Research ArticleMultiplicative Isometries on Classes119872119901

(119883) ofHolomorphic Functions

Yasuo Iida1 and Kazuhiro Kasuga2

1Department of Mathematics Iwate Medical University Yahaba Iwate 028-3694 Japan2Academic Support Center Kogakuin University Tokyo 192-0015 Japan

Correspondence should be addressed to Yasuo Iida yiidaiwate-medacjp

Received 19 March 2015 Revised 12 June 2015 Accepted 14 June 2015

Academic Editor Alberto Fiorenza

Copyright copy 2015 Y Iida and K Kasuga This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In (Iida and Kasuga 2013) the authors described multiplicative (but not necessarily linear) isometries of119872119901(119883) onto119872119901(119883) in thecase of positive integer 119901 isin N where119872119901(119883) (119901 ge 1) is included in the Smirnov class119873lowast(119883) In this paper we will generalize theresult to arbitrary (not necessarily positive integer) value of the exponents 0 lt 119901 lt infin

1 Introduction

Let 119899 be a positive integer The space of 119899-complex variables119911 = (1199111 119911119899) is denoted by C119899 The unit polydisk 119911 isin C119899

|119911119895| lt 1 1 le 119895 le 119899 is denoted by 119880119899 and the distinguishedboundary T119899 is 119911 isin C119899 |119911119895| = 1 1 le 119895 le 119899 The unitball 119911 isin C119899 sum

119899

119895=1 |119911119895|2lt 1 is denoted by 119861119899 and 119878119899 is

its boundary In this paper119883 denotes the unit polydisk or theunit ball for 119899 ge 1 and 120597119883 denotes T119899 for119883 = 119880

119899 or 119878119899 for119883 =

119861119899 The normalized (in the sense that 120590(120597119883) = 1) Lebesguemeasure on 120597119883 is denoted by 119889120590

The Hardy space on119883 is denoted by119867119902(119883) (0 lt 119902 le infin)

and sdot 119902 denotes the norm on119867119902(119883) (1 le 119902 le infin)The Nevanlinna class 119873(119883) on 119883 is defined as the set of

all holomorphic functions 119891 on119883 such that

sup0le119903lt1

int120597119883

log (1+ 1003816100381610038161003816119891 (119903119911)1003816100381610038161003816) 119889120590 (119911) lt infin (1)

holds It is known that 119891 isin 119873(119883) has a finite nontangentiallimit also denoted by 119891 almost everywhere on 120597119883

The Smirnov class 119873lowast(119883) is defined as the set of all 119891 isin

119873(119883) which satisfy the equality

sup0le119903lt1

int120597119883

log (1+ 1003816100381610038161003816119891 (119903119911)1003816100381610038161003816) 119889120590 (119911)

= int120597119883

log (1+ 1003816100381610038161003816119891 (119911)1003816100381610038161003816) 119889120590 (119911)

(2)

Define a metric

119889119873lowast(119883) (119891 119892) = int

120597119883

log (1+ 1003816100381610038161003816119891 (119911) minus 119892 (119911)1003816100381610038161003816) 119889120590 (119911) (3)

for 119891 119892 isin 119873lowast(119883) With the metric 119889119873lowast(119883)(sdot sdot) the Smirnov

class 119873lowast(119883) is an 119865-algebra Recall that an 119865-algebra isa topological algebra in which the topology arises from acomplete metric Complex-linear isometries on the Smirnovclass are characterized by Stephenson in [1]

The Privalov class 119873119901(119883) 1 lt 119901 lt infin is defined as theset of all holomorphic functions 119891 on119883 such that

sup0le119903lt1

int120597119883

(log (1 + 1003816100381610038161003816119891 (119903119911)1003816100381610038161003816))119901119889120590 (119911) lt infin (4)

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 598538 5 pageshttpdxdoiorg1011552015598538

2 Journal of Function Spaces

holds It is well-known that119873119901(119883) is a subalgebra of119873lowast(119883)hence every 119891 isin 119873

119901(119883) has a finite nontangential limit

almost everywhere on 120597119883 Under the metric defined by

119889119873119901(119883) (119891 119892)

= (int120597119883

(log (1+ 1003816100381610038161003816119891 (119911) minus 119892 (119911)1003816100381610038161003816))119901119889120590 (119911))

1119901 (5)

for 119891 119892 isin 119873119901(119883) 119873119901(119883) becomes an 119865-algebra (cf [2])

Complex-linear isometries on119873119901(119883) are investigated by IidaandMochizuki [3] for one-dimensional case and by Subbotin[2 4] for a general case

Now we define the class119872119901(119883) For 0 lt 119901 lt infin the class119872119901(119883) is defined as the set of all holomorphic functions 119891

on119883 such that

int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911) lt infin (6)

The class 119872119901(119883) with 119901 = 1 in the case where 119899 = 1 wasintroduced by Kim in [5] As for 119901 gt 0 and 119899 gt 1 the classwas considered in [6 7] For 119891 119892 isin 119872

119901(119883) define a metric

119889119872119901(119883) (119891 119892)

= int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816119891 (119903119911) minus 119892 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

120572119901119901

(7)

where 120572119901 = min(1 119901) With this metric 119872119901(119883) is also an119865-algebra (see [2]) Complex-linear surjective isometries on119872119901(119883) are investigated by Subbotin [2 4 8]It is well-known that the following inclusion relations

hold

119867119902(119883) ⊊ 119873

119901(119883) ⊊ 119872

1(119883) ⊊ 119873lowast (119883)

(0 lt 119902 le infin 119901 gt 1) (8)

Moreover it is known that 119873(X) ⊊ 119872119901(119883) (0 lt 119901 lt 1) [8]

As shown in [6] for any 119901 gt 1 the class 119872119901(119883) coincideswith the class 119873119901(119883) and the metrics 119889119872119901(119883) and 119889119873119901(119883)are equivalent Therefore the topologies induced by thesemetrics are identical on the set 119872119901(119883) = 119873

119901(119883) But we

note that in [4 Theorems 1 and 4] it is implied that the setsof linear isometries on 119872

119901(119883) and 119873

119901(119883) are different

It is known that 119867infin(119883) is a dense subalgebra of 119872119901(119883)The convergence in the metric is stronger than uniformconvergence on compact subsets of119883

In [9] the authors described multiplicative (but notnecessarily linear) isometries of 119872119901(119883) onto 119872119901(119883) in thecase of positive integer119901 isin N In this paper we will generalizethe result to arbitrary (not necessarily positive integer) valueof the exponents 0 lt 119901 lt infin

2 The Results

Proposition 1 Let 119899 be a positive integer and let 119883 be either119861119899 or 119880119899 Let 0 lt 119901 lt infin and suppose that 119860 119872

119901(119883) rarr

119872119901(119883) is a surjective isometry If 119860 is 2-homogeneous in

the sense that 119860(2119891) = 2119860(119891) holds for every 119891 isin 119872119901(119883)

then either

119860 (119891) = 120572119891 ∘Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (9)

or

119860 (119891) = 120572119891 ∘ Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (10)

where 120572 is a complex number with the unit modulus andfor 119883 = 119861119899 Φ is a unitary transformation for 119883 = 119880

119899Φ(1199111 119911119899) = (12058211199111198941 120582119899119911119894119899) where |120582119895| = 1 1 le 119895 le

119899 and (1198941 119894119899) is some permutation of the integers from 1through 119899

To prove Proposition 1 we need the following lemmas

Lemma 2 (see [4]) Let 119891 isin 119867119901(119883) 119901 ge 1 Then the norm

10038171003817100381710038171198911003817100381710038171003817119867119901119898

= int120597119883

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816119901119889120590 (119911)

1119901(11)

is equivalent to the standard norm in119867119901(119883)

Lemma 3 (see [4]) Let 0 lt 119901 lt infin Then

lim120576rarr 0+

1120576119901+1

(120576119905)119901minus (log (1 + 120576119905))119901 =

119901

2119905119901+1

119905 ge 0 (12)

We recall that the function 120593(119909) on the interval (0 +infin) issaid to be completely monotone if it is infinitely differentiableon (0 +infin) and

