





) (resp S(R119889)) and

F119863 (119891lowast119863 119892) (119910) = F119863 (119891) (119910)F119863 (119892) (119910) (20)


119901

119896(R119889


1

119896(R119889


1

119896(R119889

) for 119866 = Z119889




119901

119896(R119889

) and 119892 isin 119871119902

119896(R119889

) If120591119909119892119871

119902

119896(R119889) le 119862119892

119871119902


119871119903

119896(R119889

) and

1003817100381710038171003817119891lowast119863 1198921003817100381710038171003817119871119903119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119902

119896(R119889)

(21)




119878lowast119863 120601 (119909) = ⟨119878119910 120591minus119910120601 (119909)⟩ (22)


119896(R119889



119901

119896(R119889


)

⟨T119891lowast119863120601 120595⟩ = ⟨120601lowast119863⟩119896 (23)

where (119909) = 120595(minus119909) and

F119863 (T119891lowast119863 120601) = F119863 (T119891)F119863 (120601) (24)

For each 119906 isin S1015840(R119889


⟨119879119895119906 120595⟩ = minus⟨119906 119879119895120595⟩ (25)




F119863 (119879119895119906) = 119894119910119895F119863 (119906)

F119863 (119896119906) = minus10038171003817100381710038171199101003817100381710038171003817

2F119863 (119906)

(26)


3 B119904119896

119901119902 F119904119896

119901119902(R119889

) and H119904


Basic Properties



) such that


forall120585 isin R119889 120595 (120585) + sum

119895ge0

120593 (2minus119895120585) = 1


119895isinZ

120593 (2minus119895120585) = 1


sdot) = 0


(27)


Δ 119895119891 = Fminus1

119863(120593(

120585

2119895)F119863 (119891)) 119878119895119891 = sum

119899le119895minus1

Δ 119899119891

forall119895 isin Z

(28)



119863(120593)


119863(120595)




F119863 (119878119895119891) (120585) = (120585

2119895)F119863 (119878119895119891) (120585)

F119863 (Δ 119895119891) (120585) = 120593(120585

2119895)F119863 (Δ 119895119891) (120585)

(29)

We put

120601 = Fminus1

119863(120593) 120594 = F

minus1

119863() (30)





119878119895119906 = 0 in S1015840(R

119889) (31)



119902119886119903

119902)1119903 stands for



119901119902(R119889


ℎ119896(R119889) such that

10038171003817100381710038171198911003817100381710038171003817B119904119896119901119902

(R119889)= (sum

119895isinZ

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

)

119902

)

1119902

lt infin (32)


119901119902(R119889


ℎ119896(R119889) such that

10038171003817100381710038171198911003817100381710038171003817F119904119896119901119902

(R119889)=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

211990411989511990210038161003816100381610038161003816

Δ 11989511989110038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

119896(R119889)

lt infin (33)


1199042


(minus119896)1199042119891 = F

minus1

119863(sdot

119904F119863119891)

(119868 minus 119896)1199042119891 = F

minus1

119863((1 + sdot

2)1199042

F119863119891)

(34)



(119868 minus 119896)minus1199042

119891 = 119891lowast119863 119861119896119904(35)

where

119861119896119904 (119910) =1

Γ (1199042)int

infin

0


119905 (36)


119896(R119889

) and

119861119896119904 (119910) le 11986210038171003817100381710038171199101003817100381710038171003817


10038171003817100381710038171199101003817100381710038171003817 gt 0 (37)


119904

119901119896(R119889


1199042(119871

119901

119896(R119889


(R119889) =

(119868 minus 119896)1199042119891

119871119901

119896(R119889)


Furthermore 119901 = 2119867119904

2119896(R119889

) = 119867119904

119896(R119889

)


119896 0 lt 119904 lt 119889 + 2120574 are



119891 = 119877119896119904lowast119863119891(38)


119877119896119904 (119910) = 119862 (119896 119904 119889)10038171003817100381710038171199101003817100381710038171003817

119904minus119889minus2120574

where 119862 (119896 119904 119889) =Γ ((119889 + 2120574 minus 119904) 2)

2(119889+2120574minus119904)2Γ (1199042)

(39)


119901119896(R119889


1199042(119871

119901

119896(R119889


(R119889) =

(minus119896)1199042119891

119871119901

119896(R119889)


B119904119896

119902119902(R

119889)

= F119904119896

119902119902(R

119889) 997893rarr F

119904119896

119902infin(R

119889) 997893rarr F

119904minus(119889+2120574)119902119896

infininfin(R

119889)

(40)

H119904

119902119896(R

119889)

= F119904119896

1199022(R

119889) 997893rarr F

119904119896

119902infin(R

119889) 997893rarr F

119904minus(119889+2120574)119902119896

infininfin(R

119889)

(41)




1199021infin(R119889

) cap Fminus119887119896

1199022infin(R119889

) then one has

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

120579

(42)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

F1198861198961199021infin(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus1198871198961199022infin(R119889) (43)


1003816100381610038161003816 le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ


(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816))

(44)


2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816) le 2


(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816) (45)

and we write

1003816100381610038161003816119891 (119909)1003816100381610038161003816

le sum

119895le119873(119909)

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895isinZ

(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)


2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

119887(119886+119887)

(sup119895isinZ

2minus119887119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

119886(119886+119887)

(46)



10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus119902119901

Bminus((2120574+119889)119902minus119904)119896


10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904119896119902119902(R119889) (47)

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus119902119901

Bminus((2120574+119889)119902minus119904)119896119896


10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902119896(R119889)

(48)

where 119901 = 119902(2120574 + 119889)(2120574 + 119889 minus 119902119904)



119901119903(R119889

) cap 119871infin

119896(R119889


119906VB119904119896119901119903(R119889)

le 119862 [119906119871infin119896(R119889)VB119904119896

119901119903(R119889) + V119871infin

119896(R119889)119906B119904119896

119901119903(R119889)]

(49)


119906VB11990411989611990121199032(R119889)

le 119862[119906B1199041119896

1199011infin(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

1199011infin(R119889)

]

(50)

where 119904 = 1199041 + 1199042 minus (119889 + 2120574)1199011



119901infin(R119889)

le 119862[119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

11990111199031(R119889)

]

(51)



119906VB11990412119896

119901119903(R119889)

le 119862119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

(52)




(sum

119895isinZ

10038161003816100381610038161003816119886119895

10038161003816100381610038161003816

119902

)

1119902

lt infin (53)






119869 (119905 119886 1198600 1198601) = max (1198861198600

1199051198861198601

)

119870 (119905 119886 1198600 1198601) = min (1003817100381710038171003817119886010038171003817100381710038171198600

+ 11990510038171003817100381710038171198861

10038171003817100381710038171198601

119886 = 1198860 + 1198861)

(54)



minus119895120579119869(2

119895 119886119895 1198600 1198601))119895isinZ isin 119897

119902(Z)


119886[11986001198601]120579119902119869

= inf119886=sum119895isinZ 119886119895

(sum

119895isinZ

2minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

1119902

+ (sum

119895isinZ

2119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

1119902

(55)



minus119895120579119870(2

119895 119886 1198600 1198601))119895isinZ isin 119897

119902(Z)


119886[11986001198601]120579119902119870

= (sum

119895isinZ

2minus119895120579119902

119870(2119895 119886 1198600 1198601)

119902

)

1119902

(56)




119886[11986001198601]120579119902

le 119862 (119902 120579 984858)(sum

119895isinZ

984858minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

(1minus120579)119902

times (sum

119895isinZ

984858119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

120579119902

(57)


[[1198600 1198601]12057901199020

[1198600 1198601]12057911199021

]120579119902

= [1198600 1198601](1minus120579)1205790+1205791205791119902 (58)



0 119860

1015840

1and [1198600 1198601]

1015840

120579119902for 0 lt 120579 lt 1


1015840

120579119902=

[1198601015840

0 119860

1015840

1]1205791199021015840



119889119891119896 (120582) = int119909isinR119889 |119891(119909)|ge120582

120596119896 (119909) 119889119909

119891lowast

119896(119904) = inf 120582 119889119891119896 (120582) le 119904

(59)


1003817100381710038171003817119871119901119902

119896(R119889)

=

(int

infin

0

(1199041119901

119891lowast

119896(119904))

119902 119889119904

119904)

1119902

if 1 le 119901 119902 lt infin

sup119904gt0

1199041119901

119891lowast


(60)



119871119901119902

119896(R119889) lt infin



119871119901119902

119896(R

119889) = [119871

1

119896(R

119889) 119871

infin

119896(R

119889)]

120579119902 (61)


[11987111990101199020

119896(R

119889) 119871

11990111199021

119896(R

119889)]

120579119902= [119871

1199010

119896(R

119889) 119871

1199011

119896(R

119889)]

120579119902

= 119871119901119902

119896(R

119889)

(62)


[1198711199011199020

119896(R

119889) 119871

1199011199021

119896(R

119889)]

120579119902= 119871

119901119902

119896(R

119889) (63)


1198711199011199021

119896(R

119889) 997893rarr 119871

1199011199022

119896(R

119889) (64)



119901119902

119896(R119889


119895isinZ 119891119895 where

1003817100381710038171003817100381710038171003817(2

minus119895(119901minus1)11990110038171003817100381710038171003817119891119895

100381710038171003817100381710038171198711119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903+

1003817100381710038171003817100381710038171003817(2

11989511990110038171003817100381710038171003817119891119895

10038171003817100381710038171003817119871infin119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(65)



119896(R119889


119871119901(119901minus1)119902(119902minus1)

119896(R119889

) one has 119891119892 isin 1198711

119896(R119889

) and

1003816100381610038161003816100381610038161003816intR119889

119891 (119909) 119892 (119909) 120596119896 (119909) 119889119909

1003816100381610038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901(119901minus1)119902(119902minus1)

119896(R119889)

(66)




1

119901minus1

119902=

119904

2120574 + 119889 (67)

For 119891 isin 119871119901

119896(R119889

) one has100381710038171003817100381710038171003817(119868 minus 119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(68)



2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889)

(69)


119867119904

119896(R119889



⟨119891 119892⟩1198712119896(R119889)

= ⟨F119863(119891)F119863 (119892)⟩1198712119896(R119889)

= intR119889

(1 +10038171003817100381710038171205851003817100381710038171003817

2)1199042

F119863 (119891) (120585)F119863 (119892) (120585)(1 +10038171003817100381710038171205851003817100381710038171003817

2)minus1199042

times 120596119896 (120585) 119889120585

= ⟨(119868 minus 119896)1199042119891 (119868 minus 119896)

minus1199042119892⟩

1198712119896(R119889)

(70)

Hence100381610038161003816100381610038161003816⟨119891 119892⟩

1198712119896(R119889)

100381610038161003816100381610038161003816le100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

100381710038171003817100381710038171003817(119868 minus 119896)

minus11990421198921003817100381710038171003817100381710038171198712119896(R119889)

(71)


1198712119896(R119889)

100381610038161003816100381610038161003816le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901

119896(R119889)

(72)



10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus1

119871119902

119896(R119889)

(73)

Thus we obtain (69)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(74)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (75)


10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

1198711199010

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119901minus120579

1198711199012

119896(R119889)

(76)


where

1

1199010

=1

120579(1 minus

1

119901)

1

1199012

=1

(119901 minus 120579) 119901 (77)



exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(78)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (79)


ℎ ln(intR119889

|119891(119909)|1ℎ


intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(80)



intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

(119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

)

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

le119902120579

119901 minus 119902ln(

119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(81)



119904

119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(82)



2 If

119891 isin 11987111990111199021

119896(R119889

) 119892 isin 11987111990121199022

119896(R119889


11990131199023

119896(R119889



100381710038171003817100381711987111990131199023

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(83)





