Research ArticleWavelets Convergence and Unconditional Bases for119871119901(R119899) and1198671

(R119899)

Devendra Kumar

Department of Mathematics Faculty of Science Al-Baha University PO Box 1988 Al-Aqiq Al-Baha 65431 Saudi Arabia

Correspondence should be addressed to Devendra Kumar d kumar001rediffmailcom

Received 25 October 2013 Accepted 16 December 2013 Published 6 February 2014

Academic Editors Z Wang and J Zhang

Copyright copy 2014 Devendra Kumar 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 prove that reasonable nice wavelets form unconditional bases in function space other than 1198712(R119899 119883) Moreover characteriza-

tion of convergence of wavelets series in 119871119901(R119899 119883) space and Hardy space1198671(R119899 119883) has been obtained Here119883 is a Banach space

with boundedness of Riesz transform

1 Introduction

Let 119883 be a separable Banach space and (119890119899)119899isinN a sequence of

vectors in119883 We say that (119890119899)119899isinN is an unconditional basis for

119883 if for every 119909 isin 119883 there exists a unique sequence of scalars(119888119899)119899isinN such that

119909 = sum

119899isinN

119888119899119890119899 (1)

where the series converges in the norm and as the net ofsums over finite subsets of N In other words we require thatsuminfin

119899=1119888120590(119899)

119890120590(119899)

converges for every permutation 120590 N rarr NAgain a consequence of the uniform boundedness principleis that the coefficient functionals 119891

119899 119909 997891rarr 119888

119899are continuous

If119883 is reflexive then 119891119899is an unconditional basis for the dual

space119883lowast

Wavelet systems provide explicit unconditional bases formany function spaces For instance any orthonormal wavelet120595 such that 120595 and 120595

1015840 have a common radial decreasing 1198711-

majorant also generates an unconditional basis for 119871119901(119877) 1 lt

119901 lt infin and1198671(R) although it has less regularity

It can be shown [1] that if two coefficient spaces areequal (as subsets of CN) then their norms are equivalentand corresponding Banach spaces 119871

119901(R) 1 lt 119901 lt infin and

Hardy space 1198671(R) are (topologically) isomorphic through

an isomorphism of one basis to anotherIn this paper we will consider the spaces 119871

119901(R) 1 lt

119901 lt infin and 1198671(R119899) and try to study that the reasonable

nice wavelets form unconditional bases in these functionspaces Also the convergence of wavelet series in these spaceswill be characterizedThere are several equivalent definitionsof real Hardy space 119867

1(R119899) In the real variable theory

one finds (at least) four major types of characterizationsof 1198671 integrability of maximal functions integrability of

square functions integrability of conjugate functions (theRiesz transform or its variants) and atomic decompositions

Let 119879 be a classical Calderon-Zygmund singular integraloperator on R119899 with smooth kernel 119870 and let 119879

lowast be theassociated maximal singular integral

119879lowast119891 (119909) = sup

120576gt0

1003816100381610038161003816119879120576119891 (119909)

1003816100381610038161003816 (2)

where 119879120576119891(119909) is the truncation at level 120576

119879120576119891 (119909) = int

|119910minus119909|gt120576

119891 (119909 minus 119910)119870 (119910) 119889119910 (3)

The 119895th Riesz transform 1 le 119895 le 119899 is the singular integraloperator

R119895119891 (119909) = PV int 119891 (119909 minus 119910)

119910119895

10038161003816100381610038161199101003816100381610038161003816119899+1

119889119910

equiv lim120576rarr0

int|119909minus119910|gt120576

119891 (119909 minus 119910)119910119895

10038161003816100381610038161199101003816100381610038161003816119899+1

119889119910

(4)

Hindawi Publishing CorporationChinese Journal of MathematicsVolume 2014 Article ID 984280 7 pageshttpdxdoiorg1011552014984280

2 Chinese Journal of Mathematics

The principal value integral above exists for all 119909 if 119891 is acompactly supported smooth function and one shows forsuch functions the 119871

119901(R119899) estimates

10038171003817100381710038171003817R11989511989110038171003817100381710038171003817119871119901(R119899)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901(R119899) 1 lt 119901 lt infin (5)

for some positive constant 119862 independent of 119891 Then abounded operator119877

119895can be defined on 119871

119901(R119899) by the density

argument A subissue arises when one tries to show that theprincipal value in (4) exists for almost all 119909 in R119899 for eachfunction 119891 in 119871

119901(R119899) Following a well-known principle one

looks for 119871119901(R119899) estimates for the maximal Riesz transform

Rlowast119895119891 and one indeed proves that [2 3]

10038171003817100381710038171003817Rlowast

11989511989110038171003817100381710038171003817119871119901(R119899)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901(R119899) 1 lt 119901 lt infin (6)

The following results [4] improve inequality (6)For 1 lt 119901 le infin there exists a constant 119862 = 119862

119901119899such that

10038171003817100381710038171003817Rlowast

11989511989110038171003817100381710038171003817119871119901(R119899)

le 11986210038171003817100381710038171003817R11989511989110038171003817100381710038171003817119871119901(R119899) (7)

for each function 119891 belonging to some 119871119902(R119899) 1 le 119902 le infin

Inequality (7) still holds in the limiting case 119901 = 1 and theanswer is provided by next result

Given 119895 1 le 119895 le 119899 and a positive constant 119862 there existsa function 119891 in 119871

1(R119899) such that

10038171003817100381710038171003817Rlowast

11989511989110038171003817100381710038171003817(1198711 119871infin)(R119899)

ge 11986210038171003817100381710038171003817R119895119891100381710038171003817100381710038171198711(R119899)

(8)

Remark 1 Inequality (8) does not hold even for the Hilberttransform which is the case 119899 = 1

Stein and Weiss originally defined the real Hardy space1198671(R) to be

1198671(R) = 119891 isin 119871

1(R) 119867119891 isin 119871

1(R)

100381710038171003817100381711989110038171003817100381710038171198671(R) =

100381710038171003817100381711989110038171003817100381710038171198711(R) +

100381710038171003817100381711986711989110038171003817100381710038171198711(R)

(9)

where 119867 denotes the Hilbert transform (119867119891)(119909) =

PV intR(119891(119909minus 119905)120587119905)119889119905 for 119899 gt 1 and we use Riesz transform

In order to define real Hardy space 1198671(R119899) set 119887 =

(1 0 0) and 119886 = (minus1 0 0) and let 120583 be the lengthmeasure on the segment joining 119886 and 119887 For an appropriateconstant 119862

119899we have

120583 = 119862119899(

1

|119909|119899minus1

lowast

119899

sum

119895=1

R119895(120597119895120583)) (10)

Taking the Fourier transforms on both sides we obtain

120575119886minus 120575119887= 1205971120583 = R

1(119862119899

119899

sum

119895=1

R119895(120597119895120583)) (11)

Let 120593 be a nonnegative continuously differentiable and com-pactly supported function in the unit ball such that 120593(0) = 1and set the approximate identity as

120593120576(119909) = 120576

119899120593(

119909

120576) (12)

Convolving identity (10) by approximate identity 120593120576 we get

120593120576(119909 minus 119886) minus 120593

120576(119909 minus 119887) = R

1(119862119899

119899

sum

119895=1

R119895(120597119895120583) lowast 120593

120576) (13)

Set 119891120576= 119862119899sum119899

119895=1R119895(120597119895120583) lowast 120593

120576= 119862119899sum119899

119895=1R119895(120583 lowast 120597

119895120593120576)

Since 120583 lowast 120597119895120593120576is a compactly supported function in

119871infin(R119899) with zero integral thus 120583 lowast 120597

119895120593120576is a function in the

Hardy space 1198671(R119899) and so 119891

120576isin 1198711(R119899) From (13) we see

that1003817100381710038171003817(R1) (119891120576)

10038171003817100381710038171 le 2 (14)

Now we define

1198671(R119899) = 119891 isin 119871

1(R119899) R119895119891 isin 119871

1(R119899)

100381710038171003817100381711989110038171003817100381710038171198671(R119899) =

100381710038171003817100381711989110038171003817100381710038171198671(R119899) +

10038171003817100381710038171003817R119895119891100381710038171003817100381710038171198711(R119899)

(15)

where R119895denotes the Riesz transform

TheHardy space1198671(R119899 119883) is initially defined in terms ofthe atomic decomposition as follows

Atoms are those measurable functions 119886119894 R119899 rarr C for

which there exist balls 119861119894in R119899 such that

supp (119886119894) sub 119861119894 int 119886

119894(119909) 119889119909 = 0

infin

sum

119894=1

10038171003817100381710038171198861198941003817100381710038171003817119871119901(R119899)

100381610038161003816100381611986111989410038161003816100381610038161119901

lt infin

(16)

for some value of 119901 isin ]1infin[ being fixed and 1199011015840 denotes the

conjugate exponent 1119901 + 11199011015840= 1

We say that 119891 isin 1198671(R119899) if 119891 isin 119871

1(R119899) has an expansion

of the form 119891(119909) = suminfin

119894=1119886119894(119909) and the norm is defined as

100381710038171003817100381711989110038171003817100381710038171198671(R119899) = inf

infin

sum

119894=1

10038171003817100381710038171198861198941003817100381710038171003817119871119901(R119899)

10038161003816100381610038161003816119861119894

10038161003816100381610038161003816

11199011015840

(17)

Atomic Hardy space is a Banach space

1198671(R119899) =

infin

sum

119894=1

119886119894119909 119886119894are atoms (18)

Notice that 1198671(R119899) sube 119891 isin 1198711(R119899) int

R119899119891 = 0 but the

converse inclusion is falseIf in the definition of atoms we consider only dyadic

intervals in place of balls in R119899 we obtain a ldquosmallerrdquo spacecalled the dyadic real Hardy space 119867

1

dyadic(R) The spaces1198671(R) and119867

1

dyadic(R) are (topologically) isomorphic Banachspaces This isomorphism leads to a nonconstructive proofof the existence of an unconditional basis for 119867

1

R Namelythe Haar system provides a perfect unconditional basis for1198671

dyadic(R) because no regularity is needed in the dyadicsetting In this paper our approach is different and basedon the fact that smoother wavelets will directly provide anunconditional basis for 119867

1(R119899) we will characterize the

coefficient space by taking the wavelet basis in1198671(R119899)

Chinese Journal of Mathematics 3

It should be significant to mention here that it would bebetter to obtain constants depending only on 119901 and regularityof wavelet but no other information about it For that reasona better approach is to prove square function estimates usingCalderon-Zygmund decomposition and real interpolation[5]

A Calderon-Zygmund operator is a bounded linear oper-ator 119879

lowast 1198712(R119899) rarr 119871

2(R119899) with operator norm at most 119896

and such that there exists aC1-function 119896(119909 119910) isin 1198711(R119899timesR119899)

satisfying1003816100381610038161003816119896 (119909 119910)

1003816100381610038161003816 le 1198961003816100381610038161003816119909 minus 119910

1003816100381610038161003816minus119899

1003816100381610038161003816nabla119909119896 (119909 119910)1003816100381610038161003816 +

10038161003816100381610038161003816nabla119910119896 (119909 119910)

10038161003816100381610038161003816le 119896

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

119879lowast119891 (119909) = int

R119899119896 (119909 119910) 119891 (119910) 119889119910

for119891 isin 1198712

119888(R119899) 119909 notin supp (119891)

(19)

A function with these properties is called a Calderon-Zygmund kernel or a standard singular kernel

2 Some Definitions and Auxiliary Results

Definition 2 ABanach space119883 is UMD if for some (and thenall cf [6]) 1 lt 119901 lt infin there is a finite constant 119862 so that

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119896=1

120576119896119889119896

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(Ω119883)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119896=1

119889119896

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(Ω119883)

(20)

for all 119899 isin Z+ whenever (120576

119896)119899

119896=1isin minus1 +1

119899 and (119889119896)119899

119896=1isin

119871119901(Ω119883)

119899 a martingale difference sequence or an arbitraryprobability space (Ω 119860 120583) (ie there are sub-120590-algebras119860

0sub

1198601

sub sdot sdot sdot sub 119860119899

sub 119860) such that for all 119896 = 1 119899 thefunction 119889

119896is 119860119896-measurable and int

119860119889119896119889120583 = 0 for all 119860 isin

119860119896minus1

Remark 3 The validity of Theorem 19 of [7] on 1198671(R119899 119883)

actually characterizes the UMD-property of119883 Indeed let119883be any complex Banach space and let conclusion of Theorem19 of [7] be satisfied Since R

119895119891119871119901(R119899 119883)

le 119862119891119871119901(R119899119883)

this is equivalent to the UMD-property of 119883 We also haveR119895119891 le 119862119891

1198711(R119899 119883) This implies that R

119895is bounded on

1198671(R119899 119883)

In view of above remark we can get without modificationin [8] the following

Proposition 4 Let 119896(119909 119910) isin 1198711(R119899timesR119899) satisfy the standard

estimates1003816100381610038161003816119896 (119909 119910)

