Bounded Subsets of Classes 𝑀 𝑝 of Holomorphic Functionsdownloads.hindawi.com/journals/jfs/2017/7260602.pdf · ResearchArticle Bounded Subsets of Classes 𝑀𝑝(𝑋)of Holomorphic

Research ArticleBounded Subsets of Classes119872119901(119883) of Holomorphic Functions

Yasuo Iida

Department of Mathematics Kanazawa Medical University Uchinada Ishikawa 920-0293 Japan

Correspondence should be addressed to Yasuo Iida yiidakanazawa-medacjp

Received 19 July 2017 Accepted 5 September 2017 Published 8 October 2017

Academic Editor Kehe Zhu

Copyright copy 2017 Yasuo Iida This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Some characterizations of boundedness in119872119901(119883) will be described where119872119901(119883) (0 lt 119901 lt infin) are 119865-algebras which consist ofholomorphic functions defined by maximal functions

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 120577 isin C119899 |120577119895| = 1 1 le 119895 le 119899 The unit ball119911 isin C119899 sum119899119895=1 |119911119895|2 lt 1 is denoted by 119861119899 and 119878119899 = 120577 isin C119899 sum119899119895=1 |120577119895|2 = 1 is its boundary In this paper 119883 denotes theunit polydisk or the unit ball for 119899 ge 1 and 120597119883 denotes T119899 for119883 = 119880119899 or 119878119899 for 119883 = 119861119899 The normalized (in the sense that120590(120597119883) = 1) Lebesgue measure on 120597119883 is denoted by 119889120590

TheHardy space on119883 is denoted by119867119902(119883) (0 lt 119902 le infin)The Nevanlinna class 119873(119883) on 119883 is defined as the set of allholomorphic functions 119891 on119883 such that

sup0le119903lt1

int120597119883

log+ 1003816100381610038161003816119891 (119903120577)1003816100381610038161003816 119889120590 (120577) lt infin (1)

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

The Smirnov class 119873lowast(119883) is defined as the set of all 119891 isin119873(119883) which satisfy the equality

sup0le119903lt1

int120597119883

log+ 1003816100381610038161003816119891 (119903120577)1003816100381610038161003816 119889120590 (120577)

= int120597119883

log+ 1003816100381610038161003816119891lowast (120577)1003816100381610038161003816 119889120590 (120577) (2)

Define a metric

119889119873lowast(119883) (119891 119892) = int120597119883

log (1 + 1003816100381610038161003816119891lowast (120577) minus 119892lowast (120577)1003816100381610038161003816) 119889120590 (120577) (3)

for 119891 119892 isin 119873lowast(119883) With the metric 119889119873lowast(119883)(sdot sdot) 119873lowast(119883) is an119865-algebra Recall that an 119865-algebra is a topological algebra inwhich the topology arises from a complete metric

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+ 1003816100381610038161003816119891 (119903120577)1003816100381610038161003816)119901 119889120590 (120577) lt infin (4)

holds It is well-known that119873119901(119883) is a subalgebra of119873lowast(119883)hence every 119891 isin 119873119901(119883) has a finite nontangential limitalmost everywhere on 120597119883 Under the metric defined by

119889119873119901(119883) (119891 119892)= (int120597119883(log (1 + 1003816100381610038161003816119891lowast (120577) minus 119892lowast (120577)1003816100381610038161003816))119901 119889120590 (120577))

1119901 (5)

for 119891 119892 isin 119873119901(119883)119873119901(119883) becomes an 119865-algebra (cf [1])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+119872119891(120577))119901 119889120590 (120577) lt infin (6)

where119872119891(120577) fl sup0le119903lt1|119891(119903120577)| is the maximal functionTheclass119872119901(119883) with 119901 = 1 in the case 119899 = 1 was introduced byKim in [2] As for 119901 gt 0 and 119899 ge 1 the class was consideredin [3 4] For 119891 119892 isin 119872119901(119883) define a metric

119889119872119901(119883) (119891 119892)= int120597119883(log (1 +119872(119891 minus 119892) (120577)))119901 119889120590 (120577)120572119901119901 (7)

HindawiJournal of Function SpacesVolume 2017 Article ID 7260602 4 pageshttpsdoiorg10115520177260602

2 Journal of Function Spaces

where 120572119901 = min(1 119901) With this metric119872119901(119883) is also an 119865-algebra (see [5])