10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(84)

holds for

120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(85)


10038171003817100381710038171198911003817100381710038171003817

119902

119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

11987111990131199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus120579

11987111990121199022

119896(R119889)

(86)

where1

1199013

=1

119901minus1

119902+

1

1199021

1

1199012

=1

119901minus1

119902+

1

1199022

(87)

Note that

119891 (119909) = (119868 minus 119896)1199042119891lowast119863119861119896119904 (119909)

(88)


119871(119889+2120574)(119889+2120574minus119904)infin

119896(R119889


1003817100381710038171003817119891100381710038171003817100381711987111990131199021

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

119904211989110038171003817100381710038171003817100381711987111990111199021

119896(R119889)

(89)


for

1

1199013

=1

1199011

minus119904

119889 + 2120574 0 lt 119904 lt

119889 + 2120574

1199011

(90)




2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889) (91)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(92)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (93)


exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(94)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (95)


119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(96)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(97)

holds for120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(98)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

1198711199012

119896(R119889)

(99)





1003817100381710038171003817119871119902(119889+2120574)(119889+2120574minus119902119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(100)


100381710038171003817100381710038171003817(minus119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(101)





10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(102)



119896(R119889

) and119892 isin 119871

11990121199022

119896(R119889

) then1003817100381710038171003817119891119892

1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(103)

where 1119901 = 11199011 + 11199012 and 1119902 = 11199021 + 11199022



1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199011119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (104)


119903infin

119896(R119889

) we have10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

(105)


10038171003817100381710038171198911003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

119904119904119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(106)

for any 119891 isin 119871119901119902

119896(R119889





2

119896(R119889

) defined by

I119896 (119905) V =1

119888119896|119905|120574+1198892

119890minus119894(119889+2120574)(1205874) sgn 119905

119890119894(sdot24119905)

times [F119863 (119890119894(sdot24119905)V)] (

sdot

2119905)

(107)

1198821119903

119896(R119889



119903

119896(119877

119889)


119896(R119889


1199061198821119903

119896(R119889) = sum

|120583|le1

10038171003817100381710038171198791205831199061003817100381710038171003817119871119903119896(R119889)

(108)

1198821119903

119896119866(R119889


119896(R119889



1

119902+119889 + 2120574

2119903le119889 + 2120574

4 (109)



119875 = (22119889 + 4120574

119889 + 2120574 minus 2) (110)





L119891 (119905) = int

infin

0

119896 (119905 120591) 119891 (120591) 119889120591 (111)


10038171003817100381710038171003817119871119902119904(0infin119865)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119903(0infin119864)

(112)


L119891 (119905) = int

119905

0

119896 (119905 120591) 119891 (120591) 119889120591 (113)


1003817100381710038171003817I119896 (119905) 11989110038171003817100381710038171198711199022(0infin119871119903

119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198712119896(R119889)

(114)

10038171003817100381710038171003817100381710038171003817

int

119905

0

I119896 (119905 minus 120591) 119892 (120591) 119889120591


119896(R119889))

le 1198621003817100381710038171003817119892100381710038171003817100381711987111990210158402(0infin119871119903

1015840

119896(R119889))

(115)


1003817100381710038171003817119871119903119896(R119889)

le 119862119905minus(119889+2120574)(12minus1119903)1003817100381710038171003817119892

10038171003817100381710038171198711199031015840

119896(R119889)

(116)


119905minus(119889+2120574)(12minus1119903)

isin 1198712119903(119889+2120574)(119903minus2)infin

for any 119903 isin [22 (119889 + 2120574)

119889 + 2120574 minus 2]

(117)



119894120597119905119906 (119905 119909) + 119896119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060

(118)


119906119871119902119886(R119871119903119896(R119889)) + 119906119862(R1198712

119896(R119889))

le 119862(10038171003817100381710038171199060

10038171003817100381710038171198712119896(R119889)

+1003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(R1198711199031015840

1

119896(R119889))⋂1198712(R119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

)

(119)


119906 (119905 119909) = I119896 (119905) 1199060 (119909) + int

119905

0

I119896 (119905 minus 120591) 119891 (120591 119909) 119889120591

(119905 119909) isin R timesR119889

(120)

Let 119896(119905 120591) = I119896(119905 minus 120591) 119864 = 1198711199031015840

1

119896(R119889

) or 119871(2119889+4120574)(119889+2120574+2)2119896

(R119889)

119865 = 119871119903

119896(R119889

) and L119891(119905) = intinfin

0119896(119905 120591)119891(120591)119889120591 Then since

1199021015840


that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817119871119902119904(0infin119871119903119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(0infin119871

1199031015840

1

119896(R119889))cap1198712(0infin119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

(121)


sub 1198711199022 for all 119904 ge 2

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817

2

119871119902119904(0infin119871119903119896(R119889))

le 119862intint

infin

0

⟨I119896 (minus120591) 119891 (120591) I119896 (minus119910) 119891 (119910)⟩ 119889120591119889119910

(122)


1198712(0infin119871(2119889+4120574)(119889+2120574+2)2

119896(R119889))


11990210158401199041015840




119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)


119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889



119896119866(R119889


119902119904(0infin119882

1119903

119896119866(R119889


119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)


119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)



1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast


11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)


11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)



119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)


119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)



1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)



10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)


sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)


1)minus1199051198861)


1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)


119902119902(R119889


(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)




10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












F119863 (119879119895119906) = 119894119910119895F119863 (119906)

F119863 (119896119906) = minus10038171003817100381710038171199101003817100381710038171003817

2F119863 (119906)

(26)


3 B119904119896

119901119902 F119904119896

119901119902(R119889

) and H119904


Basic Properties



) such that


forall120585 isin R119889 120595 (120585) + sum

119895ge0

120593 (2minus119895120585) = 1


119895isinZ

120593 (2minus119895120585) = 1


sdot) = 0


(27)


Δ 119895119891 = Fminus1

119863(120593(

120585

2119895)F119863 (119891)) 119878119895119891 = sum

119899le119895minus1

Δ 119899119891

forall119895 isin Z

(28)



119863(120593)


119863(120595)




F119863 (119878119895119891) (120585) = (120585

2119895)F119863 (119878119895119891) (120585)

F119863 (Δ 119895119891) (120585) = 120593(120585

2119895)F119863 (Δ 119895119891) (120585)

(29)

We put

120601 = Fminus1

119863(120593) 120594 = F

minus1

119863() (30)





119878119895119906 = 0 in S1015840(R

119889) (31)



119902119886119903

119902)1119903 stands for



119901119902(R119889


ℎ119896(R119889) such that

10038171003817100381710038171198911003817100381710038171003817B119904119896119901119902

(R119889)= (sum

119895isinZ

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

)

119902

)

1119902

lt infin (32)


119901119902(R119889


ℎ119896(R119889) such that

10038171003817100381710038171198911003817100381710038171003817F119904119896119901119902

(R119889)=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

211990411989511990210038161003816100381610038161003816

Δ 11989511989110038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

119896(R119889)

lt infin (33)


1199042


(minus119896)1199042119891 = F

minus1

119863(sdot

119904F119863119891)

(119868 minus 119896)1199042119891 = F

minus1

119863((1 + sdot

2)1199042

F119863119891)

(34)



(119868 minus 119896)minus1199042

119891 = 119891lowast119863 119861119896119904(35)

where

119861119896119904 (119910) =1

Γ (1199042)int

infin

0


119905 (36)


119896(R119889

) and

119861119896119904 (119910) le 11986210038171003817100381710038171199101003817100381710038171003817


10038171003817100381710038171199101003817100381710038171003817 gt 0 (37)


119904

119901119896(R119889


1199042(119871

119901

119896(R119889


(R119889) =

(119868 minus 119896)1199042119891

119871119901

119896(R119889)


Furthermore 119901 = 2119867119904

2119896(R119889

) = 119867119904

119896(R119889

)


119896 0 lt 119904 lt 119889 + 2120574 are



119891 = 119877119896119904lowast119863119891(38)


119877119896119904 (119910) = 119862 (119896 119904 119889)10038171003817100381710038171199101003817100381710038171003817

119904minus119889minus2120574

where 119862 (119896 119904 119889) =Γ ((119889 + 2120574 minus 119904) 2)

2(119889+2120574minus119904)2Γ (1199042)

(39)


119901119896(R119889


1199042(119871

119901

119896(R119889


(R119889) =

(minus119896)1199042119891

119871119901

119896(R119889)


B119904119896

119902119902(R

119889)

= F119904119896

119902119902(R

119889) 997893rarr F

119904119896

119902infin(R

119889) 997893rarr F

119904minus(119889+2120574)119902119896

infininfin(R

119889)

(40)

H119904

119902119896(R

119889)

= F119904119896

1199022(R

119889) 997893rarr F

119904119896

119902infin(R

119889) 997893rarr F

119904minus(119889+2120574)119902119896

infininfin(R

119889)

(41)




1199021infin(R119889

) cap Fminus119887119896

1199022infin(R119889

) then one has

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

120579

(42)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

F1198861198961199021infin(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus1198871198961199022infin(R119889) (43)


1003816100381610038161003816 le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ


(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816))

(44)


2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816) le 2


(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816) (45)

and we write

1003816100381610038161003816119891 (119909)1003816100381610038161003816

le sum

119895le119873(119909)

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895isinZ

(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)


2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

119887(119886+119887)

(sup119895isinZ

2minus119887119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

119886(119886+119887)

(46)



10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus119902119901

Bminus((2120574+119889)119902minus119904)119896


10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904119896119902119902(R119889) (47)

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus119902119901

Bminus((2120574+119889)119902minus119904)119896119896


10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902119896(R119889)

(48)

where 119901 = 119902(2120574 + 119889)(2120574 + 119889 minus 119902119904)



119901119903(R119889

) cap 119871infin

119896(R119889


119906VB119904119896119901119903(R119889)

le 119862 [119906119871infin119896(R119889)VB119904119896

119901119903(R119889) + V119871infin

119896(R119889)119906B119904119896

119901119903(R119889)]

(49)


119906VB11990411989611990121199032(R119889)

le 119862[119906B1199041119896

1199011infin(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

1199011infin(R119889)

]

(50)

where 119904 = 1199041 + 1199042 minus (119889 + 2120574)1199011



119901infin(R119889)

le 119862[119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

11990111199031(R119889)

]

(51)



119906VB11990412119896

119901119903(R119889)

le 119862119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

(52)




(sum

119895isinZ

10038161003816100381610038161003816119886119895

10038161003816100381610038161003816

119902

)

1119902

lt infin (53)






119869 (119905 119886 1198600 1198601) = max (1198861198600

1199051198861198601

)

119870 (119905 119886 1198600 1198601) = min (1003817100381710038171003817119886010038171003817100381710038171198600

+ 11990510038171003817100381710038171198861

10038171003817100381710038171198601

119886 = 1198860 + 1198861)

(54)



minus119895120579119869(2

119895 119886119895 1198600 1198601))119895isinZ isin 119897

119902(Z)


119886[11986001198601]120579119902119869

= inf119886=sum119895isinZ 119886119895

(sum

119895isinZ

2minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

1119902

+ (sum

119895isinZ

2119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

1119902

(55)



minus119895120579119870(2

119895 119886 1198600 1198601))119895isinZ isin 119897

119902(Z)


119886[11986001198601]120579119902119870

= (sum

119895isinZ

2minus119895120579119902

119870(2119895 119886 1198600 1198601)

119902

)

1119902

(56)




119886[11986001198601]120579119902

le 119862 (119902 120579 984858)(sum

119895isinZ

984858minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

(1minus120579)119902

times (sum

119895isinZ

984858119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

120579119902

(57)