1003816100381610038161003816 le 1198961003816100381610038161003816119909 minus 119910

1003816100381610038161003816minus119899

1003816100381610038161003816nabla119909119896 (119909 119910)1003816100381610038161003816 +

10038161003816100381610038161003816nabla119910119896 (119909 119910)

10038161003816100381610038161003816le 119896

1003816100381610038161003816119909 1199101003816100381610038161003816minus119899minus1

(21)

Assume moreover that 119879 is bounded on 1198712(R119899) with

operator norm at most 119896 Then 119879 is also bounded on119871119901(R119899 119883) where

10038171003817100381710038171003817R11989511989110038171003817100381710038171003817119871119901(R119899 119883)

le 119862119901119899

(119883) 1198961003817100381710038171003817119891

1003817100381710038171003817119871119901(R119899119883) (22)

for all 119901 isin ]1infin[ and it is bounded from 1198671(R119899 119883) to

1198711(R119899 119883) with norm 119862

1119899(119883)119896 If in addition

[1198791] (119910) = intR119899

119896 (119909 119910) 119889119909 equiv 0 (23)

then 119879 is bounded on1198671(R119899 119883) with norm le 119862

0(119883)119896

Let us note that atomic definition of1198671(R119899 119883) is knownto agree with the one given in terms of various maximalfunctions even for an arbitrary Banach space 119883 One cancheck that the proof of this fact in the scalar case as givenfor example in Steinrsquos book [9] goes through word by wordin the general setting for 119899 = 1 the conjugate Hardy spacedefined as the domain of the Hilbert transform on 119871

1(R 119883)

with the graph norm is always contained in the atomic Hardyspace and agrees with it exactly when119883 is a UMD-space For119899 gt 1 we can replace Hilbert transform by Riesz transformand the condition of UMD-space for 119883 by the boundednessof Riesz transform in 119871

119901(R119899)-space

It is well-known fact that an integral operator satisfyingthe standard estimates and bounded on 119871

119901(R119899 119883) is also

bounded from 1198671(R119899 119883) to 119871

1(R119899 119883) The square function

description of 1198671(R119899 119883) that we have in mind involves thewave expansion of a function in119867

1(R119899) Recall the definition

of the wavelet [10ndash12] Let 120593119897 119897 = 1 2 2119899minus 1 be a set of

functions belonging to 1198712(R119899)

Consider

120593119897

119895119896(119909) = 2

1198951198992120593119897(2119895119909 minus 119896)

= 21198951198992

120593119897(21198951199091minus 1198961 2

119895119909119899minus 119896119899)

(119909 = (1199091 119909

119899) isin R119899)

(24)

for each 119897 = 1 2 2119899minus 1 119895 isin Z and 119896 = (119896

1 119896

119899) isin Z119899

The sequence 120593119897 119897 = 1 2 2119899minus 1 is a wavelet set if 120593119897

119895119896

119897 = 1 2 2119899minus 1 119895 isin Z 119896 isin Z119899 forms an orthonormal

basis in 1198712(R119899) Then 120593

119897

119895119896 119897 = 1 2 2

119899minus 1 119895 isin Z 119896 isin

Z119899 is a wavelet basis in 1198712(R119899) and each 120593

119897 is a waveletThe basis is called 119903-regular if |120597

120572120593119897(119909)| le 119862

119898(1 + |119909|)

minus119898

and int119909120572120593119897(119909)119889119909 = 0 for all |120572| le 119903 all 119898 isin 119873 and

119897 = 1 2 2119899minus 1

The aim of the present paper is to study that reason-able nice wavelets form unconditional bases in functionspace other than 119871

2(R119899) and characterization of convergence

of wavelets series in 119871119901(R119899 119883) and 119867

1(R119899 119883) has been

obtained For this purpose it will be enough to assume that(120593119860)119860isinΛ

where Λ is the set of dyadic cube of the form

119876119895119896

= Π119899

119894=1[2minus119895119896119894 2minus119895

(119896119894+ 1)] (25)

is an orthonormal wavelet basis such that 120593 isin C1(R119899) and

1003816100381610038161003816120593 (119909)1003816100381610038161003816 +

100381610038161003816100381610038161205931015840(119909)

10038161003816100381610038161003816le 119862(1 + |119909|)

minus119899minus2 (26)

This is certainly not the weakest possible assumption on 120593 forsomewhat less restrictive see [10] (where the class R0(R119899) isintroduced)

4 Chinese Journal of Mathematics

3 Main Results

Theorem 5 Let 1 lt 119901 lt infin and let 120593119860

be a waveletsuch that 120593 isin C1(R119899) and satisfies condition (26) Then thesystem (120593

119860)119860isinΛ

is an unconditional basis for 119871119901(R119899 119883) and

1198671(R119899 119883)

Proof For arbitrary family of ]119860

isin C 119860 isin Λ such that |]119860| le

1 we define 119879]119860 on the orthonormal basis by

119879]119860 1198712(R119899 119883) 997888rarr 119871

2(R119899 119883) 119879]119860 120593

119860997888rarr ]119860120593119860

(27)

Here 119879]119860 is not necessarily unitary but it is bounded and

10038171003817100381710038171003817119879]119860

100381710038171003817100381710038171198712rarr1198712le 1 (28)

Now we can write

(119879]119860119891) (119909) = sum

119860isinΛ

]119860⟨119891 120593119860⟩ 120593119860(119909)

= intR119899

119896]119860 (119909 119910) 119891 (119910) 119889119910

(29)

where

119896]119860 (119909 119910) = sum

119860isin119891

]119860120593119860(119909) 120593119860(119910)

= sum

119895isinZ119896isinZ119899

]119895119896

2119899119895120593 (2119895119909 minus 119896) 120593 (2119895119910 minus 119896)

(30)

where 119865 sub Λ is any finite setWe claim that 119879]119860 are Calderon-Zygmund operators with

constants uniform in ]119860and for that we have to show that 119896]119860

satisfy standard estimates uniformly in ]119860

Consider

10038161003816100381610038161003816119896]119860 (119909 119910)

10038161003816100381610038161003816le sum

119895isinZ119896isinZ119899

2119899119895(1 +

100381610038161003816100381610038162119895119909 minus 119896

10038161003816100381610038161003816)minus119899minus2

times (1 +100381610038161003816100381610038162119895119910 minus 119896

10038161003816100381610038161003816)minus119899minus2

le sum

119895isinZ

1198621198992119899119895(1 + 2

119895 1003816100381610038161003816119909 minus 1199101003816100381610038161003816)minus119899minus2

le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899

(31)

Also for derivatives

nabla119909119896]119860 (119909 119910)

= sum

119895isinZ119896isinZ119899

]119895119896

2119899119895+2119895

1205931015840(2119895119909 minus 119896) 120593(2119895119910 minus 119896)

10038161003816100381610038161003816nabla119909119896]119860 (119909 119910)

10038161003816100381610038161003816

le sum

119895isinZ119896isinZ119899

2119899119895+2119895

(1 +100381610038161003816100381610038162119895119909 minus 119896

10038161003816100381610038161003816)minus119899minus2

(1 +100381610038161003816100381610038162119895119910 minus 119896

10038161003816100381610038161003816)minus119899minus2

le sum

119895isinZ

1198621198992119899119895+2119895

(1 + 2119895 1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)minus119899minus2

le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

(32)

So we have10038161003816100381610038161003816nabla119909119896]119860 (119909 119910)

10038161003816100381610038161003816+10038161003816100381610038161003816nabla119910119896]119860 (119909 119910)

10038161003816100381610038161003816le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

(33)

Condition intR119899

119891 = 0 rArr intR119899

119879]119860119891 = 0 follows fromintR119899

119896]119860(119909 119910)119889119909 = 0 which in turn follows from intR119899

120593(119909) = 0Therefore from Proposition 4 we know that operators 119879]119860extend continuously to 119871

119901(R119899 119883) and119867

1(R119899 119883) and we get

inequalities

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899119883)le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883)

10038171003817100381710038171003817119879]119860119891

100381710038171003817100381710038171198671(R119899 119883)le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883)

(34)

and particularly for 119891 = sum119860isin119891

119862119860120593119860 119865 is a finite subset of Λ

119862119860

isin 119862 can be written as

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

]119860119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(35)

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

]119860119862119860120593119860

10038171003817100381710038171003817100381710038171003817100381710038171198671(R119899 119883)

le 119862119899

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

10038171003817100381710038171003817100381710038171003817100381710038171198671(R119899 119883)

(36)

These inequalities imply that (120593119860)119860isinΛ

is an unconditionalbasis for

span119871119901(R119899119883) (120593119860 119860 isin Λ)

span1198671(R119899 119883) (120593119860 119860 isin Λ)

(37)

However it is still not clear that (120593119860)119860isinΛ

is complete in119871119901(R119899 119883) and119867

1(R119899 119883)

In particular if we take (120576119860)119860isinΛ

isin plusmn1Λ then (34) and

1198792

120576119860imply that

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) le

1003817100381710038171003817100381711987912057611986011989110038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) (38)

119888119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) le

10038171003817100381710038171003817119879120576119860119891100381710038171003817100381710038171198671(R119899119883)

le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) (39)

Chinese Journal of Mathematics 5

Let us assume thatΩ = plusmn1Λ with the product probabil-

ity measure First we consider the function of the form 119891 =

sum119860isin119865

119862119860120593119860for some finite 119865 sub Λ and 119862

119860isin C Integrating

(38) overΩ interchanging the order of integration and usingKhintchinersquos inequality we get

(intΩ

1003817100381710038171003817100381711987912057611986011989110038171003817100381710038171003817

119901

119871119901(R119899119883)

119889120576119860)

1119901

times

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(intΩ

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119860isin119865

120576119860119862119860120593119860

1003816100381610038161003816100381610038161003816100381610038161003816

119901

119889120576119860)

1119901100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

997904rArr 119888119899119901

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(intΩ

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119860isin119865

120576119860119862119860120593119860

1003816100381610038161003816100381610038161003816100381610038161003816

119901

119889120576119860)

1119901100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(40)

In view of (38) we obtain

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883)

(41)

Similarly we get

119888119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

10038161003816100381610038161198621198601003816100381610038161003816210038161003816100381610038161003816R119895120593119860

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899 119883)

le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883)

(42)

Now we take 119891 isin 119871119901(R119899 119883) for some 119865 finite subset ofΛ and

define (]119860)119860isinΛ

by ]119860

= 1 if 119860 isin 119865 and by ]119860

= 0 if 119860 notin 119865 sothat 119879]119860119891 = sum

119860isin119865⟨119891 120593119860⟩120593119860 Applying estimate (41) to 119879]119860119891

and using (34) we get

119888119899119901

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899 119883)le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899 119883)

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899119883)le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

(43)

By virtue of monotone convergence theorem as finite sets 119865

exhaust Λ we get10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) (44)

In order to prove the completeness of (120593119860)119860isinΛ

in119871119901(R119899 119883) consider 119891 isin 119871

119901(R119899 119883) cap 119871

2(R119899 119883) such

that 119891 = suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in1198712(R119899 119883) and there exists a subsequence of partial sums

(sum119873119898

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

)119898isin119873

that converge almost every-where on R119899 By virtue of Fatoursquos lemma and (41) we obtain10038171003817100381710038171003817100381710038171003817100381710038171003817

119891 minus

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le lim inf119898rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119894=119873+1(119895119896119894)isin119865

10038161003816100381610038161003816⟨119891 120593119895119896119894

⟩120593119895119896119894

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

(45)

In view of (44) we have10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin (46)

Using dominated convergence theorem we obtain

119891 =

infin

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

(47)

with convergence in 119871119901(R119899 119883) Therefore span

119871119901(R119899119883)(120593119860

119860 isin Λ) sup 119871119901(R119899 119883)cap119871

2(R119899 119883) since 119871119901(R119899 119883)cap119871

2(R119899 119883)

is dense in 119871119901(R119899 119883) These facts lead to the conclusion

that (120593119860)119860isinΛ

is complete in 119871119901(R119899 119883) By virtue of (35) this

proves that (120593119860)119860isinΛ

is an unconditional basis for 119871119901(R119899 119883)Similarly we can prove the result for 119867

1(R119899 119883) using

(42) Hence the proof is completed

Theorem 6 Let 1 lt 119901 lt infin and let 120593119860be a wavelet such

that 120593 isin C1(R119899) and satisfies condition (26) Then for 119891 isin

119871119901(R119899 119883) one has

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

(48)

and for 119891 isin 1198671(R119899 119883) one has

1198881198991

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899 119883)

le 1198621198991

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

(49)

6 Chinese Journal of Mathematics

Proof We have

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

100381710038171003817100381712059311986010038171003817100381710038171198711199011015840(R119899 119883)