It is well-known that the following inclusion relationshold

119867119902 (119883) ⊊ 119873119901 (119883) ⊊ 1198721 (119883) ⊊ 119873lowast (119883)(0 lt 119902 le infin 119901 gt 1) (8)

Moreover it is known that119873(119883) ⊊ 119872119901(119883) (0 lt 119901 lt 1) [6]A subset 119871 of a linear topological space 119860 is said to be

bounded if for any neighborhood119880 of zero in119860 there exists areal number 120572 0 lt 120572 lt 1 such that 120572119871 = 120572119891 119891 isin 119871 sub 119880Yanagihara characterized bounded subsets of 119873lowast(119883) in thecase 119899 = 1 [7] As for119872119901(119883)with119901 = 1 in the case 119899 = 1 Kimdescribed some characterizations of boundedness (see [2])For 119901 gt 1 and 119899 = 1 these characterizations were consideredby Mestrovic [8] As for119873119901(119883) with 119901 gt 1 in the case 119899 ge 1Subbotin investigated the properties of boundedness [1]

In this paper we consider some characterizations ofboundedness in119872119901(119883) with 0 lt 119901 lt infin in the case 119899 ge 12 The Results

Theorem 1 Let 0 lt 119901 lt infin 119871 sub 119872119901(119883) is bounded if andonly if

(i) there exists 119870 lt infin such that

int120597119883(log+119872119891(120577))119901 119889120590 (120577) lt 119870 (9)

for all 119891 isin 119871(ii) for each 120576 gt 0 there exists 120575 gt 0 such that

int119864(log+119872119891(120577))119901 119889120590 (120577) lt 120576 forall119891 isin 119871 (10)

for anymeasurable set119864 sub 120597119883with the Lebesguemeasure |119864| lt120575Proof

Necessity Let 119871 be a bounded subset of119872119901(119883) We put 120573119901 =max(1 119901) = 119901120572119901

(i) For any 120578 gt 0 there is a number 1205720 = 1205720(120578) (0 lt 1205720 lt1) such that

(119889119872119901(119883) (120572119891 0))120573119901= int120597119883(log (1 + |120572|119872119891 (120577)))119901 119889120590 (120577) lt 120578120573119901 (11)

for all 119891 isin 119871 and |120572| le 1205720 It follows thatint120597119883(log+ |120572|119872119891 (120577))119901 119889120590 (120577) lt 120578120573119901 (12)

for all 119891 isin 119871 and |120572| le 1205720 Sincelog+119872119891 le log+1205720119872119891 + log 1

1205720 (13)

using the elementary inequality

(119886 + 119887)119901 le 2119901 (119886119901 + 119887119901) (119886 ge 0 119887 ge 0 119901 gt 0) (14)

we have

int120597119883(log+119872119891(120577))119901 119889120590 (120577)

le 2119901 (int120597119883(log+1205720119872119891(120577))119901 119889120590 (120577)

+ int120597119883(log 1

1205720)119901 119889120590 (120577)) = 2119901 (120578120573119901

+ (log 11205720)119901) = 119870 = constant

(15)

Thus (i) is satisfied(ii) For given 120576 gt 0 we take 120578 as 120578 lt (1205762119901+1)1120573119901 and1205720 = 1205720(120578) as above Next take 120575 gt 0 such that

120575(log 11205720)119901 lt 120576

2119901+1 (16)

Then for each set 119864 sub 120597119883 with |119864| lt 120575 and for every 119891 isin 119871we obtain

int119864(log+119872119891(120577))119901 119889120590 (120577)

le 2119901 (int119864(log+1205720119872119891(120577))119901 119889120590 (120577)

+ int119864(log 1

1205720)119901 119889120590 (120577)) le 2119901120578120573119901 + 2119901 |119864|

sdot (log 11205720)119901 lt 120576

2 +1205762 lt 120576

(17)

Therefore the condition (ii) is satisfied

Sufficiency Let

119881 = 119892 isin 119872119901 (119883) 119889119872119901(119883) (119892 0) lt 120578 (18)

be a neighborhood of 0 in119872119901(119883) Take 120576 gt 0 such that

(log (1 + 120576))119901 + 2119901 (log 2)119901 120576 + 2119901120576 lt 120578120573119901 (19)