(minus1)119899 120593(119899) (119909) ge 0 119909 gt 0 119899 isin Z+ (13)

Lemma 4 (see [8]) The functions ((log(1 + 119909))119909)119901 are

completely monotone for all 119901 gt 0

Lemma 5 (see [8]) If a completelymonotone function 120593(119909) on(0 +infin) can be continued to an infinitely differentiable functionon [0 +infin) then the inequality

(minus1)119899

119909119899120593 (119909) minus 120593 (0) minus 1205931015840 (0) 119909 minus sdot sdot sdot

minus120593(119899minus1)

(0)(119899 minus 1)

119909119899minus1

ge 0 119909 gt 0

(14)

holds for any 119899 isin N and 120593(119899)(0) = 0 if only 120593 is not constant

Lemma 6 Let 119862 be a cone of measurable functions on ameasurable space with a measure (119866 120583) and let 119860 be amapping from 119862 to the set of measurable functions Supposethat 119860 is homogeneous with positive coefficients and

int119866

(log (1+ 1003816100381610038161003816119860119891 (119909)1003816100381610038161003816))119901120583 (119889119909)

= int119866

(log (1+ 1003816100381610038161003816119891 (119909)1003816100381610038161003816))119901120583 (119889119909) 119891 isin 119862

(15)

Journal of Function Spaces 3

for some 119901 gt 0 Then

int119866

1003816100381610038161003816119860119891 (119909)1003816100381610038161003816119902120583 (119889119909) = int

119866

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902120583 (119889119909) 119891 isin 119862 (16)

for all 119902 = 119901 + 119897 where 119897 isin Z+

Proof We follow [2 Lemma 2] 119860 is homogeneous withpositive coefficients so we have using (15)

int119866

(log (1 + 119905 1003816100381610038161003816119860119891 (119909)1003816100381610038161003816))119901

119905119901120583 (119889119909)

= int119866

(log (1 + 119905 1003816100381610038161003816119891 (119909)1003816100381610038161003816))119901

119905119901120583 (119889119909) 119891 isin 119862

(17)

for any 119905 gt 0 Letting 119905 rarr 0+ we obtain

int119866

1003816100381610038161003816119860119891 (119909)1003816100381610038161003816119901120583 (119889119909) = int

119866

1003816100381610038161003816119891 (119909)1003816100381610038161003816119901120583 (119889119909) 119891 isin 119862 (18)

Next we argue by induction Let 119896 isin N and suppose that (16)holds for 119902 = 119901 + 1 119901 + 2 119901 + 119896 minus 1 Then for any 119905 gt 0and any function 119891 isin 119862 we have

int119866

1119905119901+119896

(log (1 + 119905 1003816100381610038161003816119860119891 (119909)1003816100381610038161003816))119901

minus

119896minus1sum

119897=0119888119897 (119905

1003816100381610038161003816119860119891 (119909)1003816100381610038161003816)119901+119897120583 (119889119909)

= int119866

1119905119901+119896

(log (1 + 119905 1003816100381610038161003816119891 (119909)1003816100381610038161003816))119901

minus

119896minus1sum

119897=0119888119897 (119905

1003816100381610038161003816119891 (119909)1003816100381610038161003816)119901+119897120583 (119889119909)

(19)

where 119888119897 are the Taylor coefficients of the function ((log(1 +119909))119909)

119901 at zeroAssume first that 119891 isin 119871

119901+119896(119866 120583) It is easy to see that

the integrand on the right-hand side of (19) converges to119888119896|119891(119909)|

119901+119896 as 119905 rarr 0+ this integrand is of fixed signby Lemmas 4 and 5 and is dominated by the function119862119896|119891(119909)|

119901+119896 with some constant119862119896 By the Lebesgue theoremon dominated convergence the right-hand side of (19)converges to the integral of 119888119896|119891(119909)|

119901+119896 Therefore the left-hand side of (19) has a finite limit and by the Fatou theoremthe function 119888119896|119860119891(119909)|

119901+119896 is integrable Since 119888119896 = 0 for any119896 by Lemmas 4 and 5 we deduce that 119860119891 isin 119871

119901+119896(119866 120583) and

repeating the above arguments we see that the left-hand sideof (19) converges to the integral of 119888119896|119860119891(119909)|

119901+119896 as 119905 rarr 0+Therefore passing to the limit in (19) as 119905 rarr 0+ and dividingthe result by 119888119896 = 0 we obtain (16) for 119902 = 119901 + 119896 The caseof 119860119891 isin 119871

119901+119896(119866 120583) can be considered in a similar way For

119891119860119891 notin 119871119901+119896

(119866 120583) relation (16) with 119902 = 119901 + 119896 is trivial

Proof of Proposition 1 Suppose first that 119901 ge 1 Let 119891 119892 isin

119867119901(119883) We easily confirm that by utilizing Lemma 2 and

the celebrated theorem of Mazur and Ulam [10] 119860|119867119901(119883) isa real-linear isometry in a way similar to [9 Proposition 1]

Next suppose that 0 lt 119901 lt 1 We define the class119867119901

119898(119883) (0 lt 119901 lt 1) as the set of all holomorphic functions 119891

on119883 such that

119889119867119901

119898

(119891 0) = int120597119883

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816119901119889120590 (119911) lt infin (20)

and define a metric 119889119867119901

119898

(119891 119892) = 119889119867119901

119898

(119891 minus 119892 0) for 119891 119892 isin

119867119901

119898(119883) If119860 119872

119901(119883) rarr 119872

119901(119883) is a surjective isometry and

119860 is 2-homogeneous in the sense that 119860(2119891) = 2119860(119891) holdsfor every119891 isin 119872

119901(119883) we confirm that119860 119867

119901

119898(119883) rarr 119867

119901

119898(119883)

is also a surjective isometryFor 0 lt 119901 lt 1 let 119891 119892 isin 119867

119901(119883) We have 119860(119891)2119898 =

119860(1198912119898) (119898 isin N) since 119860(2119891) = 2119860(119891) Then the followingequality holds

int120597119883

sup0le119903lt1

10038161003816100381610038161003816100381610038161003816

119891 (119903119911)

211989810038161003816100381610038161003816100381610038161003816

119901

119889120590 (119911)

minusint120597119883

(log(1+ sup0le119903lt1

10038161003816100381610038161003816100381610038161003816

119891 (119903119911)

211989810038161003816100381610038161003816100381610038161003816))

119901

119889120590 (119911)

= int120597119883

sup0le119903lt1

100381610038161003816100381610038161003816100381610038161003816

(119860119891) (119903119911)

2119898100381610038161003816100381610038161003816100381610038161003816

119901

119889120590 (119911)

minusint120597119883

(log(1+ sup0le119903lt1

100381610038161003816100381610038161003816100381610038161003816

(119860119891) (119903119911)

2119898100381610038161003816100381610038161003816100381610038161003816

))

119901

119889120590 (119911)

(21)

Therefore it follows that

int120597119883

1(12119898)119901+1

(12119898

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816)

119901

minus(log(1 + 12119898

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

= int120597119883

1(12119898)119901+1

(12119898

sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816)

119901

minus(log(1 + 12119898

sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

(22)

Using the elementary inequalities log(1 + 119909119910) le log(1 + 119909) +log(1 + 119910) (119909 ge 0 119910 ge 0) and (119886 + 119887)119901 le 2119901(119886119901 + 119887119901) (119886 ge 0119887 ge 0 119901 gt 0) we confirm that the integrand on the left-hand side of (22) is dominated by an 119871

1(120597119883)-function The

integrand on the right-hand side of (22) is also dominated byan 1198711(120597119883)-function in the same way Applying the Lebesguetheorem on dominated convergence and Lemma 3 on bothsides of (22) we have the equality

int120597119883

119901

2( sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816)

119901+1

119889120590 (119911)

= int120597119883

119901

2( sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816)

119901+1

119889120590 (119911)

(23)

4 Journal of Function Spaces

Hence we obtain

int120597119883

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816119901+1

119889120590 (119911)