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

10038171003817100381710038171205931198601003817100381710038171003817BMO

(50)

where the space BMO(R119899) is of functions with boundedmean oscillation1198671(R119899) is the predual of BMO(R119899) We seethat ⟨sdot 120593

119860⟩ 119860 isin Λ defines continuous linear functionals

on119871119901(R119899 119883) 1 lt 119901 lt infin and119867

1(R119899 119883) If119891 = sum

119860isinΛ119862119860120593119860

unconditionally in 119871119901(R119899 119883) or 119867

1(R119899 119883) then for every

1198601015840isin Λ by continuity of ⟨sdot 120593

1198601015840 ⟩

⟨119891 1205931198601015840⟩ = sum

119860isinΛ

119862119860⟨120593119860 1205931198601015840⟩ = sum

119860isinΛ

1198621198601205751198601198601015840 = 1198621198601015840 (51)

Now for every 119891 isin 119871119901(R119899 119883) we have 119891 =

suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in 119871119901(R119899 119883)

Using (41) we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) = lim

119899rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(52)

Combining (44) and (52) we get (48) Similarly we wouldhandle the case for 119867

1(R119899 119883) Hence the proof is com-

pleted

Corollary 7 For every family of scalars (119862119860)119860isinΛ

one has that

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 119871119901(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 1198671(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

lt infin

(53)

Proof Suppose that (119862119860)119860isinΛ

is a family of scalars suchthat (sum

119860isinΛ|119862119860|2|120593119860|2)12

119871119901(R119899119883)

lt infin For every120576 gt 0 we can find 119865

0finite subset of Λ such that

(sum119860isinΛ1198650

|119862119860|2|120593119860|2)12

119871119901(R119899 119883)

le 120576Then for every 119865 finite subset of Λ 119865

0 by (44) we have

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

120576 (54)

This implies that the series sum119860isinΛ

119862119860120593119860converges uncondi-

tionally in 119871119901(R119899 119883) to the same 119871

119901-function Moreover wehave proved that

sum

119860isinΛ

119862119860120593119860converges unconditionally in

119871119901(R119899 119883) lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

lt infin

(55)

This gives the characterization of the convergence of waveletseries in 119871

119901(R119899 119883) Similarly we would characterize conver-

gence in1198671(R119899 119883)

Hence the proof is completed

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by University Grants CommissionNew Delhi under the major research Project by F no 41-7922012 (SR)

References

[1] R M Young An Introduction to Nonharmonic Fourier Seriesvol 93 Academic Press New York NY USA 1980

[2] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2001

[3] E M Stein Singular Integrals and Differentiability Propertiesof Functions Princeton University Press Princeton NJ USA1970

[4] J Mateu and J Verdera ldquo119871119901 and weak 1198711 estimates for the

maximal Riesz transform and themaximal Beurling transformrdquoMathematical Research Letters vol 13 no 5-6 pp 957ndash9662006

[5] C Thiele Wave Packet Analysis vol 105 of CBMS RegionalConference Series in Mathematics 2006

[6] D L Burkholder ldquoMartingales and Fourier analysis in Banachspacesrdquo in Probability and Analysis G Letta and M PratelliEds vol 1206 of Lecture Notes in Mathematics pp 61ndash108Springer Berlin Germany 1986

[7] T Hytonen ldquoVector-valued wavelets and the Hardy space1198671(R119899 119883)rdquo Studia Mathematica vol 172 no 2 pp 125ndash147

2006[8] T Figiel ldquoSingular integral operators a martingale approachrdquo

in Geometry of Banach Spaces P F X Moller and W Schacher-mayer Eds vol 158 of London Mathematical Society LectureNote Series pp 95ndash110 Cambridge University Press Cam-bridge Mass USA 1990

[9] E M SteinHarmonic Analysis Real-Variable Methods Orthog-onality and Oscillatory Integrals vol 43 Princeton UniversityPress Princeton NJ USA 1993

[10] E Hernandez and G Weiss A First Course on Wavelets CRCPress Boca Raton Fla USA 1996

Chinese Journal of Mathematics 7

[11] YMeyerWavelets and Operators vol 37 Cambridge UniversityPress Cambridge Mass USA 1992

[12] P Wojtaszczyk ldquoThe Franklin system is an unconditional basisin1198671rdquo Arkiv for Matematik vol 20 no 2 pp 293ndash300 1982

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Stochastic AnalysisInternational Journal of

2 Chinese Journal of Mathematics

The principal value integral above exists for all 119909 if 119891 is acompactly supported smooth function and one shows forsuch functions the 119871

119901(R119899) estimates

10038171003817100381710038171003817R11989511989110038171003817100381710038171003817119871119901(R119899)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901(R119899) 1 lt 119901 lt infin (5)

for some positive constant 119862 independent of 119891 Then abounded operator119877

119895can be defined on 119871

119901(R119899) by the density

argument A subissue arises when one tries to show that theprincipal value in (4) exists for almost all 119909 in R119899 for eachfunction 119891 in 119871

119901(R119899) Following a well-known principle one

looks for 119871119901(R119899) estimates for the maximal Riesz transform

Rlowast119895119891 and one indeed proves that [2 3]

10038171003817100381710038171003817Rlowast

11989511989110038171003817100381710038171003817119871119901(R119899)

le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901(R119899) 1 lt 119901 lt infin (6)

The following results [4] improve inequality (6)For 1 lt 119901 le infin there exists a constant 119862 = 119862

119901119899such that

10038171003817100381710038171003817Rlowast

11989511989110038171003817100381710038171003817119871119901(R119899)

le 11986210038171003817100381710038171003817R11989511989110038171003817100381710038171003817119871119901(R119899) (7)

for each function 119891 belonging to some 119871119902(R119899) 1 le 119902 le infin

Inequality (7) still holds in the limiting case 119901 = 1 and theanswer is provided by next result

Given 119895 1 le 119895 le 119899 and a positive constant 119862 there existsa function 119891 in 119871

1(R119899) such that

10038171003817100381710038171003817Rlowast

11989511989110038171003817100381710038171003817(1198711 119871infin)(R119899)

ge 11986210038171003817100381710038171003817R119895119891100381710038171003817100381710038171198711(R119899)

(8)

Remark 1 Inequality (8) does not hold even for the Hilberttransform which is the case 119899 = 1

Stein and Weiss originally defined the real Hardy space1198671(R) to be

1198671(R) = 119891 isin 119871

1(R) 119867119891 isin 119871

1(R)

100381710038171003817100381711989110038171003817100381710038171198671(R) =

100381710038171003817100381711989110038171003817100381710038171198711(R) +

100381710038171003817100381711986711989110038171003817100381710038171198711(R)

(9)

where 119867 denotes the Hilbert transform (119867119891)(119909) =

PV intR(119891(119909minus 119905)120587119905)119889119905 for 119899 gt 1 and we use Riesz transform

In order to define real Hardy space 1198671(R119899) set 119887 =

(1 0 0) and 119886 = (minus1 0 0) and let 120583 be the lengthmeasure on the segment joining 119886 and 119887 For an appropriateconstant 119862

119899we have

120583 = 119862119899(

1

|119909|119899minus1

lowast

119899

sum

119895=1

R119895(120597119895120583)) (10)

Taking the Fourier transforms on both sides we obtain

120575119886minus 120575119887= 1205971120583 = R

1(119862119899

119899

sum

119895=1

R119895(120597119895120583)) (11)

Let 120593 be a nonnegative continuously differentiable and com-pactly supported function in the unit ball such that 120593(0) = 1and set the approximate identity as

120593120576(119909) = 120576

119899120593(

119909

120576) (12)

Convolving identity (10) by approximate identity 120593120576 we get

120593120576(119909 minus 119886) minus 120593

120576(119909 minus 119887) = R

1(119862119899

119899

sum

119895=1

R119895(120597119895120583) lowast 120593

120576) (13)

Set 119891120576= 119862119899sum119899

119895=1R119895(120597119895120583) lowast 120593

120576= 119862119899sum119899

119895=1R119895(120583 lowast 120597

119895120593120576)

Since 120583 lowast 120597119895120593120576is a compactly supported function in

119871infin(R119899) with zero integral thus 120583 lowast 120597

119895120593120576is a function in the

Hardy space 1198671(R119899) and so 119891

120576isin 1198711(R119899) From (13) we see

that1003817100381710038171003817(R1) (119891120576)

10038171003817100381710038171 le 2 (14)

Now we define

1198671(R119899) = 119891 isin 119871

1(R119899) R119895119891 isin 119871

1(R119899)

100381710038171003817100381711989110038171003817100381710038171198671(R119899) =

100381710038171003817100381711989110038171003817100381710038171198671(R119899) +

10038171003817100381710038171003817R119895119891100381710038171003817100381710038171198711(R119899)

(15)

where R119895denotes the Riesz transform

TheHardy space1198671(R119899 119883) is initially defined in terms ofthe atomic decomposition as follows

Atoms are those measurable functions 119886119894 R119899 rarr C for

which there exist balls 119861119894in R119899 such that

supp (119886119894) sub 119861119894 int 119886

119894(119909) 119889119909 = 0

infin

sum

119894=1

10038171003817100381710038171198861198941003817100381710038171003817119871119901(R119899)

100381610038161003816100381611986111989410038161003816100381610038161119901

lt infin

(16)

for some value of 119901 isin ]1infin[ being fixed and 1199011015840 denotes the

conjugate exponent 1119901 + 11199011015840= 1

We say that 119891 isin 1198671(R119899) if 119891 isin 119871

1(R119899) has an expansion

of the form 119891(119909) = suminfin

119894=1119886119894(119909) and the norm is defined as

100381710038171003817100381711989110038171003817100381710038171198671(R119899) = inf

infin

sum

119894=1

10038171003817100381710038171198861198941003817100381710038171003817119871119901(R119899)

10038161003816100381610038161003816119861119894

10038161003816100381610038161003816

11199011015840

(17)

Atomic Hardy space is a Banach space

1198671(R119899) =

infin

sum

119894=1

119886119894119909 119886119894are atoms (18)

Notice that 1198671(R119899) sube 119891 isin 1198711(R119899) int

R119899119891 = 0 but the

converse inclusion is falseIf in the definition of atoms we consider only dyadic

intervals in place of balls in R119899 we obtain a ldquosmallerrdquo spacecalled the dyadic real Hardy space 119867

1

dyadic(R) The spaces1198671(R) and119867

1

dyadic(R) are (topologically) isomorphic Banachspaces This isomorphism leads to a nonconstructive proofof the existence of an unconditional basis for 119867

1

R Namelythe Haar system provides a perfect unconditional basis for1198671

dyadic(R) because no regularity is needed in the dyadicsetting In this paper our approach is different and basedon the fact that smoother wavelets will directly provide anunconditional basis for 119867

1(R119899) we will characterize the

coefficient space by taking the wavelet basis in1198671(R119899)

Chinese Journal of Mathematics 3

It should be significant to mention here that it would bebetter to obtain constants depending only on 119901 and regularityof wavelet but no other information about it For that reasona better approach is to prove square function estimates usingCalderon-Zygmund decomposition and real interpolation[5]

A Calderon-Zygmund operator is a bounded linear oper-ator 119879

lowast 1198712(R119899) rarr 119871

2(R119899) with operator norm at most 119896

and such that there exists aC1-function 119896(119909 119910) isin 1198711(R119899timesR119899)

satisfying1003816100381610038161003816119896 (119909 119910)

1003816100381610038161003816 le 1198961003816100381610038161003816119909 minus 119910

1003816100381610038161003816minus119899

1003816100381610038161003816nabla119909119896 (119909 119910)1003816100381610038161003816 +

10038161003816100381610038161003816nabla119910119896 (119909 119910)

10038161003816100381610038161003816le 119896

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

119879lowast119891 (119909) = int

R119899119896 (119909 119910) 119891 (119910) 119889119910

for119891 isin 1198712

119888(R119899) 119909 notin supp (119891)

(19)

A function with these properties is called a Calderon-Zygmund kernel or a standard singular kernel

2 Some Definitions and Auxiliary Results

Definition 2 ABanach space119883 is UMD if for some (and thenall cf [6]) 1 lt 119901 lt infin there is a finite constant 119862 so that

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119896=1

120576119896119889119896

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(Ω119883)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119896=1

119889119896

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(Ω119883)

(20)

for all 119899 isin Z+ whenever (120576

119896)119899

119896=1isin minus1 +1

119899 and (119889119896)119899

119896=1isin

119871119901(Ω119883)

119899 a martingale difference sequence or an arbitraryprobability space (Ω 119860 120583) (ie there are sub-120590-algebras119860

0sub

1198601

sub sdot sdot sdot sub 119860119899

sub 119860) such that for all 119896 = 1 119899 thefunction 119889

119896is 119860119896-measurable and int

119860119889119896119889120583 = 0 for all 119860 isin

119860119896minus1

Remark 3 The validity of Theorem 19 of [7] on 1198671(R119899 119883)

actually characterizes the UMD-property of119883 Indeed let119883be any complex Banach space and let conclusion of Theorem19 of [7] be satisfied Since R