Then there is 120575 (0 lt 120575 lt 120576) such that (ii) is satisfied For119891 isin 119871 we can find 119864119891 sub 120597119883 so that

10038161003816100381610038161003816120597119883 11986411989110038161003816100381610038161003816 lt 120575(log+119872119891(120577))119901 le 119870

120575 on 119864119891(20)

by Chebyshevrsquos inequality We have

119872119891(120577) le exp(119870120575 )1119901 = 119860 (120575) = 119860 on 119864119891 (21)

Journal of Function Spaces 3

Choose 120572 such that 0 lt 120572 lt 120576119860 Then using inequality (14)and

log (1 + 119909) le log 2 + log+119909 (119909 gt 0) (22)

we obtain for every 119891 isin 119871(119889119872119901(119883) (120572119891 0))120573119901= int120597119883(log (1 + |120572|119872119891 (120577)))119901 119889120590 (120577) = int

119864119891

+int120597119883119864119891

le int119864119891

(log (1 + 120576))119901 119889120590 (120577)

+ 2119901 (int120597119883119864119891

(log 2)119901 119889120590 (120577)

+ int120597119883119864119891

(log+119872119891(120577))119901 119889120590 (120577)) le (log (1 + 120576))119901

+ 2119901 (log 2)119901 120575 + 2119901120576 lt 120578120573119901

(23)

Therefore we get 119889119872119901(119883)(120572119891 0) lt 120578 which shows 119871 is abounded subset of119872119901(119883)

The proof of the theorem is complete

Remark 2 We note that the characterization of boundednessin 119872119901(119883) (0 lt 119901 lt infin) has the same conditions as thecharacterization of boundedness in the Smirnov class119873lowast(119883)in the case 119899 = 1 ([7]Theorem 1) the class119872119901(119883)with119901 = 1in the case 119899 = 1 ([2] Theorem 41) and the Privalov class119873119901(119883) (1 lt 119901 lt infin) ([1]Theorem 5) On the other hand wesee that (ii) implies (i) in Theorem 1 Suppose that (ii) holdsThen there is a positive integer 119870 such that


for any measurable set 119864 sub 120597119883 with |119864| le 1119870 There aremeasurable sets 1198641 119864119870 sub 120597119883 such that ⋃119870119895=1 119864119895 = 120597119883119864119894 cap 119864119895 = for 119894 = 119895 and |119864119895| = 1119870 for every 119895 Then

int120597119883(log+119872119891(120577))119901 119889120590 (120577)

= 119870sum119895=1

int119864119895

(log+119872119891(120577))119901 119889120590 (120577) lt 119870(25)

holds (cf [8] (Theorem 19 and Remark 20))Next we show a standard example of a bounded set of119872119901(119883) The following theorem is easily proved in the same

way of [1] (p236) and [2] (Theorem 46) therefore we do notprove it here

Theorem 3 Let 0 lt 119901 lt infin If 119891 isin 119872119901(119883) then 119891120588(119911) =119891(120588119911) (119911 isin 119883 0 le 120588 lt 1) form a bounded set in119872119901(119883)Let 119901 gt 1 and we set |119891|119873119901(119883) fl 119889119873119901(119883)(119891 0) Subbotin

proved an equivalent condition that a subset 119871 sub 119873119901(119883) (1 lt119901 lt infin) is bounded The following is a theorem by Subbotin

Theorem 4 (see [1]) Let 119901 gt 1 A subset 119871 sub 119873119901(119883) isbounded if and only if the following two conditions are satisfied

(i)There exists119870 lt infin such that |119891|119873119901(119883) le 119870 for all119891 isin 119871(ii) For each 120576 gt 0 there exists 120575 gt 0 such that

int119864(log+ 1003816100381610038161003816119891lowast (120577)1003816100381610038161003816)119901 119889120590 (120577) lt 120576 forall119891 isin 119871 (26)

for anymeasurable set119864 sub 120597119883with the Lebesguemeasure |119864| lt120575As shown in [1 3] for any 119901 gt 1 the class 119872119901(119883)

coincides with the class 119873119901(119883) and the metrics 119889119872119901(119883) and119889119873119901(119883) are equivalent Therefore the topologies induced bythese metrics are identical on the set119872119901(119883) = 119873119901(119883)