[[1198600 1198601]12057901199020

[1198600 1198601]12057911199021

]120579119902

= [1198600 1198601](1minus120579)1205790+1205791205791119902 (58)



0 119860

1015840

1and [1198600 1198601]

1015840

120579119902for 0 lt 120579 lt 1


1015840

120579119902=

[1198601015840

0 119860

1015840

1]1205791199021015840



119889119891119896 (120582) = int119909isinR119889 |119891(119909)|ge120582

120596119896 (119909) 119889119909

119891lowast

119896(119904) = inf 120582 119889119891119896 (120582) le 119904

(59)


1003817100381710038171003817119871119901119902

119896(R119889)

=

(int

infin

0

(1199041119901

119891lowast

119896(119904))

119902 119889119904

119904)

1119902

if 1 le 119901 119902 lt infin

sup119904gt0

1199041119901

119891lowast


(60)



119871119901119902

119896(R119889) lt infin



119871119901119902

119896(R

119889) = [119871

1

119896(R

119889) 119871

infin

119896(R

119889)]

120579119902 (61)


[11987111990101199020

119896(R

119889) 119871

11990111199021

119896(R

119889)]

120579119902= [119871

1199010

119896(R

119889) 119871

1199011

119896(R

119889)]

120579119902

= 119871119901119902

119896(R

119889)

(62)


[1198711199011199020

119896(R

119889) 119871

1199011199021

119896(R

119889)]

120579119902= 119871

119901119902

119896(R

119889) (63)


1198711199011199021

119896(R

119889) 997893rarr 119871

1199011199022

119896(R

119889) (64)



119901119902

119896(R119889


119895isinZ 119891119895 where

1003817100381710038171003817100381710038171003817(2

minus119895(119901minus1)11990110038171003817100381710038171003817119891119895

100381710038171003817100381710038171198711119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903+

1003817100381710038171003817100381710038171003817(2

11989511990110038171003817100381710038171003817119891119895

10038171003817100381710038171003817119871infin119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(65)



119896(R119889


119871119901(119901minus1)119902(119902minus1)

119896(R119889

) one has 119891119892 isin 1198711

119896(R119889

) and

1003816100381610038161003816100381610038161003816intR119889

119891 (119909) 119892 (119909) 120596119896 (119909) 119889119909

1003816100381610038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901(119901minus1)119902(119902minus1)

119896(R119889)

(66)




1

119901minus1

119902=

119904

2120574 + 119889 (67)

For 119891 isin 119871119901

119896(R119889

) one has100381710038171003817100381710038171003817(119868 minus 119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(68)



2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889)

(69)


119867119904

119896(R119889



⟨119891 119892⟩1198712119896(R119889)

= ⟨F119863(119891)F119863 (119892)⟩1198712119896(R119889)

= intR119889

(1 +10038171003817100381710038171205851003817100381710038171003817

2)1199042

F119863 (119891) (120585)F119863 (119892) (120585)(1 +10038171003817100381710038171205851003817100381710038171003817

2)minus1199042

times 120596119896 (120585) 119889120585

= ⟨(119868 minus 119896)1199042119891 (119868 minus 119896)

minus1199042119892⟩

1198712119896(R119889)

(70)

Hence100381610038161003816100381610038161003816⟨119891 119892⟩

1198712119896(R119889)

100381610038161003816100381610038161003816le100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

100381710038171003817100381710038171003817(119868 minus 119896)

minus11990421198921003817100381710038171003817100381710038171198712119896(R119889)

(71)


1198712119896(R119889)

100381610038161003816100381610038161003816le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901

119896(R119889)

(72)



10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus1

119871119902

119896(R119889)

(73)

Thus we obtain (69)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(74)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (75)


10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

1198711199010

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119901minus120579

1198711199012

119896(R119889)

(76)


where

1

1199010

=1

120579(1 minus

1

119901)

1

1199012

=1

(119901 minus 120579) 119901 (77)



exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(78)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (79)


ℎ ln(intR119889

|119891(119909)|1ℎ


intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(80)



intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

(119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

)

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

le119902120579

119901 minus 119902ln(

119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(81)



119904

119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(82)



2 If

119891 isin 11987111990111199021

119896(R119889

) 119892 isin 11987111990121199022

119896(R119889


11990131199023

119896(R119889



100381710038171003817100381711987111990131199023

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(83)





10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(84)

holds for

120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(85)


10038171003817100381710038171198911003817100381710038171003817

119902

119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

11987111990131199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus120579

11987111990121199022

119896(R119889)

(86)

where1

1199013

=1

119901minus1

119902+

1

1199021

1

1199012

=1

119901minus1

119902+

1

1199022

(87)

Note that

119891 (119909) = (119868 minus 119896)1199042119891lowast119863119861119896119904 (119909)

(88)


119871(119889+2120574)(119889+2120574minus119904)infin

119896(R119889


1003817100381710038171003817119891100381710038171003817100381711987111990131199021

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

119904211989110038171003817100381710038171003817100381711987111990111199021

119896(R119889)

(89)


for

1

1199013

=1

1199011

minus119904

119889 + 2120574 0 lt 119904 lt

119889 + 2120574

1199011

(90)




2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889) (91)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(92)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (93)


exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(94)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (95)


119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(96)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(97)

holds for120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(98)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

1198711199012

119896(R119889)

(99)





1003817100381710038171003817119871119902(119889+2120574)(119889+2120574minus119902119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(100)


100381710038171003817100381710038171003817(minus119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(101)





10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(102)



119896(R119889

) and119892 isin 119871

11990121199022

119896(R119889

) then1003817100381710038171003817119891119892

1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(103)

where 1119901 = 11199011 + 11199012 and 1119902 = 11199021 + 11199022



1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199011119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (104)


119903infin

119896(R119889

) we have10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

(105)


10038171003817100381710038171198911003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

119904119904119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(106)

for any 119891 isin 119871119901119902

119896(R119889





2

119896(R119889

) defined by

I119896 (119905) V =1

119888119896|119905|120574+1198892

119890minus119894(119889+2120574)(1205874) sgn 119905

119890119894(sdot24119905)

times [F119863 (119890119894(sdot24119905)V)] (

sdot

2119905)

(107)

1198821119903

119896(R119889



119903

119896(119877

119889)


119896(R119889


1199061198821119903

119896(R119889) = sum

|120583|le1

10038171003817100381710038171198791205831199061003817100381710038171003817119871119903119896(R119889)

(108)

1198821119903

119896119866(R119889


119896(R119889



1

119902+119889 + 2120574

2119903le119889 + 2120574

4 (109)



119875 = (22119889 + 4120574

119889 + 2120574 minus 2) (110)





L119891 (119905) = int

infin

0

119896 (119905 120591) 119891 (120591) 119889120591 (111)


10038171003817100381710038171003817119871119902119904(0infin119865)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119903(0infin119864)

(112)


L119891 (119905) = int

119905

0

119896 (119905 120591) 119891 (120591) 119889120591 (113)


1003817100381710038171003817I119896 (119905) 11989110038171003817100381710038171198711199022(0infin119871119903

119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198712119896(R119889)

(114)

10038171003817100381710038171003817100381710038171003817

int

119905

0

I119896 (119905 minus 120591) 119892 (120591) 119889120591


119896(R119889))

le 1198621003817100381710038171003817119892100381710038171003817100381711987111990210158402(0infin119871119903

1015840

119896(R119889))

(115)


1003817100381710038171003817119871119903119896(R119889)

le 119862119905minus(119889+2120574)(12minus1119903)1003817100381710038171003817119892

10038171003817100381710038171198711199031015840

119896(R119889)

(116)


119905minus(119889+2120574)(12minus1119903)

isin 1198712119903(119889+2120574)(119903minus2)infin

for any 119903 isin [22 (119889 + 2120574)

119889 + 2120574 minus 2]

(117)



119894120597119905119906 (119905 119909) + 119896119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060

(118)


119906119871119902119886(R119871119903119896(R119889)) + 119906119862(R1198712

119896(R119889))

le 119862(10038171003817100381710038171199060

10038171003817100381710038171198712119896(R119889)

+1003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(R1198711199031015840

1

119896(R119889))⋂1198712(R119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

)

(119)


119906 (119905 119909) = I119896 (119905) 1199060 (119909) + int

119905

0

I119896 (119905 minus 120591) 119891 (120591 119909) 119889120591

(119905 119909) isin R timesR119889

(120)

Let 119896(119905 120591) = I119896(119905 minus 120591) 119864 = 1198711199031015840

1

119896(R119889

) or 119871(2119889+4120574)(119889+2120574+2)2119896

(R119889)

119865 = 119871119903

119896(R119889

) and L119891(119905) = intinfin

0119896(119905 120591)119891(120591)119889120591 Then since

1199021015840


that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817119871119902119904(0infin119871119903119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(0infin119871

1199031015840

1

119896(R119889))cap1198712(0infin119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

(121)


sub 1198711199022 for all 119904 ge 2

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817

2

119871119902119904(0infin119871119903119896(R119889))

le 119862intint

infin

0

⟨I119896 (minus120591) 119891 (120591) I119896 (minus119910) 119891 (119910)⟩ 119889120591119889119910

(122)


1198712(0infin119871(2119889+4120574)(119889+2120574+2)2

119896(R119889))


11990210158401199041015840




119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)


119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889



119896119866(R119889


119902119904(0infin119882

1119903

119896119866(R119889


119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)


119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)



1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast


11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)


11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)



119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)


119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)



1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)



10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)


sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)


1)minus1199051198861)


1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)


119902119902(R119889


(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)




10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Furthermore 119901 = 2119867119904

2119896(R119889

) = 119867119904

119896(R119889

)


119896 0 lt 119904 lt 119889 + 2120574 are



119891 = 119877119896119904lowast119863119891(38)


119877119896119904 (119910) = 119862 (119896 119904 119889)10038171003817100381710038171199101003817100381710038171003817

119904minus119889minus2120574

where 119862 (119896 119904 119889) =Γ ((119889 + 2120574 minus 119904) 2)

2(119889+2120574minus119904)2Γ (1199042)

(39)


119901119896(R119889


1199042(119871

119901

119896(R119889


(R119889) =

(minus119896)1199042119891

119871119901

119896(R119889)


B119904119896

119902119902(R

119889)

= F119904119896

119902119902(R

119889) 997893rarr F

119904119896

119902infin(R

119889) 997893rarr F

119904minus(119889+2120574)119902119896

infininfin(R

119889)

(40)

H119904

119902119896(R

119889)

= F119904119896

1199022(R

119889) 997893rarr F

119904119896

119902infin(R

119889) 997893rarr F

119904minus(119889+2120574)119902119896

infininfin(R

119889)

(41)




1199021infin(R119889

) cap Fminus119887119896

1199022infin(R119889

) then one has

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

120579

(42)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

F1198861198961199021infin(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus1198871198961199022infin(R119889) (43)


1003816100381610038161003816 le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ


(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816))

(44)


2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816) le 2


(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816) (45)

and we write

1003816100381610038161003816119891 (119909)1003816100381610038161003816

le sum

119895le119873(119909)

2119895119887sup

119895isinZ

(2minus119895119887 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895isinZ

(2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)


2119886119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

119887(119886+119887)

(sup119895isinZ

2minus119887119895 10038161003816100381610038161003816

Δ 119895119891 (119909)10038161003816100381610038161003816)

119886(119886+119887)

(46)



10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus119902119901

Bminus((2120574+119889)119902minus119904)119896


10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904119896119902119902(R119889) (47)