119895119891119871119901(R119899 119883)

le 119862119891119871119901(R119899119883)

this is equivalent to the UMD-property of 119883 We also haveR119895119891 le 119862119891

1198711(R119899 119883) This implies that R

119895is bounded on

1198671(R119899 119883)

In view of above remark we can get without modificationin [8] the following

Proposition 4 Let 119896(119909 119910) isin 1198711(R119899timesR119899) satisfy the standard

estimates1003816100381610038161003816119896 (119909 119910)

1003816100381610038161003816 le 1198961003816100381610038161003816119909 minus 119910

1003816100381610038161003816minus119899

1003816100381610038161003816nabla119909119896 (119909 119910)1003816100381610038161003816 +

10038161003816100381610038161003816nabla119910119896 (119909 119910)

10038161003816100381610038161003816le 119896

1003816100381610038161003816119909 1199101003816100381610038161003816minus119899minus1

(21)

Assume moreover that 119879 is bounded on 1198712(R119899) with

operator norm at most 119896 Then 119879 is also bounded on119871119901(R119899 119883) where

10038171003817100381710038171003817R11989511989110038171003817100381710038171003817119871119901(R119899 119883)

le 119862119901119899

(119883) 1198961003817100381710038171003817119891

1003817100381710038171003817119871119901(R119899119883) (22)

for all 119901 isin ]1infin[ and it is bounded from 1198671(R119899 119883) to

1198711(R119899 119883) with norm 119862

1119899(119883)119896 If in addition

[1198791] (119910) = intR119899

119896 (119909 119910) 119889119909 equiv 0 (23)

then 119879 is bounded on1198671(R119899 119883) with norm le 119862

0(119883)119896

Let us note that atomic definition of1198671(R119899 119883) is knownto agree with the one given in terms of various maximalfunctions even for an arbitrary Banach space 119883 One cancheck that the proof of this fact in the scalar case as givenfor example in Steinrsquos book [9] goes through word by wordin the general setting for 119899 = 1 the conjugate Hardy spacedefined as the domain of the Hilbert transform on 119871

1(R 119883)

with the graph norm is always contained in the atomic Hardyspace and agrees with it exactly when119883 is a UMD-space For119899 gt 1 we can replace Hilbert transform by Riesz transformand the condition of UMD-space for 119883 by the boundednessof Riesz transform in 119871

119901(R119899)-space

It is well-known fact that an integral operator satisfyingthe standard estimates and bounded on 119871

119901(R119899 119883) is also

bounded from 1198671(R119899 119883) to 119871

1(R119899 119883) The square function

description of 1198671(R119899 119883) that we have in mind involves thewave expansion of a function in119867

1(R119899) Recall the definition

of the wavelet [10ndash12] Let 120593119897 119897 = 1 2 2119899minus 1 be a set of

functions belonging to 1198712(R119899)

Consider

120593119897

119895119896(119909) = 2

1198951198992120593119897(2119895119909 minus 119896)

= 21198951198992

120593119897(21198951199091minus 1198961 2

119895119909119899minus 119896119899)

(119909 = (1199091 119909

119899) isin R119899)

(24)

for each 119897 = 1 2 2119899minus 1 119895 isin Z and 119896 = (119896

1 119896

119899) isin Z119899

The sequence 120593119897 119897 = 1 2 2119899minus 1 is a wavelet set if 120593119897

119895119896

119897 = 1 2 2119899minus 1 119895 isin Z 119896 isin Z119899 forms an orthonormal

basis in 1198712(R119899) Then 120593

119897

119895119896 119897 = 1 2 2

119899minus 1 119895 isin Z 119896 isin

Z119899 is a wavelet basis in 1198712(R119899) and each 120593

119897 is a waveletThe basis is called 119903-regular if |120597

120572120593119897(119909)| le 119862

119898(1 + |119909|)

minus119898

and int119909120572120593119897(119909)119889119909 = 0 for all |120572| le 119903 all 119898 isin 119873 and

119897 = 1 2 2119899minus 1

The aim of the present paper is to study that reason-able nice wavelets form unconditional bases in functionspace other than 119871

2(R119899) and characterization of convergence

of wavelets series in 119871119901(R119899 119883) and 119867

1(R119899 119883) has been

obtained For this purpose it will be enough to assume that(120593119860)119860isinΛ

where Λ is the set of dyadic cube of the form

119876119895119896

= Π119899

119894=1[2minus119895119896119894 2minus119895

(119896119894+ 1)] (25)

is an orthonormal wavelet basis such that 120593 isin C1(R119899) and

1003816100381610038161003816120593 (119909)1003816100381610038161003816 +

100381610038161003816100381610038161205931015840(119909)

10038161003816100381610038161003816le 119862(1 + |119909|)

minus119899minus2 (26)

This is certainly not the weakest possible assumption on 120593 forsomewhat less restrictive see [10] (where the class R0(R119899) isintroduced)

4 Chinese Journal of Mathematics

3 Main Results

Theorem 5 Let 1 lt 119901 lt infin and let 120593119860

be a waveletsuch that 120593 isin C1(R119899) and satisfies condition (26) Then thesystem (120593

119860)119860isinΛ

is an unconditional basis for 119871119901(R119899 119883) and

1198671(R119899 119883)

Proof For arbitrary family of ]119860

isin C 119860 isin Λ such that |]119860| le

1 we define 119879]119860 on the orthonormal basis by

119879]119860 1198712(R119899 119883) 997888rarr 119871

2(R119899 119883) 119879]119860 120593

119860997888rarr ]119860120593119860

(27)

Here 119879]119860 is not necessarily unitary but it is bounded and

10038171003817100381710038171003817119879]119860

100381710038171003817100381710038171198712rarr1198712le 1 (28)

Now we can write

(119879]119860119891) (119909) = sum

119860isinΛ

]119860⟨119891 120593119860⟩ 120593119860(119909)

= intR119899

119896]119860 (119909 119910) 119891 (119910) 119889119910

(29)

where

119896]119860 (119909 119910) = sum

119860isin119891

]119860120593119860(119909) 120593119860(119910)

= sum

119895isinZ119896isinZ119899

]119895119896

2119899119895120593 (2119895119909 minus 119896) 120593 (2119895119910 minus 119896)

(30)

where 119865 sub Λ is any finite setWe claim that 119879]119860 are Calderon-Zygmund operators with

constants uniform in ]119860and for that we have to show that 119896]119860

satisfy standard estimates uniformly in ]119860

Consider

10038161003816100381610038161003816119896]119860 (119909 119910)

10038161003816100381610038161003816le sum

119895isinZ119896isinZ119899

2119899119895(1 +

100381610038161003816100381610038162119895119909 minus 119896

10038161003816100381610038161003816)minus119899minus2

times (1 +100381610038161003816100381610038162119895119910 minus 119896

10038161003816100381610038161003816)minus119899minus2

le sum

119895isinZ

1198621198992119899119895(1 + 2

119895 1003816100381610038161003816119909 minus 1199101003816100381610038161003816)minus119899minus2

le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899

(31)

Also for derivatives

nabla119909119896]119860 (119909 119910)

= sum

119895isinZ119896isinZ119899

]119895119896

2119899119895+2119895

1205931015840(2119895119909 minus 119896) 120593(2119895119910 minus 119896)

10038161003816100381610038161003816nabla119909119896]119860 (119909 119910)

10038161003816100381610038161003816

le sum

119895isinZ119896isinZ119899

2119899119895+2119895

(1 +100381610038161003816100381610038162119895119909 minus 119896

10038161003816100381610038161003816)minus119899minus2

(1 +100381610038161003816100381610038162119895119910 minus 119896

10038161003816100381610038161003816)minus119899minus2

le sum

119895isinZ

1198621198992119899119895+2119895

(1 + 2119895 1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)minus119899minus2

le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

(32)

So we have10038161003816100381610038161003816nabla119909119896]119860 (119909 119910)

10038161003816100381610038161003816+10038161003816100381610038161003816nabla119910119896]119860 (119909 119910)

10038161003816100381610038161003816le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

(33)

Condition intR119899

119891 = 0 rArr intR119899

119879]119860119891 = 0 follows fromintR119899

119896]119860(119909 119910)119889119909 = 0 which in turn follows from intR119899

120593(119909) = 0Therefore from Proposition 4 we know that operators 119879]119860extend continuously to 119871

119901(R119899 119883) and119867

1(R119899 119883) and we get

inequalities

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899119883)le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883)

10038171003817100381710038171003817119879]119860119891

100381710038171003817100381710038171198671(R119899 119883)le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883)

(34)

and particularly for 119891 = sum119860isin119891

119862119860120593119860 119865 is a finite subset of Λ

119862119860

isin 119862 can be written as

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

]119860119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(35)

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

]119860119862119860120593119860

10038171003817100381710038171003817100381710038171003817100381710038171198671(R119899 119883)

le 119862119899

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

10038171003817100381710038171003817100381710038171003817100381710038171198671(R119899 119883)

(36)

These inequalities imply that (120593119860)119860isinΛ

is an unconditionalbasis for

span119871119901(R119899119883) (120593119860 119860 isin Λ)

span1198671(R119899 119883) (120593119860 119860 isin Λ)

(37)

However it is still not clear that (120593119860)119860isinΛ

is complete in119871119901(R119899 119883) and119867

1(R119899 119883)

In particular if we take (120576119860)119860isinΛ

isin plusmn1Λ then (34) and

1198792

120576119860imply that

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) le

1003817100381710038171003817100381711987912057611986011989110038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) (38)

119888119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) le

10038171003817100381710038171003817119879120576119860119891100381710038171003817100381710038171198671(R119899119883)

le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) (39)

Chinese Journal of Mathematics 5

Let us assume thatΩ = plusmn1Λ with the product probabil-

ity measure First we consider the function of the form 119891 =

sum119860isin119865

119862119860120593119860for some finite 119865 sub Λ and 119862

119860isin C Integrating

(38) overΩ interchanging the order of integration and usingKhintchinersquos inequality we get

(intΩ

1003817100381710038171003817100381711987912057611986011989110038171003817100381710038171003817

119901

119871119901(R119899119883)

119889120576119860)

1119901

times

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(intΩ

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119860isin119865

120576119860119862119860120593119860

1003816100381610038161003816100381610038161003816100381610038161003816

119901

119889120576119860)

1119901100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

997904rArr 119888119899119901

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(intΩ

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119860isin119865

120576119860119862119860120593119860

1003816100381610038161003816100381610038161003816100381610038161003816

119901

119889120576119860)

1119901100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(40)

In view of (38) we obtain

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883)

(41)

Similarly we get

119888119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

10038161003816100381610038161198621198601003816100381610038161003816210038161003816100381610038161003816R119895120593119860

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899 119883)

le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883)

(42)

Now we take 119891 isin 119871119901(R119899 119883) for some 119865 finite subset ofΛ and

define (]119860)119860isinΛ

by ]119860

= 1 if 119860 isin 119865 and by ]119860

= 0 if 119860 notin 119865 sothat 119879]119860119891 = sum

119860isin119865⟨119891 120593119860⟩120593119860 Applying estimate (41) to 119879]119860119891

and using (34) we get

119888119899119901

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899 119883)le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899 119883)

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899119883)le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

(43)

By virtue of monotone convergence theorem as finite sets 119865

exhaust Λ we get10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) (44)

In order to prove the completeness of (120593119860)119860isinΛ

in119871119901(R119899 119883) consider 119891 isin 119871

119901(R119899 119883) cap 119871

2(R119899 119883) such

that 119891 = suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in1198712(R119899 119883) and there exists a subsequence of partial sums

(sum119873119898

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

)119898isin119873

that converge almost every-where on R119899 By virtue of Fatoursquos lemma and (41) we obtain10038171003817100381710038171003817100381710038171003817100381710038171003817

119891 minus

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le lim inf119898rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119894=119873+1(119895119896119894)isin119865

10038161003816100381610038161003816⟨119891 120593119895119896119894

⟩120593119895119896119894

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

(45)

In view of (44) we have10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin (46)

Using dominated convergence theorem we obtain

119891 =

infin

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

(47)

with convergence in 119871119901(R119899 119883) Therefore span

119871119901(R119899119883)(120593119860

119860 isin Λ) sup 119871119901(R119899 119883)cap119871

2(R119899 119883) since 119871119901(R119899 119883)cap119871

2(R119899 119883)

is dense in 119871119901(R119899 119883) These facts lead to the conclusion

that (120593119860)119860isinΛ

is complete in 119871119901(R119899 119883) By virtue of (35) this

proves that (120593119860)119860isinΛ

is an unconditional basis for 119871119901(R119899 119883)Similarly we can prove the result for 119867