= int120597119883

sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816119901+1

119889120590 (119911)

(24)

The equivalence of the norms sdot 119901+1 and sdot 119867119901+1119898

guaranteesthat 119860 119867

119901+1(119883) rarr 119867

119901+1(119883) is a surjective isometry By

using Mazur-Ulam theorem again 119860|119867119901+1(119883) is a real-linearisometry since119867119901+1(119883) is a normed vector space and119860(0) =0

We consider an arbitrary function 119891 isin 119872119901(119883) (0 lt 119901 lt

infin) and the cone 119862119891 = 120582119872119891(120577) | 120582 gt 0 generated by 119891Here119872119891(120577) = sup0le119903lt1|119891(119903120577)| is the radial maximal functionfor119891Moreover consider the followingmapping on this cone

119860 120582119872119891 (120577) 997891997888rarr 120582119872(119860119891) (120577) = 119872(119860 [120582119891]) (120577)

120577 isin 120597119883 120582 ge 0(25)

Since119860 is isometric with respect to the metric 119889119872119901 it followsthat the assumptions of Lemma 6 hold on the cone 119862119891

119860 119872119901(119883) rarr 119872

119901(119883) is a surjective isometry so the

equation

int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

= int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

(26)

guarantees the equalities

int120597119883

( sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816)

119901+119897

119889120590 (119911)

= int120597119883

( sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816)

119901+119897

119889120590 (119911)

(27)

for all 119897 isin Z+Therefore119860 is isometric in the norm119867119901+119897

119898(119901 gt

0 119897 = 0 1 2 )Since 119889120590 is a finite measure we verify that

lim119897rarrinfin

10038171003817100381710038171198911003817100381710038171003817119867119901+119897119898

=10038171003817100381710038171198911003817100381710038171003817119867infin119898

(28)

holds for every 119891 isin 119867infin(119883) and it is clear that 119891119867infin

119898

=

119891infin Moreover 119891119901 = 119860(119891)119901 for every 119891 isin 119867infin(119883) and

lim119901rarrinfin119860(119891)119901 = 119860(119891)infin so we have 119860(119891) isin 119867infin(119883)

and 119891infin = 119860(119891)infin for every 119891 isin 119867infin(119883) Similarly we

see that 119891 isin 119867infin(119883) if 119860(119891) belongs to 119867infin(119883) Therefore

119860|119867infin(119883) is a surjective isometry with respect to sdot infin from119867infin(119883) onto itselfWemay suppose that119867infin(119883) is a uniform

algebra on the maximal ideal space and the maximal idealspace is connected by the Silov idempotent theorem hencewe see that 119860|119867infin(119883) is complex-linear or conjugate linear by[11Theorem] As for the rest of this proof we follow the proofof [9 Proposition 1]

Finally we consider multiplicative isometries from119872119901(119883) (0 lt 119901 lt infin) onto itself Recall that 119860 119872

119901(119883) rarr

119872119901(119883) is multiplicative if 119860(119891119892) = 119860(119891)119860(119892) for every

119891 119892 isin 119872119901(119883)

The following theorem is proved by the same method as[9 Theorem 2] therefore we do not prove it here

Theorem 7 Let 0 lt 119901 lt infin and 119860 be a multiplicative (notnecessarily linear) isometry from119872

119901(119883) onto itselfThen there

exists a holomorphic automorphismΦ on119883 such that either ofthe following holds

119860 (119891) = 119891 ∘Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (29)

or

119860 (119891) = 119891 ∘ Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (30)

where Φ is a unitary transformation for 119883 = 119861119899Φ(1199111 119911119899) = (12058211199111198941 120582119899119911119894119899) for 119883 = 119880

119899 where |120582119895| = 1for every 1 le 119895 le 119899 and (1198941 119894119899) is some permutation of theintegers from 1 through 119899

Remark 8 We note that surjective multiplicative isometriesof the class 119872119901(119883) (0 lt 119901 lt infin) have the same form assurjective multiplicative isometries of 119872119901(119883) (119901 isin N) [9Theorem 2] the Smirnov class [12 Theorem 22] and thePrivalov class [13 Corollary 34]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere gratitude to Pro-fessor Takahiko Nakazi and Professor Osamu Hatori whointroduced this subject and kindly directed themThe authorsalso would like to thank the referee for detailed commentsand valuable suggestions

References

[1] K Stephenson ldquoIsometries of the Nevanlinna classrdquo IndianaUniversity Mathematics Journal vol 26 no 2 pp 307ndash324 1977

[2] A V Subbotin ldquoIsometries of Privalov spaces of holomorphicfunctions of several variablesrdquo Journal ofMathematical Sciencesvol 135 no 1 pp 2794ndash2802 2006

[3] Y Iida and N Mochizuki ldquoIsometries of some 119865-algebras ofholomorphic functionsrdquo Archiv der Mathematik vol 71 no 4pp 297ndash300 1998

[4] A V Subbotin ldquoLinear isometry groups of Privalovrsquos spacesof holomorphic functions of several variablesrdquoDoklady Mathe-matics vol 60 no 1 pp 77ndash79 1999

[5] H O Kim ldquoOn an 119865-algebra of holomorphic functionsrdquoCanadian Journal of Mathematics vol 40 no 3 pp 718ndash7411988

[6] B R Choe and H O Kim ldquoOn the boundary behavior offunctions holomorphic on the ballrdquo Complex Variables Theoryand Application vol 20 no 1ndash4 pp 53ndash61 1992

Journal of Function Spaces 5

[7] H O Kim and Y Y Park ldquoMaximal functions of plurisubhar-monic functionsrdquo Tsukuba Journal of Mathematics vol 16 no1 pp 11ndash18 1992

[8] A V Subbotin ldquoGroups of linear isometries of spaces M119902 ofholomorphic functions of several complex variablesrdquo Mathe-matical Notes vol 83 no 3 pp 437ndash440 2008

[9] Y Iida and K Kasuga ldquoMultiplicative isometries on some 119865-algebras of holomorphic functionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 681637 3 pages 2013

[10] S Mazur and S Ulam ldquoSur les transformationes isometriquesdrsquoespaces vectoriels normesrdquo Comptes Rendus de lrsquoAcademie desSciences Paris vol 194 pp 946ndash948 1932

[11] A J Ellis ldquoReal characterizations of function algebras amongstfunction spacesrdquo Bulletin of the London Mathematical Societyvol 22 no 4 pp 381ndash385 1990

[12] O Hatori and Y Iida ldquoMultiplicative isometries on the SmirnovclassrdquoCentral European Journal ofMathematics vol 9 no 5 pp1051ndash1056 2011

[13] O Hatori Y Iida S Stevic and S-I Ueki ldquoMultiplicativeisometries on F-algebras of holomorphic functions rdquo AbstractandApplied Analysis vol 2012 Article ID 125987 16 pages 2012

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

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

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces

holds It is well-known that119873119901(119883) is a subalgebra of119873lowast(119883)hence every 119891 isin 119873

119901(119883) has a finite nontangential limit

almost everywhere on 120597119883 Under the metric defined by

119889119873119901(119883) (119891 119892)

= (int120597119883

(log (1+ 1003816100381610038161003816119891 (119911) minus 119892 (119911)1003816100381610038161003816))119901119889120590 (119911))

1119901 (5)

for 119891 119892 isin 119873119901(119883) 119873119901(119883) becomes an 119865-algebra (cf [2])

Complex-linear isometries on119873119901(119883) are investigated by IidaandMochizuki [3] for one-dimensional case and by Subbotin[2 4] for a general case

Now we define the class119872119901(119883) For 0 lt 119901 lt infin the class119872119901(119883) is defined as the set of all holomorphic functions 119891

on119883 such that

int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911) lt infin (6)

The class 119872119901(119883) with 119901 = 1 in the case where 119899 = 1 wasintroduced by Kim in [5] As for 119901 gt 0 and 119899 gt 1 the classwas considered in [6 7] For 119891 119892 isin 119872

119901(119883) define a metric

119889119872119901(119883) (119891 119892)