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus119902119901

Bminus((2120574+119889)119902minus119904)119896119896


10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902119896(R119889)

(48)

where 119901 = 119902(2120574 + 119889)(2120574 + 119889 minus 119902119904)



119901119903(R119889

) cap 119871infin

119896(R119889


119906VB119904119896119901119903(R119889)

le 119862 [119906119871infin119896(R119889)VB119904119896

119901119903(R119889) + V119871infin

119896(R119889)119906B119904119896

119901119903(R119889)]

(49)


119906VB11990411989611990121199032(R119889)

le 119862[119906B1199041119896

1199011infin(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

1199011infin(R119889)

]

(50)

where 119904 = 1199041 + 1199042 minus (119889 + 2120574)1199011



119901infin(R119889)

le 119862[119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

11990111199031(R119889)

]

(51)



119906VB11990412119896

119901119903(R119889)

le 119862119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

(52)




(sum

119895isinZ

10038161003816100381610038161003816119886119895

10038161003816100381610038161003816

119902

)

1119902

lt infin (53)






119869 (119905 119886 1198600 1198601) = max (1198861198600

1199051198861198601

)

119870 (119905 119886 1198600 1198601) = min (1003817100381710038171003817119886010038171003817100381710038171198600

+ 11990510038171003817100381710038171198861

10038171003817100381710038171198601

119886 = 1198860 + 1198861)

(54)



minus119895120579119869(2

119895 119886119895 1198600 1198601))119895isinZ isin 119897

119902(Z)


119886[11986001198601]120579119902119869

= inf119886=sum119895isinZ 119886119895

(sum

119895isinZ

2minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

1119902

+ (sum

119895isinZ

2119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

1119902

(55)



minus119895120579119870(2

119895 119886 1198600 1198601))119895isinZ isin 119897

119902(Z)


119886[11986001198601]120579119902119870

= (sum

119895isinZ

2minus119895120579119902

119870(2119895 119886 1198600 1198601)

119902

)

1119902

(56)




119886[11986001198601]120579119902

le 119862 (119902 120579 984858)(sum

119895isinZ

984858minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

(1minus120579)119902

times (sum

119895isinZ

984858119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

120579119902

(57)


[[1198600 1198601]12057901199020

[1198600 1198601]12057911199021

]120579119902

= [1198600 1198601](1minus120579)1205790+1205791205791119902 (58)



0 119860

1015840

1and [1198600 1198601]

1015840

120579119902for 0 lt 120579 lt 1


1015840

120579119902=

[1198601015840

0 119860

1015840

1]1205791199021015840



119889119891119896 (120582) = int119909isinR119889 |119891(119909)|ge120582

120596119896 (119909) 119889119909

119891lowast

119896(119904) = inf 120582 119889119891119896 (120582) le 119904

(59)


1003817100381710038171003817119871119901119902

119896(R119889)

=

(int

infin

0

(1199041119901

119891lowast

119896(119904))

119902 119889119904

119904)

1119902

if 1 le 119901 119902 lt infin

sup119904gt0

1199041119901

119891lowast


(60)



119871119901119902

119896(R119889) lt infin



119871119901119902

119896(R

119889) = [119871

1

119896(R

119889) 119871

infin

119896(R

119889)]

120579119902 (61)


[11987111990101199020

119896(R

119889) 119871

11990111199021

119896(R

119889)]

120579119902= [119871

1199010

119896(R

119889) 119871

1199011

119896(R

119889)]

120579119902

= 119871119901119902

119896(R

119889)

(62)


[1198711199011199020

119896(R

119889) 119871

1199011199021

119896(R

119889)]

120579119902= 119871

119901119902

119896(R

119889) (63)


1198711199011199021

119896(R

119889) 997893rarr 119871

1199011199022

119896(R

119889) (64)



119901119902

119896(R119889


119895isinZ 119891119895 where

1003817100381710038171003817100381710038171003817(2

minus119895(119901minus1)11990110038171003817100381710038171003817119891119895

100381710038171003817100381710038171198711119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903+

1003817100381710038171003817100381710038171003817(2

11989511990110038171003817100381710038171003817119891119895

10038171003817100381710038171003817119871infin119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(65)



119896(R119889


119871119901(119901minus1)119902(119902minus1)

119896(R119889

) one has 119891119892 isin 1198711

119896(R119889

) and

1003816100381610038161003816100381610038161003816intR119889

119891 (119909) 119892 (119909) 120596119896 (119909) 119889119909

1003816100381610038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901(119901minus1)119902(119902minus1)

119896(R119889)

(66)




1

119901minus1

119902=

119904

2120574 + 119889 (67)

For 119891 isin 119871119901

119896(R119889

) one has100381710038171003817100381710038171003817(119868 minus 119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(68)



2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889)

(69)


119867119904

119896(R119889



⟨119891 119892⟩1198712119896(R119889)

= ⟨F119863(119891)F119863 (119892)⟩1198712119896(R119889)

= intR119889

(1 +10038171003817100381710038171205851003817100381710038171003817

2)1199042

F119863 (119891) (120585)F119863 (119892) (120585)(1 +10038171003817100381710038171205851003817100381710038171003817

2)minus1199042

times 120596119896 (120585) 119889120585

= ⟨(119868 minus 119896)1199042119891 (119868 minus 119896)

minus1199042119892⟩

1198712119896(R119889)

(70)

Hence100381610038161003816100381610038161003816⟨119891 119892⟩

1198712119896(R119889)

100381610038161003816100381610038161003816le100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

100381710038171003817100381710038171003817(119868 minus 119896)

minus11990421198921003817100381710038171003817100381710038171198712119896(R119889)

(71)


1198712119896(R119889)

100381610038161003816100381610038161003816le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901

119896(R119889)

(72)



10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus1

119871119902

119896(R119889)

(73)

Thus we obtain (69)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(74)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (75)


10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

1198711199010

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119901minus120579

1198711199012

119896(R119889)

(76)


where

1

1199010

=1

120579(1 minus

1

119901)

1

1199012

=1

(119901 minus 120579) 119901 (77)



exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(78)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (79)


ℎ ln(intR119889

|119891(119909)|1ℎ


intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(80)



intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

(119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

)

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

le119902120579

119901 minus 119902ln(

119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(81)



119904

119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(82)



2 If

119891 isin 11987111990111199021

119896(R119889

) 119892 isin 11987111990121199022

119896(R119889


11990131199023

119896(R119889



100381710038171003817100381711987111990131199023

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(83)





10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(84)

holds for

120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(85)


10038171003817100381710038171198911003817100381710038171003817

119902

119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

11987111990131199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus120579

11987111990121199022

119896(R119889)

(86)

where1

1199013

=1

119901minus1

119902+

1

1199021

1

1199012

=1

119901minus1

119902+

1

1199022

(87)

Note that

119891 (119909) = (119868 minus 119896)1199042119891lowast119863119861119896119904 (119909)

(88)


119871(119889+2120574)(119889+2120574minus119904)infin

119896(R119889


1003817100381710038171003817119891100381710038171003817100381711987111990131199021

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

119904211989110038171003817100381710038171003817100381711987111990111199021

119896(R119889)

(89)


for

1

1199013

=1

1199011

minus119904

119889 + 2120574 0 lt 119904 lt

119889 + 2120574

1199011

(90)




2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889) (91)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(92)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (93)


exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(94)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (95)


119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(96)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(97)

holds for120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(98)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

1198711199012

119896(R119889)

(99)





1003817100381710038171003817119871119902(119889+2120574)(119889+2120574minus119902119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(100)


100381710038171003817100381710038171003817(minus119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(101)





10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(102)



119896(R119889

) and119892 isin 119871

11990121199022

119896(R119889

) then1003817100381710038171003817119891119892

1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(103)

where 1119901 = 11199011 + 11199012 and 1119902 = 11199021 + 11199022



1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199011119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (104)


119903infin

119896(R119889

) we have10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

(105)


10038171003817100381710038171198911003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

119904119904119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(106)

for any 119891 isin 119871119901119902

119896(R119889





2

119896(R119889

) defined by

I119896 (119905) V =1

119888119896|119905|120574+1198892

119890minus119894(119889+2120574)(1205874) sgn 119905

119890119894(sdot24119905)

times [F119863 (119890119894(sdot24119905)V)] (

sdot

2119905)

(107)

1198821119903

119896(R119889



119903

119896(119877

119889)


119896(R119889


1199061198821119903

119896(R119889) = sum

|120583|le1

10038171003817100381710038171198791205831199061003817100381710038171003817119871119903119896(R119889)

(108)

1198821119903

119896119866(R119889


119896(R119889



1

119902+119889 + 2120574

2119903le119889 + 2120574

4 (109)



119875 = (22119889 + 4120574

119889 + 2120574 minus 2) (110)





L119891 (119905) = int

infin

0

119896 (119905 120591) 119891 (120591) 119889120591 (111)


10038171003817100381710038171003817119871119902119904(0infin119865)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119903(0infin119864)

(112)


L119891 (119905) = int

119905

0

119896 (119905 120591) 119891 (120591) 119889120591 (113)


1003817100381710038171003817I119896 (119905) 11989110038171003817100381710038171198711199022(0infin119871119903

119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198712119896(R119889)

(114)

10038171003817100381710038171003817100381710038171003817

int

119905

0

I119896 (119905 minus 120591) 119892 (120591) 119889120591


119896(R119889))

le 1198621003817100381710038171003817119892100381710038171003817100381711987111990210158402(0infin119871119903

1015840

119896(R119889))

(115)


1003817100381710038171003817119871119903119896(R119889)

le 119862119905minus(119889+2120574)(12minus1119903)1003817100381710038171003817119892

10038171003817100381710038171198711199031015840

119896(R119889)

(116)


119905minus(119889+2120574)(12minus1119903)

isin 1198712119903(119889+2120574)(119903minus2)infin

for any 119903 isin [22 (119889 + 2120574)

119889 + 2120574 minus 2]

(117)



119894120597119905119906 (119905 119909) + 119896119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060

(118)


119906119871119902119886(R119871119903119896(R119889)) + 119906119862(R1198712

119896(R119889))

le 119862(10038171003817100381710038171199060

10038171003817100381710038171198712119896(R119889)

+1003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(R1198711199031015840

1

119896(R119889))⋂1198712(R119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

)

(119)


119906 (119905 119909) = I119896 (119905) 1199060 (119909) + int

119905

0

I119896 (119905 minus 120591) 119891 (120591 119909) 119889120591

(119905 119909) isin R timesR119889

(120)

Let 119896(119905 120591) = I119896(119905 minus 120591) 119864 = 1198711199031015840

1

119896(R119889

) or 119871(2119889+4120574)(119889+2120574+2)2119896

(R119889)

119865 = 119871119903

119896(R119889

) and L119891(119905) = intinfin

0119896(119905 120591)119891(120591)119889120591 Then since

1199021015840


that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817119871119902119904(0infin119871119903119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(0infin119871

1199031015840

1

119896(R119889))cap1198712(0infin119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

(121)


sub 1198711199022 for all 119904 ge 2

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817

2

119871119902119904(0infin119871119903119896(R119889))

le 119862intint

infin

0

⟨I119896 (minus120591) 119891 (120591) I119896 (minus119910) 119891 (119910)⟩ 119889120591119889119910

(122)


1198712(0infin119871(2119889+4120574)(119889+2120574+2)2

119896(R119889))


11990210158401199041015840




119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)


119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889



119896119866(R119889


119902119904(0infin119882

1119903

119896119866(R119889


119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)


119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)



1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast


11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)


11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)



119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)


119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)



1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)



10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)


sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)


1)minus1199051198861)


1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)


119902119902(R119889


(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)