1(R119899 119883) using

(42) Hence the proof is completed

Theorem 6 Let 1 lt 119901 lt infin and let 120593119860be a wavelet such

that 120593 isin C1(R119899) and satisfies condition (26) Then for 119891 isin

119871119901(R119899 119883) one has

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

(48)

and for 119891 isin 1198671(R119899 119883) one has

1198881198991

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899 119883)

le 1198621198991

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

(49)

6 Chinese Journal of Mathematics

Proof We have

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

100381710038171003817100381712059311986010038171003817100381710038171198711199011015840(R119899 119883)

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

10038171003817100381710038171205931198601003817100381710038171003817BMO

(50)

where the space BMO(R119899) is of functions with boundedmean oscillation1198671(R119899) is the predual of BMO(R119899) We seethat ⟨sdot 120593

119860⟩ 119860 isin Λ defines continuous linear functionals

on119871119901(R119899 119883) 1 lt 119901 lt infin and119867

1(R119899 119883) If119891 = sum

119860isinΛ119862119860120593119860

unconditionally in 119871119901(R119899 119883) or 119867

1(R119899 119883) then for every

1198601015840isin Λ by continuity of ⟨sdot 120593

1198601015840 ⟩

⟨119891 1205931198601015840⟩ = sum

119860isinΛ

119862119860⟨120593119860 1205931198601015840⟩ = sum

119860isinΛ

1198621198601205751198601198601015840 = 1198621198601015840 (51)

Now for every 119891 isin 119871119901(R119899 119883) we have 119891 =

suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in 119871119901(R119899 119883)

Using (41) we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) = lim

119899rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(52)

Combining (44) and (52) we get (48) Similarly we wouldhandle the case for 119867

1(R119899 119883) Hence the proof is com-

pleted

Corollary 7 For every family of scalars (119862119860)119860isinΛ

one has that

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 119871119901(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 1198671(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

lt infin

(53)

Proof Suppose that (119862119860)119860isinΛ

is a family of scalars suchthat (sum

119860isinΛ|119862119860|2|120593119860|2)12

119871119901(R119899119883)

lt infin For every120576 gt 0 we can find 119865

0finite subset of Λ such that

(sum119860isinΛ1198650

|119862119860|2|120593119860|2)12

119871119901(R119899 119883)

le 120576Then for every 119865 finite subset of Λ 119865

0 by (44) we have

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

120576 (54)

This implies that the series sum119860isinΛ

119862119860120593119860converges uncondi-

tionally in 119871119901(R119899 119883) to the same 119871

119901-function Moreover wehave proved that

sum

119860isinΛ

119862119860120593119860converges unconditionally in

119871119901(R119899 119883) lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

lt infin

(55)

This gives the characterization of the convergence of waveletseries in 119871

119901(R119899 119883) Similarly we would characterize conver-

gence in1198671(R119899 119883)

Hence the proof is completed

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by University Grants CommissionNew Delhi under the major research Project by F no 41-7922012 (SR)

References

[1] R M Young An Introduction to Nonharmonic Fourier Seriesvol 93 Academic Press New York NY USA 1980

[2] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2001

[3] E M Stein Singular Integrals and Differentiability Propertiesof Functions Princeton University Press Princeton NJ USA1970

[4] J Mateu and J Verdera ldquo119871119901 and weak 1198711 estimates for the

maximal Riesz transform and themaximal Beurling transformrdquoMathematical Research Letters vol 13 no 5-6 pp 957ndash9662006

[5] C Thiele Wave Packet Analysis vol 105 of CBMS RegionalConference Series in Mathematics 2006

[6] D L Burkholder ldquoMartingales and Fourier analysis in Banachspacesrdquo in Probability and Analysis G Letta and M PratelliEds vol 1206 of Lecture Notes in Mathematics pp 61ndash108Springer Berlin Germany 1986

[7] T Hytonen ldquoVector-valued wavelets and the Hardy space1198671(R119899 119883)rdquo Studia Mathematica vol 172 no 2 pp 125ndash147

2006[8] T Figiel ldquoSingular integral operators a martingale approachrdquo

in Geometry of Banach Spaces P F X Moller and W Schacher-mayer Eds vol 158 of London Mathematical Society LectureNote Series pp 95ndash110 Cambridge University Press Cam-bridge Mass USA 1990

[9] E M SteinHarmonic Analysis Real-Variable Methods Orthog-onality and Oscillatory Integrals vol 43 Princeton UniversityPress Princeton NJ USA 1993

[10] E Hernandez and G Weiss A First Course on Wavelets CRCPress Boca Raton Fla USA 1996

Chinese Journal of Mathematics 7

[11] YMeyerWavelets and Operators vol 37 Cambridge UniversityPress Cambridge Mass USA 1992

[12] P Wojtaszczyk ldquoThe Franklin system is an unconditional basisin1198671rdquo Arkiv for Matematik vol 20 no 2 pp 293ndash300 1982

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Stochastic AnalysisInternational Journal of

Chinese Journal of Mathematics 3

It should be significant to mention here that it would bebetter to obtain constants depending only on 119901 and regularityof wavelet but no other information about it For that reasona better approach is to prove square function estimates usingCalderon-Zygmund decomposition and real interpolation[5]

A Calderon-Zygmund operator is a bounded linear oper-ator 119879

lowast 1198712(R119899) rarr 119871

2(R119899) with operator norm at most 119896

and such that there exists aC1-function 119896(119909 119910) isin 1198711(R119899timesR119899)

satisfying1003816100381610038161003816119896 (119909 119910)

1003816100381610038161003816 le 1198961003816100381610038161003816119909 minus 119910

1003816100381610038161003816minus119899

1003816100381610038161003816nabla119909119896 (119909 119910)1003816100381610038161003816 +

10038161003816100381610038161003816nabla119910119896 (119909 119910)

10038161003816100381610038161003816le 119896

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

119879lowast119891 (119909) = int

R119899119896 (119909 119910) 119891 (119910) 119889119910

for119891 isin 1198712

119888(R119899) 119909 notin supp (119891)

(19)

A function with these properties is called a Calderon-Zygmund kernel or a standard singular kernel

2 Some Definitions and Auxiliary Results

Definition 2 ABanach space119883 is UMD if for some (and thenall cf [6]) 1 lt 119901 lt infin there is a finite constant 119862 so that

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119896=1

120576119896119889119896

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(Ω119883)

le 119862

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119896=1

119889119896

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(Ω119883)

(20)

for all 119899 isin Z+ whenever (120576

119896)119899

119896=1isin minus1 +1

119899 and (119889119896)119899

119896=1isin

119871119901(Ω119883)

119899 a martingale difference sequence or an arbitraryprobability space (Ω 119860 120583) (ie there are sub-120590-algebras119860

0sub

1198601

sub sdot sdot sdot sub 119860119899

sub 119860) such that for all 119896 = 1 119899 thefunction 119889

119896is 119860119896-measurable and int

119860119889119896119889120583 = 0 for all 119860 isin

119860119896minus1

Remark 3 The validity of Theorem 19 of [7] on 1198671(R119899 119883)

actually characterizes the UMD-property of119883 Indeed let119883be any complex Banach space and let conclusion of Theorem19 of [7] be satisfied Since R

119895119891119871119901(R119899 119883)

le 119862119891119871119901(R119899119883)

this is equivalent to the UMD-property of 119883 We also haveR119895119891 le 119862119891

1198711(R119899 119883) This implies that R

119895is bounded on

1198671(R119899 119883)

In view of above remark we can get without modificationin [8] the following

Proposition 4 Let 119896(119909 119910) isin 1198711(R119899timesR119899) satisfy the standard

estimates1003816100381610038161003816119896 (119909 119910)

1003816100381610038161003816 le 1198961003816100381610038161003816119909 minus 119910

1003816100381610038161003816minus119899

1003816100381610038161003816nabla119909119896 (119909 119910)1003816100381610038161003816 +

10038161003816100381610038161003816nabla119910119896 (119909 119910)

10038161003816100381610038161003816le 119896

1003816100381610038161003816119909 1199101003816100381610038161003816minus119899minus1

(21)

Assume moreover that 119879 is bounded on 1198712(R119899) with

operator norm at most 119896 Then 119879 is also bounded on119871119901(R119899 119883) where

10038171003817100381710038171003817R11989511989110038171003817100381710038171003817119871119901(R119899 119883)

le 119862119901119899

(119883) 1198961003817100381710038171003817119891

1003817100381710038171003817119871119901(R119899119883) (22)

for all 119901 isin ]1infin[ and it is bounded from 1198671(R119899 119883) to

1198711(R119899 119883) with norm 119862

1119899(119883)119896 If in addition

[1198791] (119910) = intR119899

119896 (119909 119910) 119889119909 equiv 0 (23)

then 119879 is bounded on1198671(R119899 119883) with norm le 119862

0(119883)119896

Let us note that atomic definition of1198671(R119899 119883) is knownto agree with the one given in terms of various maximalfunctions even for an arbitrary Banach space 119883 One cancheck that the proof of this fact in the scalar case as givenfor example in Steinrsquos book [9] goes through word by wordin the general setting for 119899 = 1 the conjugate Hardy spacedefined as the domain of the Hilbert transform on 119871

1(R 119883)

with the graph norm is always contained in the atomic Hardyspace and agrees with it exactly when119883 is a UMD-space For119899 gt 1 we can replace Hilbert transform by Riesz transformand the condition of UMD-space for 119883 by the boundednessof Riesz transform in 119871

119901(R119899)-space

It is well-known fact that an integral operator satisfyingthe standard estimates and bounded on 119871

119901(R119899 119883) is also

bounded from 1198671(R119899 119883) to 119871

1(R119899 119883) The square function

description of 1198671(R119899 119883) that we have in mind involves thewave expansion of a function in119867

1(R119899) Recall the definition

of the wavelet [10ndash12] Let 120593119897 119897 = 1 2 2119899minus 1 be a set of

functions belonging to 1198712(R119899)

Consider

120593119897

119895119896(119909) = 2

1198951198992120593119897(2119895119909 minus 119896)

= 21198951198992

120593119897(21198951199091minus 1198961 2

119895119909119899minus 119896119899)

(119909 = (1199091 119909

119899) isin R119899)

(24)

for each 119897 = 1 2 2119899minus 1 119895 isin Z and 119896 = (119896

1 119896

119899) isin Z119899

The sequence 120593119897 119897 = 1 2 2119899minus 1 is a wavelet set if 120593119897

119895119896

119897 = 1 2 2119899minus 1 119895 isin Z 119896 isin Z119899 forms an orthonormal

basis in 1198712(R119899) Then 120593

119897

119895119896 119897 = 1 2 2

119899minus 1 119895 isin Z 119896 isin

Z119899 is a wavelet basis in 1198712(R119899) and each 120593

119897 is a waveletThe basis is called 119903-regular if |120597

120572120593119897(119909)| le 119862

119898(1 + |119909|)

minus119898

and int119909120572120593119897(119909)119889119909 = 0 for all |120572| le 119903 all 119898 isin 119873 and

119897 = 1 2 2119899minus 1

The aim of the present paper is to study that reason-able nice wavelets form unconditional bases in functionspace other than 119871

2(R119899) and characterization of convergence

of wavelets series in 119871119901(R119899 119883) and 119867

1(R119899 119883) has been

obtained For this purpose it will be enough to assume that(120593119860)119860isinΛ

where Λ is the set of dyadic cube of the form

119876119895119896

= Π119899

119894=1[2minus119895119896119894 2minus119895

(119896119894+ 1)] (25)

is an orthonormal wavelet basis such that 120593 isin C1(R119899) and

1003816100381610038161003816120593 (119909)1003816100381610038161003816 +

100381610038161003816100381610038161205931015840(119909)

10038161003816100381610038161003816le 119862(1 + |119909|)

minus119899minus2 (26)

This is certainly not the weakest possible assumption on 120593 forsomewhat less restrictive see [10] (where the class R0(R119899) isintroduced)

4 Chinese Journal of Mathematics

3 Main Results

Theorem 5 Let 1 lt 119901 lt infin and let 120593119860

be a waveletsuch that 120593 isin C1(R119899) and satisfies condition (26) Then thesystem (120593

119860)119860isinΛ

is an unconditional basis for 119871119901(R119899 119883) and

1198671(R119899 119883)

Proof For arbitrary family of ]119860

isin C 119860 isin Λ such that |]119860| le

1 we define 119879]119860 on the orthonormal basis by

119879]119860 1198712(R119899 119883) 997888rarr 119871

2(R119899 119883) 119879]119860 120593

119860997888rarr ]119860120593119860

(27)

Here 119879]119860 is not necessarily unitary but it is bounded and

10038171003817100381710038171003817119879]119860

100381710038171003817100381710038171198712rarr1198712le 1 (28)

Now we can write

(119879]119860119891) (119909) = sum

119860isinΛ

]119860⟨119891 120593119860⟩ 120593119860(119909)

= intR119899

119896]119860 (119909 119910) 119891 (119910) 119889119910

(29)

where

119896]119860 (119909 119910) = sum

119860isin119891

]119860120593119860(119909) 120593119860(119910)