= int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816119891 (119903119911) minus 119892 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

120572119901119901

(7)

where 120572119901 = min(1 119901) With this metric 119872119901(119883) is also an119865-algebra (see [2]) Complex-linear surjective isometries on119872119901(119883) are investigated by Subbotin [2 4 8]It is well-known that the following inclusion relations

hold

119867119902(119883) ⊊ 119873

119901(119883) ⊊ 119872

1(119883) ⊊ 119873lowast (119883)

(0 lt 119902 le infin 119901 gt 1) (8)

Moreover it is known that 119873(X) ⊊ 119872119901(119883) (0 lt 119901 lt 1) [8]

As shown in [6] for any 119901 gt 1 the class 119872119901(119883) coincideswith the class 119873119901(119883) and the metrics 119889119872119901(119883) and 119889119873119901(119883)are equivalent Therefore the topologies induced by thesemetrics are identical on the set 119872119901(119883) = 119873

119901(119883) But we

note that in [4 Theorems 1 and 4] it is implied that the setsof linear isometries on 119872

119901(119883) and 119873

119901(119883) are different

It is known that 119867infin(119883) is a dense subalgebra of 119872119901(119883)The convergence in the metric is stronger than uniformconvergence on compact subsets of119883

In [9] the authors described multiplicative (but notnecessarily linear) isometries of 119872119901(119883) onto 119872119901(119883) in thecase of positive integer119901 isin N In this paper we will generalizethe result to arbitrary (not necessarily positive integer) valueof the exponents 0 lt 119901 lt infin

2 The Results

Proposition 1 Let 119899 be a positive integer and let 119883 be either119861119899 or 119880119899 Let 0 lt 119901 lt infin and suppose that 119860 119872

119901(119883) rarr

119872119901(119883) is a surjective isometry If 119860 is 2-homogeneous in

the sense that 119860(2119891) = 2119860(119891) holds for every 119891 isin 119872119901(119883)

then either

119860 (119891) = 120572119891 ∘Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (9)

or

119860 (119891) = 120572119891 ∘ Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (10)

where 120572 is a complex number with the unit modulus andfor 119883 = 119861119899 Φ is a unitary transformation for 119883 = 119880

119899Φ(1199111 119911119899) = (12058211199111198941 120582119899119911119894119899) where |120582119895| = 1 1 le 119895 le

119899 and (1198941 119894119899) is some permutation of the integers from 1through 119899

To prove Proposition 1 we need the following lemmas

Lemma 2 (see [4]) Let 119891 isin 119867119901(119883) 119901 ge 1 Then the norm

10038171003817100381710038171198911003817100381710038171003817119867119901119898

= int120597119883

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816119901119889120590 (119911)

1119901(11)

is equivalent to the standard norm in119867119901(119883)

Lemma 3 (see [4]) Let 0 lt 119901 lt infin Then

lim120576rarr 0+

1120576119901+1

(120576119905)119901minus (log (1 + 120576119905))119901 =

119901

2119905119901+1

119905 ge 0 (12)

We recall that the function 120593(119909) on the interval (0 +infin) issaid to be completely monotone if it is infinitely differentiableon (0 +infin) and

(minus1)119899 120593(119899) (119909) ge 0 119909 gt 0 119899 isin Z+ (13)

Lemma 4 (see [8]) The functions ((log(1 + 119909))119909)119901 are

completely monotone for all 119901 gt 0

Lemma 5 (see [8]) If a completelymonotone function 120593(119909) on(0 +infin) can be continued to an infinitely differentiable functionon [0 +infin) then the inequality

(minus1)119899

119909119899120593 (119909) minus 120593 (0) minus 1205931015840 (0) 119909 minus sdot sdot sdot

minus120593(119899minus1)

(0)(119899 minus 1)

119909119899minus1

ge 0 119909 gt 0

(14)

holds for any 119899 isin N and 120593(119899)(0) = 0 if only 120593 is not constant

Lemma 6 Let 119862 be a cone of measurable functions on ameasurable space with a measure (119866 120583) and let 119860 be amapping from 119862 to the set of measurable functions Supposethat 119860 is homogeneous with positive coefficients and

int119866

(log (1+ 1003816100381610038161003816119860119891 (119909)1003816100381610038161003816))119901120583 (119889119909)

= int119866

(log (1+ 1003816100381610038161003816119891 (119909)1003816100381610038161003816))119901120583 (119889119909) 119891 isin 119862

(15)

Journal of Function Spaces 3

for some 119901 gt 0 Then

int119866

1003816100381610038161003816119860119891 (119909)1003816100381610038161003816119902120583 (119889119909) = int

119866

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902120583 (119889119909) 119891 isin 119862 (16)

for all 119902 = 119901 + 119897 where 119897 isin Z+

Proof We follow [2 Lemma 2] 119860 is homogeneous withpositive coefficients so we have using (15)

int119866

(log (1 + 119905 1003816100381610038161003816119860119891 (119909)1003816100381610038161003816))119901

119905119901120583 (119889119909)

= int119866

(log (1 + 119905 1003816100381610038161003816119891 (119909)1003816100381610038161003816))119901

119905119901120583 (119889119909) 119891 isin 119862

(17)

for any 119905 gt 0 Letting 119905 rarr 0+ we obtain

int119866

1003816100381610038161003816119860119891 (119909)1003816100381610038161003816119901120583 (119889119909) = int

119866

1003816100381610038161003816119891 (119909)1003816100381610038161003816119901120583 (119889119909) 119891 isin 119862 (18)

Next we argue by induction Let 119896 isin N and suppose that (16)holds for 119902 = 119901 + 1 119901 + 2 119901 + 119896 minus 1 Then for any 119905 gt 0and any function 119891 isin 119862 we have

int119866

1119905119901+119896

(log (1 + 119905 1003816100381610038161003816119860119891 (119909)1003816100381610038161003816))119901

minus

119896minus1sum

119897=0119888119897 (119905

1003816100381610038161003816119860119891 (119909)1003816100381610038161003816)119901+119897120583 (119889119909)

= int119866

1119905119901+119896

(log (1 + 119905 1003816100381610038161003816119891 (119909)1003816100381610038161003816))119901

minus

119896minus1sum

119897=0119888119897 (119905

1003816100381610038161003816119891 (119909)1003816100381610038161003816)119901+119897120583 (119889119909)

(19)

where 119888119897 are the Taylor coefficients of the function ((log(1 +119909))119909)

119901 at zeroAssume first that 119891 isin 119871

119901+119896(119866 120583) It is easy to see that

the integrand on the right-hand side of (19) converges to119888119896|119891(119909)|

119901+119896 as 119905 rarr 0+ this integrand is of fixed signby Lemmas 4 and 5 and is dominated by the function119862119896|119891(119909)|

119901+119896 with some constant119862119896 By the Lebesgue theoremon dominated convergence the right-hand side of (19)converges to the integral of 119888119896|119891(119909)|

119901+119896 Therefore the left-hand side of (19) has a finite limit and by the Fatou theoremthe function 119888119896|119860119891(119909)|

119901+119896 is integrable Since 119888119896 = 0 for any119896 by Lemmas 4 and 5 we deduce that 119860119891 isin 119871

119901+119896(119866 120583) and

repeating the above arguments we see that the left-hand sideof (19) converges to the integral of 119888119896|119860119891(119909)|

119901+119896 as 119905 rarr 0+Therefore passing to the limit in (19) as 119905 rarr 0+ and dividingthe result by 119888119896 = 0 we obtain (16) for 119902 = 119901 + 119896 The caseof 119860119891 isin 119871

119901+119896(119866 120583) can be considered in a similar way For

119891119860119891 notin 119871119901+119896

(119866 120583) relation (16) with 119902 = 119901 + 119896 is trivial

Proof of Proposition 1 Suppose first that 119901 ge 1 Let 119891 119892 isin

119867119901(119883) We easily confirm that by utilizing Lemma 2 and

the celebrated theorem of Mazur and Ulam [10] 119860|119867119901(119883) isa real-linear isometry in a way similar to [9 Proposition 1]