The following theorem is clear therefore the proof maybe omitted

Theorem 5 Let 119901 gt 1 A subset 119871 sub 119872119901(119883) is bounded if andonly if the following two conditions are satisfied

(i) There exists 119870 lt infin such that

int120597119883(log+ 1003816100381610038161003816119891lowast (120577)1003816100381610038161003816)119901 119889120590 (120577) lt 119870 (27)

for all 119891 isin 119871(ii) For each 120576 gt 0 there exists 120575 gt 0 such that


for anymeasurable set119864 sub 120597119883with the Lebesguemeasure |119864| lt120575Conflicts of Interest

The author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Theauthor is partly supported by theGrant forAssist KAKENfrom Kanazawa Medical University (K2017-6)

References

[1] A V Subbotin ldquoFunctional properties of privalov spacesof holomorphic functions in several variablesrdquo MathematicalNotes vol 65 no 1-2 pp 230ndash237 1999

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

[3] 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

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

[5] V I Gavrilov and A V Subbotin ldquo119865-algebras of holomorphicfunctions in a ball containing the Nevanlinna classrdquoMathemat-ica Montisnigri vol 12 pp 17ndash31 2000 (Russian)


[6] A V Subbotin ldquoGroups of linear isometries of spaces 119872119902 ofholomorphic functions of several complex variablesrdquo Mathe-matical Notes vol 83 no 3-4 pp 437ndash440 2008

[7] N Yanagihara ldquoBounded subsets of some spaces of holo-morphic functionsrdquo Scientific Papers of the College of GeneralEducation University of Tokyo vol 23 pp 19ndash28 1973

[8] R Mestrovic ldquoOn 119865-algebras Mp (1 lt p lt infin) of holomorphicfunctionsrdquo The Scientific World Journal vol 2014 Article ID901726 10 pages 2014

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Stochastic AnalysisInternational Journal of


where 120572119901 = min(1 119901) With this metric119872119901(119883) is also an 119865-algebra (see [5])

It is well-known that the following inclusion relationshold

119867119902 (119883) ⊊ 119873119901 (119883) ⊊ 1198721 (119883) ⊊ 119873lowast (119883)(0 lt 119902 le infin 119901 gt 1) (8)

Moreover it is known that119873(119883) ⊊ 119872119901(119883) (0 lt 119901 lt 1) [6]A subset 119871 of a linear topological space 119860 is said to be

bounded if for any neighborhood119880 of zero in119860 there exists areal number 120572 0 lt 120572 lt 1 such that 120572119871 = 120572119891 119891 isin 119871 sub 119880Yanagihara characterized bounded subsets of 119873lowast(119883) in thecase 119899 = 1 [7] As for119872119901(119883)with119901 = 1 in the case 119899 = 1 Kimdescribed some characterizations of boundedness (see [2])For 119901 gt 1 and 119899 = 1 these characterizations were consideredby Mestrovic [8] As for119873119901(119883) with 119901 gt 1 in the case 119899 ge 1Subbotin investigated the properties of boundedness [1]

In this paper we consider some characterizations ofboundedness in119872119901(119883) with 0 lt 119901 lt infin in the case 119899 ge 12 The Results

Theorem 1 Let 0 lt 119901 lt infin 119871 sub 119872119901(119883) is bounded if andonly if

(i) there exists 119870 lt infin such that

int120597119883(log+119872119891(120577))119901 119889120590 (120577) lt 119870 (9)

for all 119891 isin 119871(ii) for each 120576 gt 0 there exists 120575 gt 0 such that


for anymeasurable set119864 sub 120597119883with the Lebesguemeasure |119864| lt120575Proof

Necessity Let 119871 be a bounded subset of119872119901(119883) We put 120573119901 =max(1 119901) = 119901120572119901

(i) For any 120578 gt 0 there is a number 1205720 = 1205720(120578) (0 lt 1205720 lt1) such that

(119889119872119901(119883) (120572119891 0))120573119901= int120597119883(log (1 + |120572|119872119891 (120577)))119901 119889120590 (120577) lt 120578120573119901 (11)

for all 119891 isin 119871 and |120572| le 1205720 It follows thatint120597119883(log+ |120572|119872119891 (120577))119901 119889120590 (120577) lt 120578120573119901 (12)

for all 119891 isin 119871 and |120572| le 1205720 Sincelog+119872119891 le log+1205720119872119891 + log 1