10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119901infin(R119889)

le 119862[119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

+ 119906B1199042119896

11990121199032(R119889)

VB1199041119896

11990111199031(R119889)

]

(51)



119906VB11990412119896

119901119903(R119889)

le 119862119906B1199041119896

11990111199031(R119889)

VB1199042119896

11990121199032(R119889)

(52)




(sum

119895isinZ

10038161003816100381610038161003816119886119895

10038161003816100381610038161003816

119902

)

1119902

lt infin (53)






119869 (119905 119886 1198600 1198601) = max (1198861198600

1199051198861198601

)

119870 (119905 119886 1198600 1198601) = min (1003817100381710038171003817119886010038171003817100381710038171198600

+ 11990510038171003817100381710038171198861

10038171003817100381710038171198601

119886 = 1198860 + 1198861)

(54)



minus119895120579119869(2

119895 119886119895 1198600 1198601))119895isinZ isin 119897

119902(Z)


119886[11986001198601]120579119902119869

= inf119886=sum119895isinZ 119886119895

(sum

119895isinZ

2minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

1119902

+ (sum

119895isinZ

2119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

1119902

(55)



minus119895120579119870(2

119895 119886 1198600 1198601))119895isinZ isin 119897

119902(Z)


119886[11986001198601]120579119902119870

= (sum

119895isinZ

2minus119895120579119902

119870(2119895 119886 1198600 1198601)

119902

)

1119902

(56)




119886[11986001198601]120579119902

le 119862 (119902 120579 984858)(sum

119895isinZ

984858minus11989512057911990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198600

)

(1minus120579)119902

times (sum

119895isinZ

984858119895(1minus120579)11990210038171003817100381710038171003817

119886119895

10038171003817100381710038171003817

119902

1198601

)

120579119902

(57)


[[1198600 1198601]12057901199020

[1198600 1198601]12057911199021

]120579119902

= [1198600 1198601](1minus120579)1205790+1205791205791119902 (58)



0 119860

1015840

1and [1198600 1198601]

1015840

120579119902for 0 lt 120579 lt 1


1015840

120579119902=

[1198601015840

0 119860

1015840

1]1205791199021015840



119889119891119896 (120582) = int119909isinR119889 |119891(119909)|ge120582

120596119896 (119909) 119889119909

119891lowast

119896(119904) = inf 120582 119889119891119896 (120582) le 119904

(59)


1003817100381710038171003817119871119901119902

119896(R119889)

=

(int

infin

0

(1199041119901

119891lowast

119896(119904))

119902 119889119904

119904)

1119902

if 1 le 119901 119902 lt infin

sup119904gt0

1199041119901

119891lowast


(60)



119871119901119902

119896(R119889) lt infin



119871119901119902

119896(R

119889) = [119871

1

119896(R

119889) 119871

infin

119896(R

119889)]

120579119902 (61)


[11987111990101199020

119896(R

119889) 119871

11990111199021

119896(R

119889)]

120579119902= [119871

1199010

119896(R

119889) 119871

1199011

119896(R

119889)]

120579119902

= 119871119901119902

119896(R

119889)

(62)


[1198711199011199020

119896(R

119889) 119871

1199011199021

119896(R

119889)]

120579119902= 119871

119901119902

119896(R

119889) (63)


1198711199011199021

119896(R

119889) 997893rarr 119871

1199011199022

119896(R

119889) (64)



119901119902

119896(R119889


119895isinZ 119891119895 where

1003817100381710038171003817100381710038171003817(2

minus119895(119901minus1)11990110038171003817100381710038171003817119891119895

100381710038171003817100381710038171198711119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903+

1003817100381710038171003817100381710038171003817(2

11989511990110038171003817100381710038171003817119891119895

10038171003817100381710038171003817119871infin119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(65)



119896(R119889


119871119901(119901minus1)119902(119902minus1)

119896(R119889

) one has 119891119892 isin 1198711

119896(R119889

) and

1003816100381610038161003816100381610038161003816intR119889

119891 (119909) 119892 (119909) 120596119896 (119909) 119889119909

1003816100381610038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901(119901minus1)119902(119902minus1)

119896(R119889)

(66)




1

119901minus1

119902=

119904

2120574 + 119889 (67)

For 119891 isin 119871119901

119896(R119889

) one has100381710038171003817100381710038171003817(119868 minus 119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(68)



2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889)

(69)


119867119904

119896(R119889



⟨119891 119892⟩1198712119896(R119889)

= ⟨F119863(119891)F119863 (119892)⟩1198712119896(R119889)

= intR119889

(1 +10038171003817100381710038171205851003817100381710038171003817

2)1199042

F119863 (119891) (120585)F119863 (119892) (120585)(1 +10038171003817100381710038171205851003817100381710038171003817

2)minus1199042

times 120596119896 (120585) 119889120585

= ⟨(119868 minus 119896)1199042119891 (119868 minus 119896)

minus1199042119892⟩

1198712119896(R119889)

(70)

Hence100381610038161003816100381610038161003816⟨119891 119892⟩

1198712119896(R119889)

100381610038161003816100381610038161003816le100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

100381710038171003817100381710038171003817(119868 minus 119896)

minus11990421198921003817100381710038171003817100381710038171198712119896(R119889)

(71)


1198712119896(R119889)

100381610038161003816100381610038161003816le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901

119896(R119889)

(72)



10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus1

119871119902

119896(R119889)

(73)

Thus we obtain (69)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(74)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (75)


10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

1198711199010

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119901minus120579

1198711199012

119896(R119889)

(76)


where

1

1199010

=1

120579(1 minus

1

119901)

1

1199012

=1

(119901 minus 120579) 119901 (77)



exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(78)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (79)


ℎ ln(intR119889

|119891(119909)|1ℎ


intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(80)



intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

(119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

)

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

le119902120579

119901 minus 119902ln(

119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(81)



119904

119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(82)



2 If

119891 isin 11987111990111199021

119896(R119889

) 119892 isin 11987111990121199022

119896(R119889


11990131199023

119896(R119889



100381710038171003817100381711987111990131199023

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(83)





10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(84)

holds for

120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(85)


10038171003817100381710038171198911003817100381710038171003817

119902

119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

11987111990131199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus120579

11987111990121199022

119896(R119889)

(86)

where1

1199013

=1

119901minus1

119902+

1

1199021

1

1199012

=1

119901minus1

119902+

1

1199022

(87)

Note that

119891 (119909) = (119868 minus 119896)1199042119891lowast119863119861119896119904 (119909)

(88)


119871(119889+2120574)(119889+2120574minus119904)infin

119896(R119889


1003817100381710038171003817119891100381710038171003817100381711987111990131199021

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

119904211989110038171003817100381710038171003817100381711987111990111199021

119896(R119889)

(89)


for

1

1199013

=1

1199011

minus119904

119889 + 2120574 0 lt 119904 lt

119889 + 2120574

1199011

(90)




2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889) (91)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(92)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (93)


exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(94)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (95)


119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(96)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(97)

holds for120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(98)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

1198711199012

119896(R119889)

(99)





1003817100381710038171003817119871119902(119889+2120574)(119889+2120574minus119902119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(100)


100381710038171003817100381710038171003817(minus119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(101)





10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(102)



119896(R119889

) and119892 isin 119871

11990121199022

119896(R119889

) then1003817100381710038171003817119891119892

1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(103)

where 1119901 = 11199011 + 11199012 and 1119902 = 11199021 + 11199022



1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199011119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (104)


119903infin

119896(R119889

) we have10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

(105)


10038171003817100381710038171198911003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

119904119904119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(106)

for any 119891 isin 119871119901119902

119896(R119889





2

119896(R119889

) defined by

I119896 (119905) V =1

119888119896|119905|120574+1198892

119890minus119894(119889+2120574)(1205874) sgn 119905

119890119894(sdot24119905)

times [F119863 (119890119894(sdot24119905)V)] (

sdot

2119905)

(107)

1198821119903

119896(R119889



119903

119896(119877

119889)


119896(R119889


1199061198821119903

119896(R119889) = sum

|120583|le1

10038171003817100381710038171198791205831199061003817100381710038171003817119871119903119896(R119889)

(108)

1198821119903

119896119866(R119889


119896(R119889



1

119902+119889 + 2120574

2119903le119889 + 2120574

4 (109)



119875 = (22119889 + 4120574

119889 + 2120574 minus 2) (110)





L119891 (119905) = int

infin

0

119896 (119905 120591) 119891 (120591) 119889120591 (111)


10038171003817100381710038171003817119871119902119904(0infin119865)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119903(0infin119864)

(112)


L119891 (119905) = int

119905

0

119896 (119905 120591) 119891 (120591) 119889120591 (113)


1003817100381710038171003817I119896 (119905) 11989110038171003817100381710038171198711199022(0infin119871119903

119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198712119896(R119889)

(114)

10038171003817100381710038171003817100381710038171003817

int

119905

0

I119896 (119905 minus 120591) 119892 (120591) 119889120591


119896(R119889))

le 1198621003817100381710038171003817119892100381710038171003817100381711987111990210158402(0infin119871119903

1015840

119896(R119889))

(115)


1003817100381710038171003817119871119903119896(R119889)

le 119862119905minus(119889+2120574)(12minus1119903)1003817100381710038171003817119892

10038171003817100381710038171198711199031015840

119896(R119889)

(116)


119905minus(119889+2120574)(12minus1119903)

isin 1198712119903(119889+2120574)(119903minus2)infin

for any 119903 isin [22 (119889 + 2120574)

119889 + 2120574 minus 2]

(117)



119894120597119905119906 (119905 119909) + 119896119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060

(118)


119906119871119902119886(R119871119903119896(R119889)) + 119906119862(R1198712

119896(R119889))

le 119862(10038171003817100381710038171199060

10038171003817100381710038171198712119896(R119889)

+1003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(R1198711199031015840

1

119896(R119889))⋂1198712(R119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

)

(119)


119906 (119905 119909) = I119896 (119905) 1199060 (119909) + int

119905

0

I119896 (119905 minus 120591) 119891 (120591 119909) 119889120591

(119905 119909) isin R timesR119889

(120)

Let 119896(119905 120591) = I119896(119905 minus 120591) 119864 = 1198711199031015840

1

119896(R119889

) or 119871(2119889+4120574)(119889+2120574+2)2119896

(R119889)

119865 = 119871119903

119896(R119889

) and L119891(119905) = intinfin

0119896(119905 120591)119891(120591)119889120591 Then since

1199021015840


that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817119871119902119904(0infin119871119903119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(0infin119871

1199031015840

1

119896(R119889))cap1198712(0infin119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

(121)


sub 1198711199022 for all 119904 ge 2

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817

2

119871119902119904(0infin119871119903119896(R119889))

le 119862intint

infin

0

⟨I119896 (minus120591) 119891 (120591) I119896 (minus119910) 119891 (119910)⟩ 119889120591119889119910

(122)


1198712(0infin119871(2119889+4120574)(119889+2120574+2)2

119896(R119889))


11990210158401199041015840




119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)


119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889



119896119866(R119889


119902119904(0infin119882

1119903

119896119866(R119889


119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)


119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)



1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast


11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)


11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)



119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)


119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)



1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)



10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)


sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)


1)minus1199051198861)


1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)


119902119902(R119889


(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)




10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119871119901119902

119896(R

119889) = [119871

1

119896(R

119889) 119871

infin

119896(R

119889)]

120579119902 (61)


[11987111990101199020

119896(R

119889) 119871

11990111199021

119896(R

119889)]

120579119902= [119871

1199010

119896(R

119889) 119871

1199011

119896(R

119889)]

120579119902

= 119871119901119902

119896(R

119889)

(62)