Next suppose that 0 lt 119901 lt 1 We define the class119867119901

119898(119883) (0 lt 119901 lt 1) as the set of all holomorphic functions 119891

on119883 such that

119889119867119901

119898

(119891 0) = int120597119883

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816119901119889120590 (119911) lt infin (20)

and define a metric 119889119867119901

119898

(119891 119892) = 119889119867119901

119898

(119891 minus 119892 0) for 119891 119892 isin

119867119901

119898(119883) If119860 119872

119901(119883) rarr 119872

119901(119883) is a surjective isometry and

119860 is 2-homogeneous in the sense that 119860(2119891) = 2119860(119891) holdsfor every119891 isin 119872

119901(119883) we confirm that119860 119867

119901

119898(119883) rarr 119867

119901

119898(119883)

is also a surjective isometryFor 0 lt 119901 lt 1 let 119891 119892 isin 119867

119901(119883) We have 119860(119891)2119898 =

119860(1198912119898) (119898 isin N) since 119860(2119891) = 2119860(119891) Then the followingequality holds

int120597119883

sup0le119903lt1

10038161003816100381610038161003816100381610038161003816

119891 (119903119911)

211989810038161003816100381610038161003816100381610038161003816

119901

119889120590 (119911)

minusint120597119883

(log(1+ sup0le119903lt1

10038161003816100381610038161003816100381610038161003816

119891 (119903119911)

211989810038161003816100381610038161003816100381610038161003816))

119901

119889120590 (119911)

= int120597119883

sup0le119903lt1

100381610038161003816100381610038161003816100381610038161003816

(119860119891) (119903119911)

2119898100381610038161003816100381610038161003816100381610038161003816

119901

119889120590 (119911)

minusint120597119883

(log(1+ sup0le119903lt1

100381610038161003816100381610038161003816100381610038161003816

(119860119891) (119903119911)

2119898100381610038161003816100381610038161003816100381610038161003816

))

119901

119889120590 (119911)

(21)

Therefore it follows that

int120597119883

1(12119898)119901+1

(12119898

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816)

119901

minus(log(1 + 12119898

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

= int120597119883

1(12119898)119901+1

(12119898

sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816)

119901

minus(log(1 + 12119898

sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

(22)

Using the elementary inequalities log(1 + 119909119910) le log(1 + 119909) +log(1 + 119910) (119909 ge 0 119910 ge 0) and (119886 + 119887)119901 le 2119901(119886119901 + 119887119901) (119886 ge 0119887 ge 0 119901 gt 0) we confirm that the integrand on the left-hand side of (22) is dominated by an 119871

1(120597119883)-function The

integrand on the right-hand side of (22) is also dominated byan 1198711(120597119883)-function in the same way Applying the Lebesguetheorem on dominated convergence and Lemma 3 on bothsides of (22) we have the equality

int120597119883

119901

2( sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816)

119901+1

119889120590 (119911)

= int120597119883

119901

2( sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816)

119901+1

119889120590 (119911)

(23)

4 Journal of Function Spaces

Hence we obtain

int120597119883

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816119901+1

119889120590 (119911)

= int120597119883

sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816119901+1

119889120590 (119911)

(24)

The equivalence of the norms sdot 119901+1 and sdot 119867119901+1119898

guaranteesthat 119860 119867

119901+1(119883) rarr 119867

119901+1(119883) is a surjective isometry By

using Mazur-Ulam theorem again 119860|119867119901+1(119883) is a real-linearisometry since119867119901+1(119883) is a normed vector space and119860(0) =0

We consider an arbitrary function 119891 isin 119872119901(119883) (0 lt 119901 lt

infin) and the cone 119862119891 = 120582119872119891(120577) | 120582 gt 0 generated by 119891Here119872119891(120577) = sup0le119903lt1|119891(119903120577)| is the radial maximal functionfor119891Moreover consider the followingmapping on this cone

119860 120582119872119891 (120577) 997891997888rarr 120582119872(119860119891) (120577) = 119872(119860 [120582119891]) (120577)

120577 isin 120597119883 120582 ge 0(25)

Since119860 is isometric with respect to the metric 119889119872119901 it followsthat the assumptions of Lemma 6 hold on the cone 119862119891

119860 119872119901(119883) rarr 119872

119901(119883) is a surjective isometry so the

equation

int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

= int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

(26)

guarantees the equalities

int120597119883

( sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816)

119901+119897

119889120590 (119911)

= int120597119883

( sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816)

119901+119897

119889120590 (119911)

(27)

for all 119897 isin Z+Therefore119860 is isometric in the norm119867119901+119897

119898(119901 gt

0 119897 = 0 1 2 )Since 119889120590 is a finite measure we verify that

lim119897rarrinfin

10038171003817100381710038171198911003817100381710038171003817119867119901+119897119898

=10038171003817100381710038171198911003817100381710038171003817119867infin119898

(28)

holds for every 119891 isin 119867infin(119883) and it is clear that 119891119867infin

119898

=

119891infin Moreover 119891119901 = 119860(119891)119901 for every 119891 isin 119867infin(119883) and

lim119901rarrinfin119860(119891)119901 = 119860(119891)infin so we have 119860(119891) isin 119867infin(119883)

and 119891infin = 119860(119891)infin for every 119891 isin 119867infin(119883) Similarly we

see that 119891 isin 119867infin(119883) if 119860(119891) belongs to 119867infin(119883) Therefore

119860|119867infin(119883) is a surjective isometry with respect to sdot infin from119867infin(119883) onto itselfWemay suppose that119867infin(119883) is a uniform

algebra on the maximal ideal space and the maximal idealspace is connected by the Silov idempotent theorem hencewe see that 119860|119867infin(119883) is complex-linear or conjugate linear by[11Theorem] As for the rest of this proof we follow the proofof [9 Proposition 1]

Finally we consider multiplicative isometries from119872119901(119883) (0 lt 119901 lt infin) onto itself Recall that 119860 119872

119901(119883) rarr

119872119901(119883) is multiplicative if 119860(119891119892) = 119860(119891)119860(119892) for every

119891 119892 isin 119872119901(119883)

The following theorem is proved by the same method as[9 Theorem 2] therefore we do not prove it here

Theorem 7 Let 0 lt 119901 lt infin and 119860 be a multiplicative (notnecessarily linear) isometry from119872

119901(119883) onto itselfThen there

exists a holomorphic automorphismΦ on119883 such that either ofthe following holds

119860 (119891) = 119891 ∘Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (29)

or

119860 (119891) = 119891 ∘ Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (30)

where Φ is a unitary transformation for 119883 = 119861119899Φ(1199111 119911119899) = (12058211199111198941 120582119899119911119894119899) for 119883 = 119880

119899 where |120582119895| = 1for every 1 le 119895 le 119899 and (1198941 119894119899) is some permutation of theintegers from 1 through 119899

Remark 8 We note that surjective multiplicative isometriesof the class 119872119901(119883) (0 lt 119901 lt infin) have the same form assurjective multiplicative isometries of 119872119901(119883) (119901 isin N) [9Theorem 2] the Smirnov class [12 Theorem 22] and thePrivalov class [13 Corollary 34]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere gratitude to Pro-fessor Takahiko Nakazi and Professor Osamu Hatori whointroduced this subject and kindly directed themThe authorsalso would like to thank the referee for detailed commentsand valuable suggestions

References

[1] K Stephenson ldquoIsometries of the Nevanlinna classrdquo IndianaUniversity Mathematics Journal vol 26 no 2 pp 307ndash324 1977

[2] A V Subbotin ldquoIsometries of Privalov spaces of holomorphicfunctions of several variablesrdquo Journal ofMathematical Sciencesvol 135 no 1 pp 2794ndash2802 2006

[3] Y Iida and N Mochizuki ldquoIsometries of some 119865-algebras ofholomorphic functionsrdquo Archiv der Mathematik vol 71 no 4pp 297ndash300 1998

[4] A V Subbotin ldquoLinear isometry groups of Privalovrsquos spacesof holomorphic functions of several variablesrdquoDoklady Mathe-matics vol 60 no 1 pp 77ndash79 1999