= sum

119895isinZ119896isinZ119899

]119895119896

2119899119895120593 (2119895119909 minus 119896) 120593 (2119895119910 minus 119896)

(30)

where 119865 sub Λ is any finite setWe claim that 119879]119860 are Calderon-Zygmund operators with

constants uniform in ]119860and for that we have to show that 119896]119860

satisfy standard estimates uniformly in ]119860

Consider

10038161003816100381610038161003816119896]119860 (119909 119910)

10038161003816100381610038161003816le sum

119895isinZ119896isinZ119899

2119899119895(1 +

100381610038161003816100381610038162119895119909 minus 119896

10038161003816100381610038161003816)minus119899minus2

times (1 +100381610038161003816100381610038162119895119910 minus 119896

10038161003816100381610038161003816)minus119899minus2

le sum

119895isinZ

1198621198992119899119895(1 + 2

119895 1003816100381610038161003816119909 minus 1199101003816100381610038161003816)minus119899minus2

le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899

(31)

Also for derivatives

nabla119909119896]119860 (119909 119910)

= sum

119895isinZ119896isinZ119899

]119895119896

2119899119895+2119895

1205931015840(2119895119909 minus 119896) 120593(2119895119910 minus 119896)

10038161003816100381610038161003816nabla119909119896]119860 (119909 119910)

10038161003816100381610038161003816

le sum

119895isinZ119896isinZ119899

2119899119895+2119895

(1 +100381610038161003816100381610038162119895119909 minus 119896

10038161003816100381610038161003816)minus119899minus2

(1 +100381610038161003816100381610038162119895119910 minus 119896

10038161003816100381610038161003816)minus119899minus2

le sum

119895isinZ

1198621198992119899119895+2119895

(1 + 2119895 1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)minus119899minus2

le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

(32)

So we have10038161003816100381610038161003816nabla119909119896]119860 (119909 119910)

10038161003816100381610038161003816+10038161003816100381610038161003816nabla119910119896]119860 (119909 119910)

10038161003816100381610038161003816le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

(33)

Condition intR119899

119891 = 0 rArr intR119899

119879]119860119891 = 0 follows fromintR119899

119896]119860(119909 119910)119889119909 = 0 which in turn follows from intR119899

120593(119909) = 0Therefore from Proposition 4 we know that operators 119879]119860extend continuously to 119871

119901(R119899 119883) and119867

1(R119899 119883) and we get

inequalities

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899119883)le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883)

10038171003817100381710038171003817119879]119860119891

100381710038171003817100381710038171198671(R119899 119883)le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883)

(34)

and particularly for 119891 = sum119860isin119891

119862119860120593119860 119865 is a finite subset of Λ

119862119860

isin 119862 can be written as

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

]119860119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(35)

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

]119860119862119860120593119860

10038171003817100381710038171003817100381710038171003817100381710038171198671(R119899 119883)

le 119862119899

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

10038171003817100381710038171003817100381710038171003817100381710038171198671(R119899 119883)

(36)

These inequalities imply that (120593119860)119860isinΛ

is an unconditionalbasis for

span119871119901(R119899119883) (120593119860 119860 isin Λ)

span1198671(R119899 119883) (120593119860 119860 isin Λ)

(37)

However it is still not clear that (120593119860)119860isinΛ

is complete in119871119901(R119899 119883) and119867

1(R119899 119883)

In particular if we take (120576119860)119860isinΛ

isin plusmn1Λ then (34) and

1198792

120576119860imply that

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) le

1003817100381710038171003817100381711987912057611986011989110038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) (38)

119888119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) le

10038171003817100381710038171003817119879120576119860119891100381710038171003817100381710038171198671(R119899119883)

le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) (39)

Chinese Journal of Mathematics 5

Let us assume thatΩ = plusmn1Λ with the product probabil-

ity measure First we consider the function of the form 119891 =

sum119860isin119865

119862119860120593119860for some finite 119865 sub Λ and 119862

119860isin C Integrating

(38) overΩ interchanging the order of integration and usingKhintchinersquos inequality we get

(intΩ

1003817100381710038171003817100381711987912057611986011989110038171003817100381710038171003817

119901

119871119901(R119899119883)

119889120576119860)

1119901

times

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(intΩ

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119860isin119865

120576119860119862119860120593119860

1003816100381610038161003816100381610038161003816100381610038161003816

119901

119889120576119860)

1119901100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

997904rArr 119888119899119901

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(intΩ

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119860isin119865

120576119860119862119860120593119860

1003816100381610038161003816100381610038161003816100381610038161003816

119901

119889120576119860)

1119901100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(40)

In view of (38) we obtain

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883)

(41)

Similarly we get

119888119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

10038161003816100381610038161198621198601003816100381610038161003816210038161003816100381610038161003816R119895120593119860

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899 119883)

le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883)

(42)

Now we take 119891 isin 119871119901(R119899 119883) for some 119865 finite subset ofΛ and

define (]119860)119860isinΛ

by ]119860

= 1 if 119860 isin 119865 and by ]119860

= 0 if 119860 notin 119865 sothat 119879]119860119891 = sum

119860isin119865⟨119891 120593119860⟩120593119860 Applying estimate (41) to 119879]119860119891

and using (34) we get

119888119899119901

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899 119883)le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899 119883)

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899119883)le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

(43)

By virtue of monotone convergence theorem as finite sets 119865

exhaust Λ we get10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) (44)

In order to prove the completeness of (120593119860)119860isinΛ

in119871119901(R119899 119883) consider 119891 isin 119871

119901(R119899 119883) cap 119871

2(R119899 119883) such

that 119891 = suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in1198712(R119899 119883) and there exists a subsequence of partial sums

(sum119873119898

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

)119898isin119873

that converge almost every-where on R119899 By virtue of Fatoursquos lemma and (41) we obtain10038171003817100381710038171003817100381710038171003817100381710038171003817

119891 minus

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le lim inf119898rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119894=119873+1(119895119896119894)isin119865

10038161003816100381610038161003816⟨119891 120593119895119896119894

⟩120593119895119896119894

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

(45)

In view of (44) we have10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin (46)

Using dominated convergence theorem we obtain

119891 =

infin

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

(47)

with convergence in 119871119901(R119899 119883) Therefore span

119871119901(R119899119883)(120593119860

119860 isin Λ) sup 119871119901(R119899 119883)cap119871

2(R119899 119883) since 119871119901(R119899 119883)cap119871

2(R119899 119883)

is dense in 119871119901(R119899 119883) These facts lead to the conclusion

that (120593119860)119860isinΛ

is complete in 119871119901(R119899 119883) By virtue of (35) this

proves that (120593119860)119860isinΛ

is an unconditional basis for 119871119901(R119899 119883)Similarly we can prove the result for 119867

1(R119899 119883) using

(42) Hence the proof is completed

Theorem 6 Let 1 lt 119901 lt infin and let 120593119860be a wavelet such

that 120593 isin C1(R119899) and satisfies condition (26) Then for 119891 isin

119871119901(R119899 119883) one has

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

(48)

and for 119891 isin 1198671(R119899 119883) one has

1198881198991

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899 119883)

le 1198621198991

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

(49)

6 Chinese Journal of Mathematics

Proof We have

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

100381710038171003817100381712059311986010038171003817100381710038171198711199011015840(R119899 119883)

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

10038171003817100381710038171205931198601003817100381710038171003817BMO

(50)

where the space BMO(R119899) is of functions with boundedmean oscillation1198671(R119899) is the predual of BMO(R119899) We seethat ⟨sdot 120593

119860⟩ 119860 isin Λ defines continuous linear functionals

on119871119901(R119899 119883) 1 lt 119901 lt infin and119867

1(R119899 119883) If119891 = sum

119860isinΛ119862119860120593119860

unconditionally in 119871119901(R119899 119883) or 119867

1(R119899 119883) then for every

1198601015840isin Λ by continuity of ⟨sdot 120593

1198601015840 ⟩

⟨119891 1205931198601015840⟩ = sum

119860isinΛ

119862119860⟨120593119860 1205931198601015840⟩ = sum

119860isinΛ

1198621198601205751198601198601015840 = 1198621198601015840 (51)

Now for every 119891 isin 119871119901(R119899 119883) we have 119891 =

suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in 119871119901(R119899 119883)

Using (41) we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) = lim

119899rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(52)

Combining (44) and (52) we get (48) Similarly we wouldhandle the case for 119867

1(R119899 119883) Hence the proof is com-

pleted

Corollary 7 For every family of scalars (119862119860)119860isinΛ

one has that

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 119871119901(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 1198671(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

lt infin

(53)

Proof Suppose that (119862119860)119860isinΛ

is a family of scalars suchthat (sum

119860isinΛ|119862119860|2|120593119860|2)12

119871119901(R119899119883)

lt infin For every120576 gt 0 we can find 119865

0finite subset of Λ such that

(sum119860isinΛ1198650

|119862119860|2|120593119860|2)12

119871119901(R119899 119883)

le 120576Then for every 119865 finite subset of Λ 119865

0 by (44) we have

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

120576 (54)

This implies that the series sum119860isinΛ

119862119860120593119860converges uncondi-

tionally in 119871119901(R119899 119883) to the same 119871

119901-function Moreover wehave proved that

sum

119860isinΛ

119862119860120593119860converges unconditionally in

119871119901(R119899 119883) lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

lt infin

(55)

This gives the characterization of the convergence of waveletseries in 119871

119901(R119899 119883) Similarly we would characterize conver-

gence in1198671(R119899 119883)

Hence the proof is completed

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by University Grants CommissionNew Delhi under the major research Project by F no 41-7922012 (SR)

References

[1] R M Young An Introduction to Nonharmonic Fourier Seriesvol 93 Academic Press New York NY USA 1980

[2] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2001

[3] E M Stein Singular Integrals and Differentiability Propertiesof Functions Princeton University Press Princeton NJ USA1970

[4] J Mateu and J Verdera ldquo119871119901 and weak 1198711 estimates for the

maximal Riesz transform and themaximal Beurling transformrdquoMathematical Research Letters vol 13 no 5-6 pp 957ndash9662006

[5] C Thiele Wave Packet Analysis vol 105 of CBMS RegionalConference Series in Mathematics 2006

[6] D L Burkholder ldquoMartingales and Fourier analysis in Banachspacesrdquo in Probability and Analysis G Letta and M PratelliEds vol 1206 of Lecture Notes in Mathematics pp 61ndash108Springer Berlin Germany 1986

[7] T Hytonen ldquoVector-valued wavelets and the Hardy space1198671(R119899 119883)rdquo Studia Mathematica vol 172 no 2 pp 125ndash147

2006[8] T Figiel ldquoSingular integral operators a martingale approachrdquo

in Geometry of Banach Spaces P F X Moller and W Schacher-mayer Eds vol 158 of London Mathematical Society LectureNote Series pp 95ndash110 Cambridge University Press Cam-bridge Mass USA 1990

[9] E M SteinHarmonic Analysis Real-Variable Methods Orthog-onality and Oscillatory Integrals vol 43 Princeton UniversityPress Princeton NJ USA 1993

[10] E Hernandez and G Weiss A First Course on Wavelets CRCPress Boca Raton Fla USA 1996

Chinese Journal of Mathematics 7

[11] YMeyerWavelets and Operators vol 37 Cambridge UniversityPress Cambridge Mass USA 1992

[12] P Wojtaszczyk ldquoThe Franklin system is an unconditional basisin1198671rdquo Arkiv for Matematik vol 20 no 2 pp 293ndash300 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Chinese Journal of Mathematics

3 Main Results

Theorem 5 Let 1 lt 119901 lt infin and let 120593119860

be a waveletsuch that 120593 isin C1(R119899) and satisfies condition (26) Then thesystem (120593

119860)119860isinΛ

is an unconditional basis for 119871119901(R119899 119883) and

1198671(R119899 119883)

Proof For arbitrary family of ]119860

isin C 119860 isin Λ such that |]119860| le

1 we define 119879]119860 on the orthonormal basis by

119879]119860 1198712(R119899 119883) 997888rarr 119871

2(R119899 119883) 119879]119860 120593

119860997888rarr ]119860120593119860

(27)

Here 119879]119860 is not necessarily unitary but it is bounded and

10038171003817100381710038171003817119879]119860

100381710038171003817100381710038171198712rarr1198712le 1 (28)

Now we can write

(119879]119860119891) (119909) = sum

119860isinΛ

]119860⟨119891 120593119860⟩ 120593119860(119909)

= intR119899

119896]119860 (119909 119910) 119891 (119910) 119889119910

(29)

where

119896]119860 (119909 119910) = sum

119860isin119891

]119860120593119860(119909) 120593119860(119910)

= sum

119895isinZ119896isinZ119899

]119895119896

2119899119895120593 (2119895119909 minus 119896) 120593 (2119895119910 minus 119896)