[1198711199011199020

119896(R

119889) 119871

1199011199021

119896(R

119889)]

120579119902= 119871

119901119902

119896(R

119889) (63)


1198711199011199021

119896(R

119889) 997893rarr 119871

1199011199022

119896(R

119889) (64)



119901119902

119896(R119889


119895isinZ 119891119895 where

1003817100381710038171003817100381710038171003817(2

minus119895(119901minus1)11990110038171003817100381710038171003817119891119895

100381710038171003817100381710038171198711119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903+

1003817100381710038171003817100381710038171003817(2

11989511990110038171003817100381710038171003817119891119895

10038171003817100381710038171003817119871infin119896(R119889)

)

1003817100381710038171003817100381710038171003817119897119903

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(65)



119896(R119889


119871119901(119901minus1)119902(119902minus1)

119896(R119889

) one has 119891119892 isin 1198711

119896(R119889

) and

1003816100381610038161003816100381610038161003816intR119889

119891 (119909) 119892 (119909) 120596119896 (119909) 119889119909

1003816100381610038161003816100381610038161003816le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901(119901minus1)119902(119902minus1)

119896(R119889)

(66)




1

119901minus1

119902=

119904

2120574 + 119889 (67)

For 119891 isin 119871119901

119896(R119889

) one has100381710038171003817100381710038171003817(119868 minus 119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(68)



2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889)

(69)


119867119904

119896(R119889



⟨119891 119892⟩1198712119896(R119889)

= ⟨F119863(119891)F119863 (119892)⟩1198712119896(R119889)

= intR119889

(1 +10038171003817100381710038171205851003817100381710038171003817

2)1199042

F119863 (119891) (120585)F119863 (119892) (120585)(1 +10038171003817100381710038171205851003817100381710038171003817

2)minus1199042

times 120596119896 (120585) 119889120585

= ⟨(119868 minus 119896)1199042119891 (119868 minus 119896)

minus1199042119892⟩

1198712119896(R119889)

(70)

Hence100381610038161003816100381610038161003816⟨119891 119892⟩

1198712119896(R119889)

100381610038161003816100381610038161003816le100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

100381710038171003817100381710038171003817(119868 minus 119896)

minus11990421198921003817100381710038171003817100381710038171198712119896(R119889)

(71)


1198712119896(R119889)

100381610038161003816100381610038161003816le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119901

119896(R119889)

(72)



10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus1

119871119902

119896(R119889)

(73)

Thus we obtain (69)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(74)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (75)


10038171003817100381710038171198911003817100381710038171003817

119901

119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

1198711199010

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119901minus120579

1198711199012

119896(R119889)

(76)


where

1

1199010

=1

120579(1 minus

1

119901)

1

1199012

=1

(119901 minus 120579) 119901 (77)



exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(78)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (79)


ℎ ln(intR119889

|119891(119909)|1ℎ


intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(80)



intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

(119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

)

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

le119902120579

119901 minus 119902ln(

119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(81)



119904

119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(82)



2 If

119891 isin 11987111990111199021

119896(R119889

) 119892 isin 11987111990121199022

119896(R119889


11990131199023

119896(R119889



100381710038171003817100381711987111990131199023

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(83)





10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(84)

holds for

120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(85)


10038171003817100381710038171198911003817100381710038171003817

119902

119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

11987111990131199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus120579

11987111990121199022

119896(R119889)

(86)

where1

1199013

=1

119901minus1

119902+

1

1199021

1

1199012

=1

119901minus1

119902+

1

1199022

(87)

Note that

119891 (119909) = (119868 minus 119896)1199042119891lowast119863119861119896119904 (119909)

(88)


119871(119889+2120574)(119889+2120574minus119904)infin

119896(R119889


1003817100381710038171003817119891100381710038171003817100381711987111990131199021

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

119904211989110038171003817100381710038171003817100381711987111990111199021

119896(R119889)

(89)


for

1

1199013

=1

1199011

minus119904

119889 + 2120574 0 lt 119904 lt

119889 + 2120574

1199011

(90)




2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889) (91)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(92)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (93)


exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(94)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (95)


119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(96)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(97)

holds for120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(98)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

1198711199012

119896(R119889)

(99)





1003817100381710038171003817119871119902(119889+2120574)(119889+2120574minus119902119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(100)


100381710038171003817100381710038171003817(minus119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(101)





10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(102)



119896(R119889

) and119892 isin 119871

11990121199022

119896(R119889

) then1003817100381710038171003817119891119892

1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(103)

where 1119901 = 11199011 + 11199012 and 1119902 = 11199021 + 11199022



1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199011119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (104)


119903infin

119896(R119889

) we have10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

(105)


10038171003817100381710038171198911003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

119904119904119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(106)

for any 119891 isin 119871119901119902

119896(R119889





2

119896(R119889

) defined by

I119896 (119905) V =1

119888119896|119905|120574+1198892

119890minus119894(119889+2120574)(1205874) sgn 119905

119890119894(sdot24119905)

times [F119863 (119890119894(sdot24119905)V)] (

sdot

2119905)

(107)

1198821119903

119896(R119889



119903

119896(119877

119889)


119896(R119889


1199061198821119903

119896(R119889) = sum

|120583|le1

10038171003817100381710038171198791205831199061003817100381710038171003817119871119903119896(R119889)

(108)

1198821119903

119896119866(R119889


119896(R119889



1

119902+119889 + 2120574

2119903le119889 + 2120574

4 (109)



119875 = (22119889 + 4120574

119889 + 2120574 minus 2) (110)





L119891 (119905) = int

infin

0

119896 (119905 120591) 119891 (120591) 119889120591 (111)


10038171003817100381710038171003817119871119902119904(0infin119865)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119903(0infin119864)

(112)


L119891 (119905) = int

119905

0

119896 (119905 120591) 119891 (120591) 119889120591 (113)


1003817100381710038171003817I119896 (119905) 11989110038171003817100381710038171198711199022(0infin119871119903

119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198712119896(R119889)

(114)

10038171003817100381710038171003817100381710038171003817

int

119905

0

I119896 (119905 minus 120591) 119892 (120591) 119889120591


119896(R119889))

le 1198621003817100381710038171003817119892100381710038171003817100381711987111990210158402(0infin119871119903

1015840

119896(R119889))

(115)


1003817100381710038171003817119871119903119896(R119889)

le 119862119905minus(119889+2120574)(12minus1119903)1003817100381710038171003817119892

10038171003817100381710038171198711199031015840

119896(R119889)

(116)


119905minus(119889+2120574)(12minus1119903)

isin 1198712119903(119889+2120574)(119903minus2)infin

for any 119903 isin [22 (119889 + 2120574)

119889 + 2120574 minus 2]

(117)



119894120597119905119906 (119905 119909) + 119896119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060

(118)


119906119871119902119886(R119871119903119896(R119889)) + 119906119862(R1198712

119896(R119889))

le 119862(10038171003817100381710038171199060

10038171003817100381710038171198712119896(R119889)

+1003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(R1198711199031015840

1

119896(R119889))⋂1198712(R119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

)

(119)


119906 (119905 119909) = I119896 (119905) 1199060 (119909) + int

119905

0

I119896 (119905 minus 120591) 119891 (120591 119909) 119889120591

(119905 119909) isin R timesR119889

(120)

Let 119896(119905 120591) = I119896(119905 minus 120591) 119864 = 1198711199031015840

1

119896(R119889

) or 119871(2119889+4120574)(119889+2120574+2)2119896

(R119889)

119865 = 119871119903

119896(R119889

) and L119891(119905) = intinfin

0119896(119905 120591)119891(120591)119889120591 Then since

1199021015840


that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817119871119902119904(0infin119871119903119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(0infin119871

1199031015840

1

119896(R119889))cap1198712(0infin119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

(121)


sub 1198711199022 for all 119904 ge 2

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817

2

119871119902119904(0infin119871119903119896(R119889))

le 119862intint

infin

0

⟨I119896 (minus120591) 119891 (120591) I119896 (minus119910) 119891 (119910)⟩ 119889120591119889119910

(122)


1198712(0infin119871(2119889+4120574)(119889+2120574+2)2

119896(R119889))


11990210158401199041015840




119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)


119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889



119896119866(R119889


119902119904(0infin119882

1119903

119896119866(R119889


119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)


119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)



1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast


11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)


11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)



119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)


119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)



1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)



10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)


sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)


1)minus1199051198861)


1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)


119902119902(R119889


(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)




10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

1

1199010

=1

120579(1 minus

1

119901)

1

1199012

=1

(119901 minus 120579) 119901 (77)



exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(78)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (79)


ℎ ln(intR119889

|119891(119909)|1ℎ


intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(80)



intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909

le119901

119901 minus 119902ln(

(119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

)

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

le119902120579

119901 minus 119902ln(

119862100381710038171003817100381710038171003817(119868 minus 119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)

(81)



119904

119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(82)



2 If

119891 isin 11987111990111199021

119896(R119889

) 119892 isin 11987111990121199022

119896(R119889


11990131199023

119896(R119889



100381710038171003817100381711987111990131199023

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(83)





10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(84)

holds for

120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(85)


10038171003817100381710038171198911003817100381710038171003817

119902

119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

11987111990131199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

119902minus120579

11987111990121199022

119896(R119889)

(86)

where1

1199013

=1

119901minus1

119902+

1

1199021

1

1199012

=1

119901minus1

119902+

1

1199022

(87)

Note that

119891 (119909) = (119868 minus 119896)1199042119891lowast119863119861119896119904 (119909)

(88)


119871(119889+2120574)(119889+2120574minus119904)infin

119896(R119889


1003817100381710038171003817119891100381710038171003817100381711987111990131199021

119896(R119889)

le 119862100381710038171003817100381710038171003817(119868 minus 119896)

119904211989110038171003817100381710038171003817100381711987111990111199021

119896(R119889)

(89)


for

1

1199013

=1

1199011

minus119904

119889 + 2120574 0 lt 119904 lt

119889 + 2120574

1199011

(90)




2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889) (91)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(92)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (93)


exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(94)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (95)


119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(96)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(97)

holds for120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(98)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

1198711199012

119896(R119889)

(99)





1003817100381710038171003817119871119902(119889+2120574)(119889+2120574minus119902119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(100)


100381710038171003817100381710038171003817(minus119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(101)





10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(102)



119896(R119889

) and119892 isin 119871

11990121199022

119896(R119889

) then1003817100381710038171003817119891119892

1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(103)

where 1119901 = 11199011 + 11199012 and 1119902 = 11199021 + 11199022



1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199011119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (104)


119903infin

119896(R119889

) we have10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

(105)


10038171003817100381710038171198911003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

119904119904119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(106)

for any 119891 isin 119871119901119902

119896(R119889





2

119896(R119889

) defined by

I119896 (119905) V =1

119888119896|119905|120574+1198892

119890minus119894(119889+2120574)(1205874) sgn 119905

119890119894(sdot24119905)

times [F119863 (119890119894(sdot24119905)V)] (

sdot

2119905)

(107)

1198821119903

119896(R119889



119903

119896(119877

119889)


119896(R119889


1199061198821119903

119896(R119889) = sum

|120583|le1

10038171003817100381710038171198791205831199061003817100381710038171003817119871119903119896(R119889)

(108)

1198821119903

119896119866(R119889


119896(R119889



1

119902+119889 + 2120574

2119903le119889 + 2120574

4 (109)



119875 = (22119889 + 4120574

119889 + 2120574 minus 2) (110)





L119891 (119905) = int

infin

0

119896 (119905 120591) 119891 (120591) 119889120591 (111)


10038171003817100381710038171003817119871119902119904(0infin119865)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119903(0infin119864)