[5] H O Kim ldquoOn an 119865-algebra of holomorphic functionsrdquoCanadian Journal of Mathematics vol 40 no 3 pp 718ndash7411988

[6] B R Choe and H O Kim ldquoOn the boundary behavior offunctions holomorphic on the ballrdquo Complex Variables Theoryand Application vol 20 no 1ndash4 pp 53ndash61 1992

Journal of Function Spaces 5

[7] H O Kim and Y Y Park ldquoMaximal functions of plurisubhar-monic functionsrdquo Tsukuba Journal of Mathematics vol 16 no1 pp 11ndash18 1992

[8] A V Subbotin ldquoGroups of linear isometries of spaces M119902 ofholomorphic functions of several complex variablesrdquo Mathe-matical Notes vol 83 no 3 pp 437ndash440 2008

[9] Y Iida and K Kasuga ldquoMultiplicative isometries on some 119865-algebras of holomorphic functionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 681637 3 pages 2013

[10] S Mazur and S Ulam ldquoSur les transformationes isometriquesdrsquoespaces vectoriels normesrdquo Comptes Rendus de lrsquoAcademie desSciences Paris vol 194 pp 946ndash948 1932

[11] A J Ellis ldquoReal characterizations of function algebras amongstfunction spacesrdquo Bulletin of the London Mathematical Societyvol 22 no 4 pp 381ndash385 1990

[12] O Hatori and Y Iida ldquoMultiplicative isometries on the SmirnovclassrdquoCentral European Journal ofMathematics vol 9 no 5 pp1051ndash1056 2011

[13] O Hatori Y Iida S Stevic and S-I Ueki ldquoMultiplicativeisometries on F-algebras of holomorphic functions rdquo AbstractandApplied Analysis vol 2012 Article ID 125987 16 pages 2012

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

Journal of Function Spaces 3

for some 119901 gt 0 Then

int119866

1003816100381610038161003816119860119891 (119909)1003816100381610038161003816119902120583 (119889119909) = int

119866

1003816100381610038161003816119891 (119909)1003816100381610038161003816119902120583 (119889119909) 119891 isin 119862 (16)

for all 119902 = 119901 + 119897 where 119897 isin Z+

Proof We follow [2 Lemma 2] 119860 is homogeneous withpositive coefficients so we have using (15)

int119866

(log (1 + 119905 1003816100381610038161003816119860119891 (119909)1003816100381610038161003816))119901

119905119901120583 (119889119909)

= int119866

(log (1 + 119905 1003816100381610038161003816119891 (119909)1003816100381610038161003816))119901

119905119901120583 (119889119909) 119891 isin 119862

(17)

for any 119905 gt 0 Letting 119905 rarr 0+ we obtain

int119866

1003816100381610038161003816119860119891 (119909)1003816100381610038161003816119901120583 (119889119909) = int

119866

1003816100381610038161003816119891 (119909)1003816100381610038161003816119901120583 (119889119909) 119891 isin 119862 (18)

Next we argue by induction Let 119896 isin N and suppose that (16)holds for 119902 = 119901 + 1 119901 + 2 119901 + 119896 minus 1 Then for any 119905 gt 0and any function 119891 isin 119862 we have

int119866

1119905119901+119896

(log (1 + 119905 1003816100381610038161003816119860119891 (119909)1003816100381610038161003816))119901

minus

119896minus1sum

119897=0119888119897 (119905

1003816100381610038161003816119860119891 (119909)1003816100381610038161003816)119901+119897120583 (119889119909)

= int119866

1119905119901+119896

(log (1 + 119905 1003816100381610038161003816119891 (119909)1003816100381610038161003816))119901

minus

119896minus1sum

119897=0119888119897 (119905

1003816100381610038161003816119891 (119909)1003816100381610038161003816)119901+119897120583 (119889119909)

(19)

where 119888119897 are the Taylor coefficients of the function ((log(1 +119909))119909)

119901 at zeroAssume first that 119891 isin 119871

119901+119896(119866 120583) It is easy to see that

the integrand on the right-hand side of (19) converges to119888119896|119891(119909)|

119901+119896 as 119905 rarr 0+ this integrand is of fixed signby Lemmas 4 and 5 and is dominated by the function119862119896|119891(119909)|

119901+119896 with some constant119862119896 By the Lebesgue theoremon dominated convergence the right-hand side of (19)converges to the integral of 119888119896|119891(119909)|

119901+119896 Therefore the left-hand side of (19) has a finite limit and by the Fatou theoremthe function 119888119896|119860119891(119909)|

119901+119896 is integrable Since 119888119896 = 0 for any119896 by Lemmas 4 and 5 we deduce that 119860119891 isin 119871

119901+119896(119866 120583) and

repeating the above arguments we see that the left-hand sideof (19) converges to the integral of 119888119896|119860119891(119909)|

119901+119896 as 119905 rarr 0+Therefore passing to the limit in (19) as 119905 rarr 0+ and dividingthe result by 119888119896 = 0 we obtain (16) for 119902 = 119901 + 119896 The caseof 119860119891 isin 119871

119901+119896(119866 120583) can be considered in a similar way For

119891119860119891 notin 119871119901+119896

(119866 120583) relation (16) with 119902 = 119901 + 119896 is trivial

Proof of Proposition 1 Suppose first that 119901 ge 1 Let 119891 119892 isin

119867119901(119883) We easily confirm that by utilizing Lemma 2 and

the celebrated theorem of Mazur and Ulam [10] 119860|119867119901(119883) isa real-linear isometry in a way similar to [9 Proposition 1]

Next suppose that 0 lt 119901 lt 1 We define the class119867119901

119898(119883) (0 lt 119901 lt 1) as the set of all holomorphic functions 119891

on119883 such that

119889119867119901

119898

(119891 0) = int120597119883

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816119901119889120590 (119911) lt infin (20)

and define a metric 119889119867119901

119898

(119891 119892) = 119889119867119901

119898

(119891 minus 119892 0) for 119891 119892 isin

119867119901

119898(119883) If119860 119872

119901(119883) rarr 119872

119901(119883) is a surjective isometry and

119860 is 2-homogeneous in the sense that 119860(2119891) = 2119860(119891) holdsfor every119891 isin 119872

119901(119883) we confirm that119860 119867

119901

119898(119883) rarr 119867

119901

119898(119883)

is also a surjective isometryFor 0 lt 119901 lt 1 let 119891 119892 isin 119867

119901(119883) We have 119860(119891)2119898 =

119860(1198912119898) (119898 isin N) since 119860(2119891) = 2119860(119891) Then the followingequality holds

int120597119883

sup0le119903lt1

10038161003816100381610038161003816100381610038161003816

119891 (119903119911)

211989810038161003816100381610038161003816100381610038161003816

119901

119889120590 (119911)

minusint120597119883

(log(1+ sup0le119903lt1

10038161003816100381610038161003816100381610038161003816

119891 (119903119911)

211989810038161003816100381610038161003816100381610038161003816))

119901

119889120590 (119911)

= int120597119883

sup0le119903lt1

100381610038161003816100381610038161003816100381610038161003816

(119860119891) (119903119911)

2119898100381610038161003816100381610038161003816100381610038161003816

119901

119889120590 (119911)

minusint120597119883

(log(1+ sup0le119903lt1

100381610038161003816100381610038161003816100381610038161003816

(119860119891) (119903119911)

2119898100381610038161003816100381610038161003816100381610038161003816

))

119901

119889120590 (119911)

(21)

Therefore it follows that

int120597119883

1(12119898)119901+1

(12119898

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816)

119901

minus(log(1 + 12119898

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

= int120597119883

1(12119898)119901+1

(12119898

sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816)

119901

minus(log(1 + 12119898

sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

(22)

Using the elementary inequalities log(1 + 119909119910) le log(1 + 119909) +log(1 + 119910) (119909 ge 0 119910 ge 0) and (119886 + 119887)119901 le 2119901(119886119901 + 119887119901) (119886 ge 0119887 ge 0 119901 gt 0) we confirm that the integrand on the left-hand side of (22) is dominated by an 119871