(30)

where 119865 sub Λ is any finite setWe claim that 119879]119860 are Calderon-Zygmund operators with

constants uniform in ]119860and for that we have to show that 119896]119860

satisfy standard estimates uniformly in ]119860

Consider

10038161003816100381610038161003816119896]119860 (119909 119910)

10038161003816100381610038161003816le sum

119895isinZ119896isinZ119899

2119899119895(1 +

100381610038161003816100381610038162119895119909 minus 119896

10038161003816100381610038161003816)minus119899minus2

times (1 +100381610038161003816100381610038162119895119910 minus 119896

10038161003816100381610038161003816)minus119899minus2

le sum

119895isinZ

1198621198992119899119895(1 + 2

119895 1003816100381610038161003816119909 minus 1199101003816100381610038161003816)minus119899minus2

le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899

(31)

Also for derivatives

nabla119909119896]119860 (119909 119910)

= sum

119895isinZ119896isinZ119899

]119895119896

2119899119895+2119895

1205931015840(2119895119909 minus 119896) 120593(2119895119910 minus 119896)

10038161003816100381610038161003816nabla119909119896]119860 (119909 119910)

10038161003816100381610038161003816

le sum

119895isinZ119896isinZ119899

2119899119895+2119895

(1 +100381610038161003816100381610038162119895119909 minus 119896

10038161003816100381610038161003816)minus119899minus2

(1 +100381610038161003816100381610038162119895119910 minus 119896

10038161003816100381610038161003816)minus119899minus2

le sum

119895isinZ

1198621198992119899119895+2119895

(1 + 2119895 1003816100381610038161003816119909 minus 119910

1003816100381610038161003816)minus119899minus2

le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

(32)

So we have10038161003816100381610038161003816nabla119909119896]119860 (119909 119910)

10038161003816100381610038161003816+10038161003816100381610038161003816nabla119910119896]119860 (119909 119910)

10038161003816100381610038161003816le 119862119899

1003816100381610038161003816119909 minus 1199101003816100381610038161003816minus119899minus1

(33)

Condition intR119899

119891 = 0 rArr intR119899

119879]119860119891 = 0 follows fromintR119899

119896]119860(119909 119910)119889119909 = 0 which in turn follows from intR119899

120593(119909) = 0Therefore from Proposition 4 we know that operators 119879]119860extend continuously to 119871

119901(R119899 119883) and119867

1(R119899 119883) and we get

inequalities

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899119883)le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883)

10038171003817100381710038171003817119879]119860119891

100381710038171003817100381710038171198671(R119899 119883)le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883)

(34)

and particularly for 119891 = sum119860isin119891

119862119860120593119860 119865 is a finite subset of Λ

119862119860

isin 119862 can be written as

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

]119860119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(35)

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

]119860119862119860120593119860

10038171003817100381710038171003817100381710038171003817100381710038171198671(R119899 119883)

le 119862119899

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

10038171003817100381710038171003817100381710038171003817100381710038171198671(R119899 119883)

(36)

These inequalities imply that (120593119860)119860isinΛ

is an unconditionalbasis for

span119871119901(R119899119883) (120593119860 119860 isin Λ)

span1198671(R119899 119883) (120593119860 119860 isin Λ)

(37)

However it is still not clear that (120593119860)119860isinΛ

is complete in119871119901(R119899 119883) and119867

1(R119899 119883)

In particular if we take (120576119860)119860isinΛ

isin plusmn1Λ then (34) and

1198792

120576119860imply that

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) le

1003817100381710038171003817100381711987912057611986011989110038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) (38)

119888119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) le

10038171003817100381710038171003817119879120576119860119891100381710038171003817100381710038171198671(R119899119883)

le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) (39)

Chinese Journal of Mathematics 5

Let us assume thatΩ = plusmn1Λ with the product probabil-

ity measure First we consider the function of the form 119891 =

sum119860isin119865

119862119860120593119860for some finite 119865 sub Λ and 119862

119860isin C Integrating

(38) overΩ interchanging the order of integration and usingKhintchinersquos inequality we get

(intΩ

1003817100381710038171003817100381711987912057611986011989110038171003817100381710038171003817

119901

119871119901(R119899119883)

119889120576119860)

1119901

times

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(intΩ

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119860isin119865

120576119860119862119860120593119860

1003816100381610038161003816100381610038161003816100381610038161003816

119901

119889120576119860)

1119901100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

997904rArr 119888119899119901

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(intΩ

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119860isin119865

120576119860119862119860120593119860

1003816100381610038161003816100381610038161003816100381610038161003816

119901

119889120576119860)

1119901100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(40)

In view of (38) we obtain

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883)

(41)

Similarly we get

119888119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

10038161003816100381610038161198621198601003816100381610038161003816210038161003816100381610038161003816R119895120593119860

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899 119883)

le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883)

(42)

Now we take 119891 isin 119871119901(R119899 119883) for some 119865 finite subset ofΛ and

define (]119860)119860isinΛ

by ]119860

= 1 if 119860 isin 119865 and by ]119860

= 0 if 119860 notin 119865 sothat 119879]119860119891 = sum

119860isin119865⟨119891 120593119860⟩120593119860 Applying estimate (41) to 119879]119860119891

and using (34) we get

119888119899119901

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899 119883)le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899 119883)

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899119883)le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

(43)

By virtue of monotone convergence theorem as finite sets 119865

exhaust Λ we get10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) (44)

In order to prove the completeness of (120593119860)119860isinΛ

in119871119901(R119899 119883) consider 119891 isin 119871

119901(R119899 119883) cap 119871

2(R119899 119883) such

that 119891 = suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in1198712(R119899 119883) and there exists a subsequence of partial sums

(sum119873119898

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

)119898isin119873

that converge almost every-where on R119899 By virtue of Fatoursquos lemma and (41) we obtain10038171003817100381710038171003817100381710038171003817100381710038171003817

119891 minus

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le lim inf119898rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119894=119873+1(119895119896119894)isin119865

10038161003816100381610038161003816⟨119891 120593119895119896119894

⟩120593119895119896119894

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

(45)

In view of (44) we have10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin (46)

Using dominated convergence theorem we obtain

119891 =

infin

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

(47)

with convergence in 119871119901(R119899 119883) Therefore span

119871119901(R119899119883)(120593119860

119860 isin Λ) sup 119871119901(R119899 119883)cap119871

2(R119899 119883) since 119871119901(R119899 119883)cap119871

2(R119899 119883)

is dense in 119871119901(R119899 119883) These facts lead to the conclusion

that (120593119860)119860isinΛ

is complete in 119871119901(R119899 119883) By virtue of (35) this

proves that (120593119860)119860isinΛ

is an unconditional basis for 119871119901(R119899 119883)Similarly we can prove the result for 119867

1(R119899 119883) using

(42) Hence the proof is completed

Theorem 6 Let 1 lt 119901 lt infin and let 120593119860be a wavelet such

that 120593 isin C1(R119899) and satisfies condition (26) Then for 119891 isin

119871119901(R119899 119883) one has

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

(48)

and for 119891 isin 1198671(R119899 119883) one has

1198881198991

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899 119883)

le 1198621198991

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

(49)

6 Chinese Journal of Mathematics

Proof We have

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

100381710038171003817100381712059311986010038171003817100381710038171198711199011015840(R119899 119883)

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

10038171003817100381710038171205931198601003817100381710038171003817BMO

(50)

where the space BMO(R119899) is of functions with boundedmean oscillation1198671(R119899) is the predual of BMO(R119899) We seethat ⟨sdot 120593

119860⟩ 119860 isin Λ defines continuous linear functionals

on119871119901(R119899 119883) 1 lt 119901 lt infin and119867

1(R119899 119883) If119891 = sum

119860isinΛ119862119860120593119860

unconditionally in 119871119901(R119899 119883) or 119867

1(R119899 119883) then for every

1198601015840isin Λ by continuity of ⟨sdot 120593

1198601015840 ⟩

⟨119891 1205931198601015840⟩ = sum

119860isinΛ

119862119860⟨120593119860 1205931198601015840⟩ = sum

119860isinΛ

1198621198601205751198601198601015840 = 1198621198601015840 (51)

Now for every 119891 isin 119871119901(R119899 119883) we have 119891 =

suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in 119871119901(R119899 119883)

Using (41) we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) = lim

119899rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(52)

Combining (44) and (52) we get (48) Similarly we wouldhandle the case for 119867

1(R119899 119883) Hence the proof is com-

pleted

Corollary 7 For every family of scalars (119862119860)119860isinΛ

one has that

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 119871119901(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 1198671(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

lt infin

(53)

Proof Suppose that (119862119860)119860isinΛ

is a family of scalars suchthat (sum

119860isinΛ|119862119860|2|120593119860|2)12

119871119901(R119899119883)

lt infin For every120576 gt 0 we can find 119865

0finite subset of Λ such that

(sum119860isinΛ1198650

|119862119860|2|120593119860|2)12

119871119901(R119899 119883)

le 120576Then for every 119865 finite subset of Λ 119865

0 by (44) we have

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

120576 (54)

This implies that the series sum119860isinΛ

119862119860120593119860converges uncondi-

tionally in 119871119901(R119899 119883) to the same 119871

119901-function Moreover wehave proved that

sum

119860isinΛ

119862119860120593119860converges unconditionally in

119871119901(R119899 119883) lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

lt infin

(55)

This gives the characterization of the convergence of waveletseries in 119871

119901(R119899 119883) Similarly we would characterize conver-

gence in1198671(R119899 119883)

Hence the proof is completed

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by University Grants CommissionNew Delhi under the major research Project by F no 41-7922012 (SR)

References

[1] R M Young An Introduction to Nonharmonic Fourier Seriesvol 93 Academic Press New York NY USA 1980

[2] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2001

[3] E M Stein Singular Integrals and Differentiability Propertiesof Functions Princeton University Press Princeton NJ USA1970

[4] J Mateu and J Verdera ldquo119871119901 and weak 1198711 estimates for the

maximal Riesz transform and themaximal Beurling transformrdquoMathematical Research Letters vol 13 no 5-6 pp 957ndash9662006

[5] C Thiele Wave Packet Analysis vol 105 of CBMS RegionalConference Series in Mathematics 2006

[6] D L Burkholder ldquoMartingales and Fourier analysis in Banachspacesrdquo in Probability and Analysis G Letta and M PratelliEds vol 1206 of Lecture Notes in Mathematics pp 61ndash108Springer Berlin Germany 1986

[7] T Hytonen ldquoVector-valued wavelets and the Hardy space1198671(R119899 119883)rdquo Studia Mathematica vol 172 no 2 pp 125ndash147

2006[8] T Figiel ldquoSingular integral operators a martingale approachrdquo

in Geometry of Banach Spaces P F X Moller and W Schacher-mayer Eds vol 158 of London Mathematical Society LectureNote Series pp 95ndash110 Cambridge University Press Cam-bridge Mass USA 1990

[9] E M SteinHarmonic Analysis Real-Variable Methods Orthog-onality and Oscillatory Integrals vol 43 Princeton UniversityPress Princeton NJ USA 1993

[10] E Hernandez and G Weiss A First Course on Wavelets CRCPress Boca Raton Fla USA 1996

Chinese Journal of Mathematics 7

[11] YMeyerWavelets and Operators vol 37 Cambridge UniversityPress Cambridge Mass USA 1992

[12] P Wojtaszczyk ldquoThe Franklin system is an unconditional basisin1198671rdquo Arkiv for Matematik vol 20 no 2 pp 293ndash300 1982

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Chinese Journal of Mathematics 5

Let us assume thatΩ = plusmn1Λ with the product probabil-

ity measure First we consider the function of the form 119891 =

sum119860isin119865

119862119860120593119860for some finite 119865 sub Λ and 119862

119860isin C Integrating

(38) overΩ interchanging the order of integration and usingKhintchinersquos inequality we get

(intΩ

1003817100381710038171003817100381711987912057611986011989110038171003817100381710038171003817

119901

119871119901(R119899119883)

119889120576119860)

1119901

times

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(intΩ

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119860isin119865

120576119860119862119860120593119860

1003816100381610038161003816100381610038161003816100381610038161003816

119901

119889120576119860)

1119901100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

997904rArr 119888119899119901

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(intΩ

1003816100381610038161003816100381610038161003816100381610038161003816

sum

119860isin119865

120576119860119862119860120593119860

1003816100381610038161003816100381610038161003816100381610038161003816

119901

119889120576119860)

1119901100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(40)

In view of (38) we obtain

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883)

(41)

Similarly we get

119888119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

+

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

10038161003816100381610038161198621198601003816100381610038161003816210038161003816100381610038161003816R119895120593119860

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899 119883)

le 119862119899

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883)

(42)

Now we take 119891 isin 119871119901(R119899 119883) for some 119865 finite subset ofΛ and

define (]119860)119860isinΛ

by ]119860

= 1 if 119860 isin 119865 and by ]119860

= 0 if 119860 notin 119865 sothat 119879]119860119891 = sum

119860isin119865⟨119891 120593119860⟩120593119860 Applying estimate (41) to 119879]119860119891

and using (34) we get

119888119899119901

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899 119883)le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899 119883)