(112)


L119891 (119905) = int

119905

0

119896 (119905 120591) 119891 (120591) 119889120591 (113)


1003817100381710038171003817I119896 (119905) 11989110038171003817100381710038171198711199022(0infin119871119903

119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198712119896(R119889)

(114)

10038171003817100381710038171003817100381710038171003817

int

119905

0

I119896 (119905 minus 120591) 119892 (120591) 119889120591


119896(R119889))

le 1198621003817100381710038171003817119892100381710038171003817100381711987111990210158402(0infin119871119903

1015840

119896(R119889))

(115)


1003817100381710038171003817119871119903119896(R119889)

le 119862119905minus(119889+2120574)(12minus1119903)1003817100381710038171003817119892

10038171003817100381710038171198711199031015840

119896(R119889)

(116)


119905minus(119889+2120574)(12minus1119903)

isin 1198712119903(119889+2120574)(119903minus2)infin

for any 119903 isin [22 (119889 + 2120574)

119889 + 2120574 minus 2]

(117)



119894120597119905119906 (119905 119909) + 119896119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060

(118)


119906119871119902119886(R119871119903119896(R119889)) + 119906119862(R1198712

119896(R119889))

le 119862(10038171003817100381710038171199060

10038171003817100381710038171198712119896(R119889)

+1003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(R1198711199031015840

1

119896(R119889))⋂1198712(R119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

)

(119)


119906 (119905 119909) = I119896 (119905) 1199060 (119909) + int

119905

0

I119896 (119905 minus 120591) 119891 (120591 119909) 119889120591

(119905 119909) isin R timesR119889

(120)

Let 119896(119905 120591) = I119896(119905 minus 120591) 119864 = 1198711199031015840

1

119896(R119889

) or 119871(2119889+4120574)(119889+2120574+2)2119896

(R119889)

119865 = 119871119903

119896(R119889

) and L119891(119905) = intinfin

0119896(119905 120591)119891(120591)119889120591 Then since

1199021015840


that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817119871119902119904(0infin119871119903119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(0infin119871

1199031015840

1

119896(R119889))cap1198712(0infin119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

(121)


sub 1198711199022 for all 119904 ge 2

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817

2

119871119902119904(0infin119871119903119896(R119889))

le 119862intint

infin

0

⟨I119896 (minus120591) 119891 (120591) I119896 (minus119910) 119891 (119910)⟩ 119889120591119889119910

(122)


1198712(0infin119871(2119889+4120574)(119889+2120574+2)2

119896(R119889))


11990210158401199041015840




119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)


119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889



119896119866(R119889


119902119904(0infin119882

1119903

119896119866(R119889


119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)


119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)



1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast


11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)


11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)



119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)


119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)



1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)



10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)


sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)


1)minus1199051198861)


1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)


119902119902(R119889


(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)




10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










for

1

1199013

=1

1199011

minus119904

119889 + 2120574 0 lt 119904 lt

119889 + 2120574

1199011

(90)




2120574 minus 2119904) Then

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198712119896(R119889)

119891 isin 119867119904

119896(R

119889) (91)


10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119901

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119901minus120579)119901

1198711199012

119896(R119889)

(92)

with

120579(1

1199011

minus119904

119889 + 2120574) +

119901 minus 120579

1199012

= 1 (93)


exp((1

119902+

119904

119889 + 2120574minus

1

1199011

)

timesintR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

ln(1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119902

10038171003817100381710038171198911003817100381710038171003817

119902

119871119902

119896(R119889)

)120596119896 (119909) 119889119909)

le 119862

100381710038171003817100381710038171003817(minus119896)

11990421198911003817100381710038171003817100381710038171198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817119871119902

119896(R119889)

(94)

holds for

1

119902+

119904

119889 + 2120574minus

1

1199011

gt 0 (95)


119902119896(R119889

) such that 119891119871119902

119896(R119889) = 1 one has

exp( 119904

119889 + 2120574intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902 ln (1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902) 120596119896 (119909) 119889119909)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(96)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

11987111990111199021

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

11987111990121199022

119896(R119889)

(97)

holds for120579

1199021

+119902 minus 120579

1199022

= 1

120579 (1

1199011

minus119904

119889 + 2120574) +

119902 minus 120579

1199012

=119902

119901

(98)




10038171003817100381710038171198911003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817

120579119902

1198711199011

119896(R119889)

10038171003817100381710038171198911003817100381710038171003817

(119902minus120579)119902

1198711199012

119896(R119889)

(99)





1003817100381710038171003817119871119902(119889+2120574)(119889+2120574minus119902119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

(100)


100381710038171003817100381710038171003817(minus119896)

minus1199042119891100381710038171003817100381710038171003817119871119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901

119896(R119889)

(101)





10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

1199042119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(102)



119896(R119889

) and119892 isin 119871

11990121199022

119896(R119889

) then1003817100381710038171003817119891119892

1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

100381710038171003817100381711987111990111199021

119896(R119889)

1003817100381710038171003817119892100381710038171003817100381711987111990121199022

119896(R119889)

(103)

where 1119901 = 11199011 + 11199012 and 1119902 = 11199021 + 11199022



1003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199011119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (104)


119903infin

119896(R119889

) we have10038171003817100381710038171003817100381710038171003817

119891 (119909)

119909119904

10038171003817100381710038171003817100381710038171003817119871119901119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

(105)


10038171003817100381710038171198911003817100381710038171003817119871((119889+2120574)119901)(119889+2120574minus119901119904)119902

119896(R119889)

le 119862100381710038171003817100381710038171003817(minus119896)

119904119904119891100381710038171003817100381710038171003817119871119901119902

119896(R119889)

(106)

for any 119891 isin 119871119901119902

119896(R119889





2

119896(R119889

) defined by

I119896 (119905) V =1

119888119896|119905|120574+1198892

119890minus119894(119889+2120574)(1205874) sgn 119905

119890119894(sdot24119905)

times [F119863 (119890119894(sdot24119905)V)] (

sdot

2119905)

(107)

1198821119903

119896(R119889



119903

119896(119877

119889)


119896(R119889


1199061198821119903

119896(R119889) = sum

|120583|le1

10038171003817100381710038171198791205831199061003817100381710038171003817119871119903119896(R119889)

(108)

1198821119903

119896119866(R119889


119896(R119889



1

119902+119889 + 2120574

2119903le119889 + 2120574

4 (109)



119875 = (22119889 + 4120574

119889 + 2120574 minus 2) (110)





L119891 (119905) = int

infin

0

119896 (119905 120591) 119891 (120591) 119889120591 (111)


10038171003817100381710038171003817119871119902119904(0infin119865)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119903(0infin119864)

(112)


L119891 (119905) = int

119905

0

119896 (119905 120591) 119891 (120591) 119889120591 (113)


1003817100381710038171003817I119896 (119905) 11989110038171003817100381710038171198711199022(0infin119871119903

119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198712119896(R119889)

(114)

10038171003817100381710038171003817100381710038171003817

int

119905

0

I119896 (119905 minus 120591) 119892 (120591) 119889120591


119896(R119889))

le 1198621003817100381710038171003817119892100381710038171003817100381711987111990210158402(0infin119871119903

1015840

119896(R119889))

(115)


1003817100381710038171003817119871119903119896(R119889)

le 119862119905minus(119889+2120574)(12minus1119903)1003817100381710038171003817119892

10038171003817100381710038171198711199031015840

119896(R119889)

(116)


119905minus(119889+2120574)(12minus1119903)

isin 1198712119903(119889+2120574)(119903minus2)infin

for any 119903 isin [22 (119889 + 2120574)

119889 + 2120574 minus 2]

(117)



119894120597119905119906 (119905 119909) + 119896119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060

(118)


119906119871119902119886(R119871119903119896(R119889)) + 119906119862(R1198712

119896(R119889))

le 119862(10038171003817100381710038171199060

10038171003817100381710038171198712119896(R119889)

+1003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(R1198711199031015840

1

119896(R119889))⋂1198712(R119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

)

(119)


119906 (119905 119909) = I119896 (119905) 1199060 (119909) + int

119905

0

I119896 (119905 minus 120591) 119891 (120591 119909) 119889120591

(119905 119909) isin R timesR119889

(120)

Let 119896(119905 120591) = I119896(119905 minus 120591) 119864 = 1198711199031015840

1

119896(R119889

) or 119871(2119889+4120574)(119889+2120574+2)2119896

(R119889)

119865 = 119871119903

119896(R119889

) and L119891(119905) = intinfin

0119896(119905 120591)119891(120591)119889120591 Then since

1199021015840


that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817119871119902119904(0infin119871119903119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(0infin119871

1199031015840

1

119896(R119889))cap1198712(0infin119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

(121)


sub 1198711199022 for all 119904 ge 2

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817

2

119871119902119904(0infin119871119903119896(R119889))

le 119862intint

infin

0

⟨I119896 (minus120591) 119891 (120591) I119896 (minus119910) 119891 (119910)⟩ 119889120591119889119910

(122)


1198712(0infin119871(2119889+4120574)(119889+2120574+2)2

119896(R119889))


11990210158401199041015840




119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)


119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889



119896119866(R119889


119902119904(0infin119882

1119903

119896119866(R119889


119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)


119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)



1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast


11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)


11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)



119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)


119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)



1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)



10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)


sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)


1)minus1199051198861)


1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)


119902119902(R119889


(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)




10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












2

119896(R119889

) defined by

I119896 (119905) V =1

119888119896|119905|120574+1198892

119890minus119894(119889+2120574)(1205874) sgn 119905

119890119894(sdot24119905)

times [F119863 (119890119894(sdot24119905)V)] (

sdot

2119905)

(107)

1198821119903

119896(R119889



119903

119896(119877

119889)


119896(R119889


1199061198821119903

119896(R119889) = sum

|120583|le1

10038171003817100381710038171198791205831199061003817100381710038171003817119871119903119896(R119889)

(108)

1198821119903

119896119866(R119889


119896(R119889



1

119902+119889 + 2120574

2119903le119889 + 2120574

4 (109)



119875 = (22119889 + 4120574

119889 + 2120574 minus 2) (110)





L119891 (119905) = int

infin

0

119896 (119905 120591) 119891 (120591) 119889120591 (111)


10038171003817100381710038171003817119871119902119904(0infin119865)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901119903(0infin119864)

(112)


L119891 (119905) = int

119905

0

119896 (119905 120591) 119891 (120591) 119889120591 (113)


1003817100381710038171003817I119896 (119905) 11989110038171003817100381710038171198711199022(0infin119871119903

119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198712119896(R119889)

(114)

10038171003817100381710038171003817100381710038171003817

int

119905

0

I119896 (119905 minus 120591) 119892 (120591) 119889120591


119896(R119889))

le 1198621003817100381710038171003817119892100381710038171003817100381711987111990210158402(0infin119871119903

1015840

119896(R119889))

(115)


1003817100381710038171003817119871119903119896(R119889)

le 119862119905minus(119889+2120574)(12minus1119903)1003817100381710038171003817119892

10038171003817100381710038171198711199031015840

119896(R119889)

(116)


119905minus(119889+2120574)(12minus1119903)

isin 1198712119903(119889+2120574)(119903minus2)infin

for any 119903 isin [22 (119889 + 2120574)

119889 + 2120574 minus 2]

(117)



119894120597119905119906 (119905 119909) + 119896119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060

(118)


119906119871119902119886(R119871119903119896(R119889)) + 119906119862(R1198712

119896(R119889))

le 119862(10038171003817100381710038171199060

10038171003817100381710038171198712119896(R119889)

+1003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(R1198711199031015840

1

119896(R119889))⋂1198712(R119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

)