1(120597119883)-function The

integrand on the right-hand side of (22) is also dominated byan 1198711(120597119883)-function in the same way Applying the Lebesguetheorem on dominated convergence and Lemma 3 on bothsides of (22) we have the equality

int120597119883

119901

2( sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816)

119901+1

119889120590 (119911)

= int120597119883

119901

2( sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816)

119901+1

119889120590 (119911)

(23)

4 Journal of Function Spaces

Hence we obtain

int120597119883

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816119901+1

119889120590 (119911)

= int120597119883

sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816119901+1

119889120590 (119911)

(24)

The equivalence of the norms sdot 119901+1 and sdot 119867119901+1119898

guaranteesthat 119860 119867

119901+1(119883) rarr 119867

119901+1(119883) is a surjective isometry By

using Mazur-Ulam theorem again 119860|119867119901+1(119883) is a real-linearisometry since119867119901+1(119883) is a normed vector space and119860(0) =0

We consider an arbitrary function 119891 isin 119872119901(119883) (0 lt 119901 lt

infin) and the cone 119862119891 = 120582119872119891(120577) | 120582 gt 0 generated by 119891Here119872119891(120577) = sup0le119903lt1|119891(119903120577)| is the radial maximal functionfor119891Moreover consider the followingmapping on this cone

119860 120582119872119891 (120577) 997891997888rarr 120582119872(119860119891) (120577) = 119872(119860 [120582119891]) (120577)

120577 isin 120597119883 120582 ge 0(25)

Since119860 is isometric with respect to the metric 119889119872119901 it followsthat the assumptions of Lemma 6 hold on the cone 119862119891

119860 119872119901(119883) rarr 119872

119901(119883) is a surjective isometry so the

equation

int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

= int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

(26)

guarantees the equalities

int120597119883

( sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816)

119901+119897

119889120590 (119911)

= int120597119883

( sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816)

119901+119897

119889120590 (119911)

(27)

for all 119897 isin Z+Therefore119860 is isometric in the norm119867119901+119897

119898(119901 gt

0 119897 = 0 1 2 )Since 119889120590 is a finite measure we verify that

lim119897rarrinfin

10038171003817100381710038171198911003817100381710038171003817119867119901+119897119898

=10038171003817100381710038171198911003817100381710038171003817119867infin119898

(28)

holds for every 119891 isin 119867infin(119883) and it is clear that 119891119867infin

119898

=

119891infin Moreover 119891119901 = 119860(119891)119901 for every 119891 isin 119867infin(119883) and

lim119901rarrinfin119860(119891)119901 = 119860(119891)infin so we have 119860(119891) isin 119867infin(119883)

and 119891infin = 119860(119891)infin for every 119891 isin 119867infin(119883) Similarly we

see that 119891 isin 119867infin(119883) if 119860(119891) belongs to 119867infin(119883) Therefore

119860|119867infin(119883) is a surjective isometry with respect to sdot infin from119867infin(119883) onto itselfWemay suppose that119867infin(119883) is a uniform

algebra on the maximal ideal space and the maximal idealspace is connected by the Silov idempotent theorem hencewe see that 119860|119867infin(119883) is complex-linear or conjugate linear by[11Theorem] As for the rest of this proof we follow the proofof [9 Proposition 1]

Finally we consider multiplicative isometries from119872119901(119883) (0 lt 119901 lt infin) onto itself Recall that 119860 119872

119901(119883) rarr

119872119901(119883) is multiplicative if 119860(119891119892) = 119860(119891)119860(119892) for every

119891 119892 isin 119872119901(119883)

The following theorem is proved by the same method as[9 Theorem 2] therefore we do not prove it here

Theorem 7 Let 0 lt 119901 lt infin and 119860 be a multiplicative (notnecessarily linear) isometry from119872

119901(119883) onto itselfThen there

exists a holomorphic automorphismΦ on119883 such that either ofthe following holds

119860 (119891) = 119891 ∘Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (29)

or

119860 (119891) = 119891 ∘ Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (30)

where Φ is a unitary transformation for 119883 = 119861119899Φ(1199111 119911119899) = (12058211199111198941 120582119899119911119894119899) for 119883 = 119880

119899 where |120582119895| = 1for every 1 le 119895 le 119899 and (1198941 119894119899) is some permutation of theintegers from 1 through 119899

Remark 8 We note that surjective multiplicative isometriesof the class 119872119901(119883) (0 lt 119901 lt infin) have the same form assurjective multiplicative isometries of 119872119901(119883) (119901 isin N) [9Theorem 2] the Smirnov class [12 Theorem 22] and thePrivalov class [13 Corollary 34]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere gratitude to Pro-fessor Takahiko Nakazi and Professor Osamu Hatori whointroduced this subject and kindly directed themThe authorsalso would like to thank the referee for detailed commentsand valuable suggestions

References

[1] K Stephenson ldquoIsometries of the Nevanlinna classrdquo IndianaUniversity Mathematics Journal vol 26 no 2 pp 307ndash324 1977

[2] A V Subbotin ldquoIsometries of Privalov spaces of holomorphicfunctions of several variablesrdquo Journal ofMathematical Sciencesvol 135 no 1 pp 2794ndash2802 2006

[3] Y Iida and N Mochizuki ldquoIsometries of some 119865-algebras ofholomorphic functionsrdquo Archiv der Mathematik vol 71 no 4pp 297ndash300 1998

[4] A V Subbotin ldquoLinear isometry groups of Privalovrsquos spacesof holomorphic functions of several variablesrdquoDoklady Mathe-matics vol 60 no 1 pp 77ndash79 1999

[5] H O Kim ldquoOn an 119865-algebra of holomorphic functionsrdquoCanadian Journal of Mathematics vol 40 no 3 pp 718ndash7411988

[6] B R Choe and H O Kim ldquoOn the boundary behavior offunctions holomorphic on the ballrdquo Complex Variables Theoryand Application vol 20 no 1ndash4 pp 53ndash61 1992

Journal of Function Spaces 5

[7] H O Kim and Y Y Park ldquoMaximal functions of plurisubhar-monic functionsrdquo Tsukuba Journal of Mathematics vol 16 no1 pp 11ndash18 1992

[8] A V Subbotin ldquoGroups of linear isometries of spaces M119902 ofholomorphic functions of several complex variablesrdquo Mathe-matical Notes vol 83 no 3 pp 437ndash440 2008

[9] Y Iida and K Kasuga ldquoMultiplicative isometries on some 119865-algebras of holomorphic functionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 681637 3 pages 2013

[10] S Mazur and S Ulam ldquoSur les transformationes isometriquesdrsquoespaces vectoriels normesrdquo Comptes Rendus de lrsquoAcademie desSciences Paris vol 194 pp 946ndash948 1932

[11] A J Ellis ldquoReal characterizations of function algebras amongstfunction spacesrdquo Bulletin of the London Mathematical Societyvol 22 no 4 pp 381ndash385 1990

[12] O Hatori and Y Iida ldquoMultiplicative isometries on the SmirnovclassrdquoCentral European Journal ofMathematics vol 9 no 5 pp1051ndash1056 2011

[13] O Hatori Y Iida S Stevic and S-I Ueki ldquoMultiplicativeisometries on F-algebras of holomorphic functions rdquo AbstractandApplied Analysis vol 2012 Article ID 125987 16 pages 2012

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

4 Journal of Function Spaces

Hence we obtain

int120597119883

sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816119901+1

119889120590 (119911)

= int120597119883

sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816119901+1

119889120590 (119911)

(24)

The equivalence of the norms sdot 119901+1 and sdot 119867119901+1119898

guaranteesthat 119860 119867

119901+1(119883) rarr 119867

119901+1(119883) is a surjective isometry By

using Mazur-Ulam theorem again 119860|119867119901+1(119883) is a real-linearisometry since119867119901+1(119883) is a normed vector space and119860(0) =0