1205720 (13)

using the elementary inequality

(119886 + 119887)119901 le 2119901 (119886119901 + 119887119901) (119886 ge 0 119887 ge 0 119901 gt 0) (14)

we have

int120597119883(log+119872119891(120577))119901 119889120590 (120577)

le 2119901 (int120597119883(log+1205720119872119891(120577))119901 119889120590 (120577)

+ int120597119883(log 1

1205720)119901 119889120590 (120577)) = 2119901 (120578120573119901

+ (log 11205720)119901) = 119870 = constant

(15)

Thus (i) is satisfied(ii) For given 120576 gt 0 we take 120578 as 120578 lt (1205762119901+1)1120573119901 and1205720 = 1205720(120578) as above Next take 120575 gt 0 such that

120575(log 11205720)119901 lt 120576

2119901+1 (16)

Then for each set 119864 sub 120597119883 with |119864| lt 120575 and for every 119891 isin 119871we obtain

int119864(log+119872119891(120577))119901 119889120590 (120577)

le 2119901 (int119864(log+1205720119872119891(120577))119901 119889120590 (120577)

+ int119864(log 1

1205720)119901 119889120590 (120577)) le 2119901120578120573119901 + 2119901 |119864|

sdot (log 11205720)119901 lt 120576

2 +1205762 lt 120576

(17)

Therefore the condition (ii) is satisfied

Sufficiency Let

119881 = 119892 isin 119872119901 (119883) 119889119872119901(119883) (119892 0) lt 120578 (18)

be a neighborhood of 0 in119872119901(119883) Take 120576 gt 0 such that

(log (1 + 120576))119901 + 2119901 (log 2)119901 120576 + 2119901120576 lt 120578120573119901 (19)

Then there is 120575 (0 lt 120575 lt 120576) such that (ii) is satisfied For119891 isin 119871 we can find 119864119891 sub 120597119883 so that

10038161003816100381610038161003816120597119883 11986411989110038161003816100381610038161003816 lt 120575(log+119872119891(120577))119901 le 119870

120575 on 119864119891(20)

by Chebyshevrsquos inequality We have

119872119891(120577) le exp(119870120575 )1119901 = 119860 (120575) = 119860 on 119864119891 (21)



log (1 + 119909) le log 2 + log+119909 (119909 gt 0) (22)


119864119891

+int120597119883119864119891

le int119864119891

(log (1 + 120576))119901 119889120590 (120577)

+ 2119901 (int120597119883119864119891

(log 2)119901 119889120590 (120577)

+ int120597119883119864119891

(log+119872119891(120577))119901 119889120590 (120577)) le (log (1 + 120576))119901

+ 2119901 (log 2)119901 120575 + 2119901120576 lt 120578120573119901

(23)






int120597119883(log+119872119891(120577))119901 119889120590 (120577)

= 119870sum119895=1

int119864119895

(log+119872119891(120577))119901 119889120590 (120577) lt 119870(25)













int120597119883(log+ 1003816100381610038161003816119891lowast (120577)1003816100381610038161003816)119901 119889120590 (120577) lt 119870 (27)





Acknowledgments


References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






log (1 + 119909) le log 2 + log+119909 (119909 gt 0) (22)


119864119891

+int120597119883119864119891

le int119864119891

(log (1 + 120576))119901 119889120590 (120577)

+ 2119901 (int120597119883119864119891

(log 2)119901 119889120590 (120577)

+ int120597119883119864119891

(log+119872119891(120577))119901 119889120590 (120577)) le (log (1 + 120576))119901

+ 2119901 (log 2)119901 120575 + 2119901120576 lt 120578120573119901

(23)






int120597119883(log+119872119891(120577))119901 119889120590 (120577)

= 119870sum119895=1

int119864119895

(log+119872119891(120577))119901 119889120590 (120577) lt 119870(25)













int120597119883(log+ 1003816100381610038161003816119891lowast (120577)1003816100381610038161003816)119901 119889120590 (120577) lt 119870 (27)





Acknowledgments


References

















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of















Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Documents

Bounded Subsets of Classes 𝑀 𝑝 of Holomorphic Functionsdownloads.hindawi.com/journals/jfs/2017/7260602.pdf · ResearchArticle Bounded Subsets of Classes 𝑀𝑝(𝑋)of Holomorphic