(119)


119906 (119905 119909) = I119896 (119905) 1199060 (119909) + int

119905

0

I119896 (119905 minus 120591) 119891 (120591 119909) 119889120591

(119905 119909) isin R timesR119889

(120)

Let 119896(119905 120591) = I119896(119905 minus 120591) 119864 = 1198711199031015840

1

119896(R119889

) or 119871(2119889+4120574)(119889+2120574+2)2119896

(R119889)

119865 = 119871119903

119896(R119889

) and L119891(119905) = intinfin

0119896(119905 120591)119891(120591)119889120591 Then since

1199021015840


that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817119871119902119904(0infin119871119903119896(R119889))

le 1198621003817100381710038171003817119891

10038171003817100381710038171198711199021015840

12(0infin119871

1199031015840

1

119896(R119889))cap1198712(0infin119871

(2119889+4120574)(119889+2120574+2)2

119896(R119889))

(121)


sub 1198711199022 for all 119904 ge 2

that10038171003817100381710038171003817100381710038171003817

int

infin

0

119896(119905 120591)119891(120591)119889120591

10038171003817100381710038171003817100381710038171003817

2

119871119902119904(0infin119871119903119896(R119889))

le 119862intint

infin

0

⟨I119896 (minus120591) 119891 (120591) I119896 (minus119910) 119891 (119910)⟩ 119889120591119889119910

(122)


1198712(0infin119871(2119889+4120574)(119889+2120574+2)2

119896(R119889))


11990210158401199041015840




119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)


119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889



119896119866(R119889


119902119904(0infin119882

1119903

119896119866(R119889


119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)


119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)



1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast


11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)


11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)



119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)


119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)



1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)



10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)


sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)


1)minus1199051198861)


1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)


119902119902(R119889


(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)




10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894120597119905119906 (119905 119909) + 119896119906 (119905 119909)

= minus|119906 (119905 119909)|4(119889+2120574minus2)

119906 (119905 119909) (119905 119909) isin R timesR119889

119906|119905=0 = 1199060 isin 1198671

119896(R

119889) in R

119889

(123)


119871119902119904(0infin119882

1119903

119896119866(R119889

)) cap 1198712(0infin119882

1(2119889+4120574)(119889+2120574minus2)

119896119866(R119889

)) cap

119862([0infin)1198671

119896119866(R119889



119896119866(R119889


119902119904(0infin119882

1119903

119896119866(R119889


119906 (119905 119909) = I119896 (119905) 1199060 (119909)

+ int

119905

0

I119896 (119905 minus 120591) (|119906 (120591 119909)|4(119889+2120574minus2)

119906 (120591 119909)) 119889120591

(124)


119879119906119871119902119904(R119871119903119896(R119889))

le 119862 (10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171003817119879 (|119906|

4(119889+2120574minus2)119906)100381710038171003817100381710038171198711199021015840

12(R11987111990310158401

119896(R119889))

)

(125)



1

1199021

=4

(119889 + 2120574 minus 2) 1199020

+1

1199020

1

1199031

=4

(119889 + 2120574 minus 2) 1199031

+1

1199030

1

1199041

=4

(119889 + 2120574 minus 2) 1199040

+1

1199040

(126)

where 119903lowast


11987911990611987111990201199040 (R119871

1199030

119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

(127)


11987911990611987111990201199040 (R119871

1199030

119896(R119889)) le 119862

1003817100381710038171003817119879119906010038171003817100381710038171198712119896(R119889)

(128)



119879119906119871119902119904(R119871119903119896(R119889))

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+ 119879119906(119889+2120574+2)(119889+2120574minus2)

11987111990201199040(R1198711199030

119896(R119889))

)

le 119862(10038171003817100381710038171198791199060

10038171003817100381710038171198712119896(R119889)

+10038171003817100381710038171198791199060

1003817100381710038171003817

(119889+2120574+2)(119889+2120574minus2)

1198712119896(R119889)

)

(129)


119896(R119889)

119906119871119902119904(R119871119903119896(R119889)) le 119862

1003817100381710038171003817119906010038171003817100381710038171198712119896(R119889)

(130)



1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

) then 119891 isin 119871119901119903

119896(R119889

) and one has1003817100381710038171003817119891

1003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(131)



10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus119886119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

119886119894

1198711199022

119896(R119889)

(132)


sum

119895isinZ

984858minus119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199011

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198861)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198861

Bminus1199051198961199022119903(R119889)

sum

119895isinZ

984858119895119903210038171003817100381710038171003817

Δ 11989511989110038171003817100381710038171003817

119903

1198711199012

119896(R119889)

le1003817100381710038171003817119891

1003817100381710038171003817

(1minus1198862)119903

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

1199031198862

Bminus1199051198961199022119903(R119889)

(133)


1)minus1199051198861)


1199021119903(R119889

) cap Bminus119905119896

1199022119903(R119889

)then 119891 isin [119871

1199011

119896(R119889

) 1198711199012

119896(R119889

)]12119903 = 119871119901119903

119896(R119889


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B1199041198961199021119903(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus1199051198961199022119903(R119889)

(134)


119902119902(R119889


(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817

120579

B119904119896119902119902(R119889)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B119904minus(119889+2120574)119902119896

infin119902 (R119889)

(135)

where 120579 = 1 minus 119902119904(119889 + 2120574)




10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












10038171003817100381710038171198911198921003817100381710038171003817119871119902119902

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

10038171003817100381710038171198921003817100381710038171003817119871119903infin119896

(R119889) (136)


119903infin

119896(R119889

) we have

(intR119889

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119902

119909119904119902

120596119896 (119909) 119889119909)

1119902

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901119902

119896(R119889)

(137)



2119896(R119889

)one has

intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 le 1198621199062

H1199042119896(R119889)

(138)




1infin(R119889

)


119868119904119896 (119906) = intR119889

|119906 (119909)|2

1199092119904

120596119896 (119909) 119889119909 = ⟨sdotminus2119904

1199062⟩ (139)



119868119904119896 (119906) = sum

|119899minus119898|le2

⟨Δ 119899 (sdotminus2119904

) Δ119898 (1199062)⟩

le 119862 sum

|119899minus119898|le2

⟨2119899((119889+2120574)2minus2119904)


) 2minus119898((119889+2120574)2minus2119904)

Δ119898 (1199062)⟩

(140)


2infin(R119889

)Theorem 17 yields

10038171003817100381710038171003817119906210038171003817100381710038171003817B2119904minus(119889+2120574)2119896

21(R119889)

le 1198621199062

H1199042119896(R119889)

(141)

Thus

119868119904119896 (119906) le 1198621199062

H1199042119896(R119889)

(142)




(i) For every 119891 isin B119904119896

11990211199031

(R119889) cap Bminus119905119896

11990221199032

(R119889) and if 119903 gt 1199030

one has 119891 isin 119871119901119903

119896(R119889

) and

10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(143)


(a) 119903 = 1199031 = 1199032(b) 1199031 = 1199021 and 1199032 = 1199022(c) 1 lt 119901 le 2 and 1199030 = 119901



1

1199022

lt1

119901minus 120576 (

1

1199021

minus1

1199022

) =1

1199012

lt1

119901+ 120576(

1

1199021

minus1

1199022

)

=1

1199011

lt1

1199021

(144)


119895119904Δ 119895119891119871

1199021

119896(R119889)

and119873119895 = 2minus119895119905

Δ 1198951198911198711199022

119896(R119889)

and for 1205760 = 1 and 1205761 = minus1

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901119894

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579+120576120576119894

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579minus120576120576119894

1198711199022

119896(R119889)

= 1198721minus120579+120576120576

119894

119895119873

120579minus120576120576119894

1198952minus119895120576120576119894(119904+119905)

(145)


119895119873

120579minus120576120576119894

119895)119895isinZ isin 119897

984858119894


119901984858

119896(R119889

) = [1198711199011

119896(R119889

) 1198711199012

119896(R119889


(ii) Case 119903 = 1199030


(43) since we have1003817100381710038171003817119891

1003817100381710038171003817B119904119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817F119904119896119902119894infin(R119889)

10038171003817100381710038171198911003817100381710038171003817Bminus119905119896119902119894119902119894(R119889)

=1003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894119902119894(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817Fminus119905119896119902119894infin(R119889)

(146)

we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(147)


(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(c) Case 1 lt 119901 le 2 and 1199030 = 119901

We just write

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

119896(R119889)

le10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1minus120579

1198711199021

119896(R119889)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

120579

1198711199022

119896(R119889)

= (211989511990410038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199021

119896(R119889)

)

1minus120579

(2minus11989511990510038171003817100381710038171003817

Δ 119895119891100381710038171003817100381710038171198711199022

119896(R119889)

)

120579

(148)


10038171003817100381710038171198911003817100381710038171003817B0119896119901119901

(R119889)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(149)


119901119901(R119889

) sub 119871119901

119896(R119889

) = 119871119901119901

119896(R119889

)




10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

(150)


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(151)


11990211199031


120582119895 = 2119904119895Δ 119895119891119871

1199021

119896(R119889)



1198971199031 = [119897

1199021 119897

1199022]

119886119903 (152)


119895isinZ 119885119895 such that

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199022

119899)

111990221003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(153)


11990221199022

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (154)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

100381710038171003817100381711989111989910038171003817100381710038171198711199011199022

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199022(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199022(R119889)

le 1198621198711minus120579

119899119872

120579


(155)



119896(R119889

) 1198711199011199022

119896(R119889



119896(R119889

) 1198711199011199022

119896(R119889

)]119886119903 =

119871119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199022


10038171003817100381710038171198911003817100381710038171003817119871119901119903

119896(R119889)

le 1198621003817100381710038171003817119891

1003817100381710038171003817

1minus120579

B11990411989611990211199031(R119889)

10038171003817100381710038171198911003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

(156)


1198971199031 = [119897

1199021 119897

1199032]

1198861199031

(157)


that100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2minus119886119895

( sum

119899isin119885119895

1205821199021

119899)

111990211003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

+

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

2(1minus119886)119895

( sum

119899isin119885119895

1205821199032

119899)

111990321003817100381710038171003817100381710038171003817100381710038171003817100381710038171198971199031

le 11986210038171003817100381710038171003817120582119895

100381710038171003817100381710038171198971199031

(158)


1199021

119896(R119889)


11990211199031


Bminus119905119896

11990221199032

(R119889) we have

((sum

119895isin119885119899

2minus119895119902211990510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817

1199022

1198711199022

119896(R119889)

)

11199022

)

119899isinZ

isin 1198971199022 (159)


2minus1198951199022119905Δ 119895119891

1199022

1198711199022

119896(R119889)

)11199022 119873119899 =

2minus119886119899

(sum119895isin119885119899

1205821199021

119895)11199021

119871119899 = 2(1minus119886)119899

(sum119895isin119885119899

1205821199022

119895)11199022 and 119891119899 =

sum119895isin119885119899


10038171003817100381710038171198911198991003817100381710038171003817119871119901119887

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199021(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579

1198992119899119886(1minus120579)

(160)


where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where 1119887 = (1 minus 120579)1199021 + 1205791199032 and

100381710038171003817100381711989111989910038171003817100381710038171198711199011199032

119896(R119889)

le 1198621003817100381710038171003817119891119899

1003817100381710038171003817

1minus120579

B11990411989611990211199032(R119889)

10038171003817100381710038171198911198991003817100381710038171003817

120579

Bminus11990511989611990221199032(R119889)

le 1198621198731minus120579

119899119872

120579


(161)


119896(R119889

) 1198711199011199032

119896(R119889


119901119903

119896(R119889

) with 1119903 = (1 minus 120579)1199031 + 1205791199032


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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