We consider an arbitrary function 119891 isin 119872119901(119883) (0 lt 119901 lt

infin) and the cone 119862119891 = 120582119872119891(120577) | 120582 gt 0 generated by 119891Here119872119891(120577) = sup0le119903lt1|119891(119903120577)| is the radial maximal functionfor119891Moreover consider the followingmapping on this cone

119860 120582119872119891 (120577) 997891997888rarr 120582119872(119860119891) (120577) = 119872(119860 [120582119891]) (120577)

120577 isin 120597119883 120582 ge 0(25)

Since119860 is isometric with respect to the metric 119889119872119901 it followsthat the assumptions of Lemma 6 hold on the cone 119862119891

119860 119872119901(119883) rarr 119872

119901(119883) is a surjective isometry so the

equation

int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

= int120597119883

(log(1+ sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816))

119901

119889120590 (119911)

(26)

guarantees the equalities

int120597119883

( sup0le119903lt1

1003816100381610038161003816119891 (119903119911)1003816100381610038161003816)

119901+119897

119889120590 (119911)

= int120597119883

( sup0le119903lt1

1003816100381610038161003816(119860119891) (119903119911)1003816100381610038161003816)

119901+119897

119889120590 (119911)

(27)

for all 119897 isin Z+Therefore119860 is isometric in the norm119867119901+119897

119898(119901 gt

0 119897 = 0 1 2 )Since 119889120590 is a finite measure we verify that

lim119897rarrinfin

10038171003817100381710038171198911003817100381710038171003817119867119901+119897119898

=10038171003817100381710038171198911003817100381710038171003817119867infin119898

(28)

holds for every 119891 isin 119867infin(119883) and it is clear that 119891119867infin

119898

=

119891infin Moreover 119891119901 = 119860(119891)119901 for every 119891 isin 119867infin(119883) and

lim119901rarrinfin119860(119891)119901 = 119860(119891)infin so we have 119860(119891) isin 119867infin(119883)

and 119891infin = 119860(119891)infin for every 119891 isin 119867infin(119883) Similarly we

see that 119891 isin 119867infin(119883) if 119860(119891) belongs to 119867infin(119883) Therefore

119860|119867infin(119883) is a surjective isometry with respect to sdot infin from119867infin(119883) onto itselfWemay suppose that119867infin(119883) is a uniform

algebra on the maximal ideal space and the maximal idealspace is connected by the Silov idempotent theorem hencewe see that 119860|119867infin(119883) is complex-linear or conjugate linear by[11Theorem] As for the rest of this proof we follow the proofof [9 Proposition 1]

Finally we consider multiplicative isometries from119872119901(119883) (0 lt 119901 lt infin) onto itself Recall that 119860 119872

119901(119883) rarr

119872119901(119883) is multiplicative if 119860(119891119892) = 119860(119891)119860(119892) for every

119891 119892 isin 119872119901(119883)

The following theorem is proved by the same method as[9 Theorem 2] therefore we do not prove it here

Theorem 7 Let 0 lt 119901 lt infin and 119860 be a multiplicative (notnecessarily linear) isometry from119872

119901(119883) onto itselfThen there

exists a holomorphic automorphismΦ on119883 such that either ofthe following holds

119860 (119891) = 119891 ∘Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (29)

or

119860 (119891) = 119891 ∘ Φ 119891119900119903 119890V119890119903119910 119891 isin 119872119901(119883) (30)

where Φ is a unitary transformation for 119883 = 119861119899Φ(1199111 119911119899) = (12058211199111198941 120582119899119911119894119899) for 119883 = 119880

119899 where |120582119895| = 1for every 1 le 119895 le 119899 and (1198941 119894119899) is some permutation of theintegers from 1 through 119899

Remark 8 We note that surjective multiplicative isometriesof the class 119872119901(119883) (0 lt 119901 lt infin) have the same form assurjective multiplicative isometries of 119872119901(119883) (119901 isin N) [9Theorem 2] the Smirnov class [12 Theorem 22] and thePrivalov class [13 Corollary 34]

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors wish to express their sincere gratitude to Pro-fessor Takahiko Nakazi and Professor Osamu Hatori whointroduced this subject and kindly directed themThe authorsalso would like to thank the referee for detailed commentsand valuable suggestions

References

[1] K Stephenson ldquoIsometries of the Nevanlinna classrdquo IndianaUniversity Mathematics Journal vol 26 no 2 pp 307ndash324 1977

[2] A V Subbotin ldquoIsometries of Privalov spaces of holomorphicfunctions of several variablesrdquo Journal ofMathematical Sciencesvol 135 no 1 pp 2794ndash2802 2006

[3] Y Iida and N Mochizuki ldquoIsometries of some 119865-algebras ofholomorphic functionsrdquo Archiv der Mathematik vol 71 no 4pp 297ndash300 1998

[4] A V Subbotin ldquoLinear isometry groups of Privalovrsquos spacesof holomorphic functions of several variablesrdquoDoklady Mathe-matics vol 60 no 1 pp 77ndash79 1999

[5] H O Kim ldquoOn an 119865-algebra of holomorphic functionsrdquoCanadian Journal of Mathematics vol 40 no 3 pp 718ndash7411988

[6] B R Choe and H O Kim ldquoOn the boundary behavior offunctions holomorphic on the ballrdquo Complex Variables Theoryand Application vol 20 no 1ndash4 pp 53ndash61 1992

Journal of Function Spaces 5

[7] H O Kim and Y Y Park ldquoMaximal functions of plurisubhar-monic functionsrdquo Tsukuba Journal of Mathematics vol 16 no1 pp 11ndash18 1992

[8] A V Subbotin ldquoGroups of linear isometries of spaces M119902 ofholomorphic functions of several complex variablesrdquo Mathe-matical Notes vol 83 no 3 pp 437ndash440 2008

[9] Y Iida and K Kasuga ldquoMultiplicative isometries on some 119865-algebras of holomorphic functionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 681637 3 pages 2013

[10] S Mazur and S Ulam ldquoSur les transformationes isometriquesdrsquoespaces vectoriels normesrdquo Comptes Rendus de lrsquoAcademie desSciences Paris vol 194 pp 946ndash948 1932

[11] A J Ellis ldquoReal characterizations of function algebras amongstfunction spacesrdquo Bulletin of the London Mathematical Societyvol 22 no 4 pp 381ndash385 1990

[12] O Hatori and Y Iida ldquoMultiplicative isometries on the SmirnovclassrdquoCentral European Journal ofMathematics vol 9 no 5 pp1051ndash1056 2011

[13] O Hatori Y Iida S Stevic and S-I Ueki ldquoMultiplicativeisometries on F-algebras of holomorphic functions rdquo AbstractandApplied Analysis vol 2012 Article ID 125987 16 pages 2012

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

Journal of Function Spaces 5

[7] H O Kim and Y Y Park ldquoMaximal functions of plurisubhar-monic functionsrdquo Tsukuba Journal of Mathematics vol 16 no1 pp 11ndash18 1992

[8] A V Subbotin ldquoGroups of linear isometries of spaces M119902 ofholomorphic functions of several complex variablesrdquo Mathe-matical Notes vol 83 no 3 pp 437ndash440 2008

[9] Y Iida and K Kasuga ldquoMultiplicative isometries on some 119865-algebras of holomorphic functionsrdquo Journal of Function Spacesand Applications vol 2013 Article ID 681637 3 pages 2013

[10] S Mazur and S Ulam ldquoSur les transformationes isometriquesdrsquoespaces vectoriels normesrdquo Comptes Rendus de lrsquoAcademie desSciences Paris vol 194 pp 946ndash948 1932

[11] A J Ellis ldquoReal characterizations of function algebras amongstfunction spacesrdquo Bulletin of the London Mathematical Societyvol 22 no 4 pp 381ndash385 1990

[12] O Hatori and Y Iida ldquoMultiplicative isometries on the SmirnovclassrdquoCentral European Journal ofMathematics vol 9 no 5 pp1051ndash1056 2011

[13] O Hatori Y Iida S Stevic and S-I Ueki ldquoMultiplicativeisometries on F-algebras of holomorphic functions rdquo AbstractandApplied Analysis vol 2012 Article ID 125987 16 pages 2012

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