10038171003817100381710038171003817119879]119860119891

10038171003817100381710038171003817119871119901(R119899119883)le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

(43)

By virtue of monotone convergence theorem as finite sets 119865

exhaust Λ we get10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899119883) (44)

In order to prove the completeness of (120593119860)119860isinΛ

in119871119901(R119899 119883) consider 119891 isin 119871

119901(R119899 119883) cap 119871

2(R119899 119883) such

that 119891 = suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in1198712(R119899 119883) and there exists a subsequence of partial sums

(sum119873119898

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

)119898isin119873

that converge almost every-where on R119899 By virtue of Fatoursquos lemma and (41) we obtain10038171003817100381710038171003817100381710038171003817100381710038171003817

119891 minus

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le lim inf119898rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873119898

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119894=119873+1(119895119896119894)isin119865

10038161003816100381610038161003816⟨119891 120593119895119896119894

⟩120593119895119896119894

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

(45)

In view of (44) we have10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isin119865

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin (46)

Using dominated convergence theorem we obtain

119891 =

infin

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

(47)

with convergence in 119871119901(R119899 119883) Therefore span

119871119901(R119899119883)(120593119860

119860 isin Λ) sup 119871119901(R119899 119883)cap119871

2(R119899 119883) since 119871119901(R119899 119883)cap119871

2(R119899 119883)

is dense in 119871119901(R119899 119883) These facts lead to the conclusion

that (120593119860)119860isinΛ

is complete in 119871119901(R119899 119883) By virtue of (35) this

proves that (120593119860)119860isinΛ

is an unconditional basis for 119871119901(R119899 119883)Similarly we can prove the result for 119867

1(R119899 119883) using

(42) Hence the proof is completed

Theorem 6 Let 1 lt 119901 lt infin and let 120593119860be a wavelet such

that 120593 isin C1(R119899) and satisfies condition (26) Then for 119891 isin

119871119901(R119899 119883) one has

119888119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

(48)

and for 119891 isin 1198671(R119899 119883) one has

1198881198991

100381710038171003817100381711989110038171003817100381710038171198671(R119899119883) le

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899 119883)

le 1198621198991

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

(49)

6 Chinese Journal of Mathematics

Proof We have

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

100381710038171003817100381712059311986010038171003817100381710038171198711199011015840(R119899 119883)

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

10038171003817100381710038171205931198601003817100381710038171003817BMO

(50)

where the space BMO(R119899) is of functions with boundedmean oscillation1198671(R119899) is the predual of BMO(R119899) We seethat ⟨sdot 120593

119860⟩ 119860 isin Λ defines continuous linear functionals

on119871119901(R119899 119883) 1 lt 119901 lt infin and119867

1(R119899 119883) If119891 = sum

119860isinΛ119862119860120593119860

unconditionally in 119871119901(R119899 119883) or 119867

1(R119899 119883) then for every

1198601015840isin Λ by continuity of ⟨sdot 120593

1198601015840 ⟩

⟨119891 1205931198601015840⟩ = sum

119860isinΛ

119862119860⟨120593119860 1205931198601015840⟩ = sum

119860isinΛ

1198621198601205751198601198601015840 = 1198621198601015840 (51)

Now for every 119891 isin 119871119901(R119899 119883) we have 119891 =

suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in 119871119901(R119899 119883)

Using (41) we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) = lim

119899rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(52)

Combining (44) and (52) we get (48) Similarly we wouldhandle the case for 119867

1(R119899 119883) Hence the proof is com-

pleted

Corollary 7 For every family of scalars (119862119860)119860isinΛ

one has that

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 119871119901(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 1198671(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

lt infin

(53)

Proof Suppose that (119862119860)119860isinΛ

is a family of scalars suchthat (sum

119860isinΛ|119862119860|2|120593119860|2)12

119871119901(R119899119883)

lt infin For every120576 gt 0 we can find 119865

0finite subset of Λ such that

(sum119860isinΛ1198650

|119862119860|2|120593119860|2)12

119871119901(R119899 119883)

le 120576Then for every 119865 finite subset of Λ 119865

0 by (44) we have

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

120576 (54)

This implies that the series sum119860isinΛ

119862119860120593119860converges uncondi-

tionally in 119871119901(R119899 119883) to the same 119871

119901-function Moreover wehave proved that

sum

119860isinΛ

119862119860120593119860converges unconditionally in

119871119901(R119899 119883) lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

lt infin

(55)

This gives the characterization of the convergence of waveletseries in 119871

119901(R119899 119883) Similarly we would characterize conver-

gence in1198671(R119899 119883)

Hence the proof is completed

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by University Grants CommissionNew Delhi under the major research Project by F no 41-7922012 (SR)

References

[1] R M Young An Introduction to Nonharmonic Fourier Seriesvol 93 Academic Press New York NY USA 1980

[2] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2001

[3] E M Stein Singular Integrals and Differentiability Propertiesof Functions Princeton University Press Princeton NJ USA1970

[4] J Mateu and J Verdera ldquo119871119901 and weak 1198711 estimates for the

maximal Riesz transform and themaximal Beurling transformrdquoMathematical Research Letters vol 13 no 5-6 pp 957ndash9662006

[5] C Thiele Wave Packet Analysis vol 105 of CBMS RegionalConference Series in Mathematics 2006

[6] D L Burkholder ldquoMartingales and Fourier analysis in Banachspacesrdquo in Probability and Analysis G Letta and M PratelliEds vol 1206 of Lecture Notes in Mathematics pp 61ndash108Springer Berlin Germany 1986

[7] T Hytonen ldquoVector-valued wavelets and the Hardy space1198671(R119899 119883)rdquo Studia Mathematica vol 172 no 2 pp 125ndash147

2006[8] T Figiel ldquoSingular integral operators a martingale approachrdquo

in Geometry of Banach Spaces P F X Moller and W Schacher-mayer Eds vol 158 of London Mathematical Society LectureNote Series pp 95ndash110 Cambridge University Press Cam-bridge Mass USA 1990

[9] E M SteinHarmonic Analysis Real-Variable Methods Orthog-onality and Oscillatory Integrals vol 43 Princeton UniversityPress Princeton NJ USA 1993

[10] E Hernandez and G Weiss A First Course on Wavelets CRCPress Boca Raton Fla USA 1996

Chinese Journal of Mathematics 7

[11] YMeyerWavelets and Operators vol 37 Cambridge UniversityPress Cambridge Mass USA 1992

[12] P Wojtaszczyk ldquoThe Franklin system is an unconditional basisin1198671rdquo Arkiv for Matematik vol 20 no 2 pp 293ndash300 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Chinese Journal of Mathematics

Proof We have

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883)

100381710038171003817100381712059311986010038171003817100381710038171198711199011015840(R119899 119883)

1003816100381610038161003816⟨119891 120593119860⟩1003816100381610038161003816 le

100381710038171003817100381711989110038171003817100381710038171198671(R119899 119883)

10038171003817100381710038171205931198601003817100381710038171003817BMO

(50)

where the space BMO(R119899) is of functions with boundedmean oscillation1198671(R119899) is the predual of BMO(R119899) We seethat ⟨sdot 120593

119860⟩ 119860 isin Λ defines continuous linear functionals

on119871119901(R119899 119883) 1 lt 119901 lt infin and119867

1(R119899 119883) If119891 = sum

119860isinΛ119862119860120593119860

unconditionally in 119871119901(R119899 119883) or 119867

1(R119899 119883) then for every

1198601015840isin Λ by continuity of ⟨sdot 120593

1198601015840 ⟩

⟨119891 1205931198601015840⟩ = sum

119860isinΛ

119862119860⟨120593119860 1205931198601015840⟩ = sum

119860isinΛ

1198621198601205751198601198601015840 = 1198621198601015840 (51)

Now for every 119891 isin 119871119901(R119899 119883) we have 119891 =

suminfin

119894=1(119895119896119894)isin119865⟨119891 120593119895119896119894

⟩120593119895119896119894

with convergence in 119871119901(R119899 119883)

Using (41) we obtain

10038171003817100381710038171198911003817100381710038171003817119871119901(R119899 119883) = lim

119899rarrinfin

10038171003817100381710038171003817100381710038171003817100381710038171003817

119873

sum

119894=1(119895119896119894)isin119865

⟨119891 120593119895119896119894

⟩120593119895119896119894

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

le 119862119899119901

1003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816⟨119891 120593119860⟩100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

(52)

Combining (44) and (52) we get (48) Similarly we wouldhandle the case for 119867

1(R119899 119883) Hence the proof is com-

pleted

Corollary 7 For every family of scalars (119862119860)119860isinΛ

one has that

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 119871119901(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899 119883)

lt infin

sum

119860isinΛ

119862119860120593119860

119888119900119899V119890119903119892119890119904 119906119899119888119900119899119889119894119905119894119900119899119886119897119897119910 119894119899 1198671(R119899 119883)

lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171198711(R119899119883)

lt infin

(53)

Proof Suppose that (119862119860)119860isinΛ

is a family of scalars suchthat (sum

119860isinΛ|119862119860|2|120593119860|2)12

119871119901(R119899119883)

lt infin For every120576 gt 0 we can find 119865

0finite subset of Λ such that

(sum119860isinΛ1198650

|119862119860|2|120593119860|2)12

119871119901(R119899 119883)

le 120576Then for every 119865 finite subset of Λ 119865

0 by (44) we have

1003817100381710038171003817100381710038171003817100381710038171003817

sum

119860isin119865

119862119860120593119860

1003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

le 119862119899119901

120576 (54)

This implies that the series sum119860isinΛ

119862119860120593119860converges uncondi-

tionally in 119871119901(R119899 119883) to the same 119871

119901-function Moreover wehave proved that

sum

119860isinΛ

119862119860120593119860converges unconditionally in

119871119901(R119899 119883) lArrrArr

10038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119860isinΛ

1003816100381610038161003816119862119860100381610038161003816100381621003816100381610038161003816120593119860

10038161003816100381610038162

)

1210038171003817100381710038171003817100381710038171003817100381710038171003817119871119901(R119899119883)

lt infin

(55)

This gives the characterization of the convergence of waveletseries in 119871

119901(R119899 119883) Similarly we would characterize conver-

gence in1198671(R119899 119883)

Hence the proof is completed

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by University Grants CommissionNew Delhi under the major research Project by F no 41-7922012 (SR)

References

[1] R M Young An Introduction to Nonharmonic Fourier Seriesvol 93 Academic Press New York NY USA 1980

[2] J Duoandikoetxea Fourier Analysis vol 29 ofGraduate Studiesin Mathematics American Mathematical Society ProvidenceRI USA 2001

[3] E M Stein Singular Integrals and Differentiability Propertiesof Functions Princeton University Press Princeton NJ USA1970

[4] J Mateu and J Verdera ldquo119871119901 and weak 1198711 estimates for the

maximal Riesz transform and themaximal Beurling transformrdquoMathematical Research Letters vol 13 no 5-6 pp 957ndash9662006

[5] C Thiele Wave Packet Analysis vol 105 of CBMS RegionalConference Series in Mathematics 2006

[6] D L Burkholder ldquoMartingales and Fourier analysis in Banachspacesrdquo in Probability and Analysis G Letta and M PratelliEds vol 1206 of Lecture Notes in Mathematics pp 61ndash108Springer Berlin Germany 1986

[7] T Hytonen ldquoVector-valued wavelets and the Hardy space1198671(R119899 119883)rdquo Studia Mathematica vol 172 no 2 pp 125ndash147

2006[8] T Figiel ldquoSingular integral operators a martingale approachrdquo

in Geometry of Banach Spaces P F X Moller and W Schacher-mayer Eds vol 158 of London Mathematical Society LectureNote Series pp 95ndash110 Cambridge University Press Cam-bridge Mass USA 1990

[9] E M SteinHarmonic Analysis Real-Variable Methods Orthog-onality and Oscillatory Integrals vol 43 Princeton UniversityPress Princeton NJ USA 1993

[10] E Hernandez and G Weiss A First Course on Wavelets CRCPress Boca Raton Fla USA 1996

Chinese Journal of Mathematics 7

[11] YMeyerWavelets and Operators vol 37 Cambridge UniversityPress Cambridge Mass USA 1992

[12] P Wojtaszczyk ldquoThe Franklin system is an unconditional basisin1198671rdquo Arkiv for Matematik vol 20 no 2 pp 293ndash300 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Chinese Journal of Mathematics 7

[11] YMeyerWavelets and Operators vol 37 Cambridge UniversityPress Cambridge Mass USA 1992

[12] P Wojtaszczyk ldquoThe Franklin system is an unconditional basisin1198671rdquo Arkiv for Matematik vol 20 no 2 pp 293ndash300 1982

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of