# Bounded Subsets of ClassesMpX of Holomorphic Functions

Authors: Yasuo Iida
Journal: Journal of Function Spaces
 (2017)

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## Abstract

Some characterizations of boundedness inMp(X) will be described, where Mp(X)(0<p<∞) are F-algebras which consist of holomorphic functions defined by maximal functions.


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## Body

## 1. Introduction

Letn be a positive integer. The space of n-complex variables z=(z1,…,zn) is denoted by Cn. The unit polydisk z∈Cn:zj<1,1≤j≤n is denoted by Un and the distinguished boundary Tn is ζ∈Cn:ζj=1,1≤j≤n. The unit ball z∈Cn:∑j=1n|zj|2<1 is denoted by Bn and Sn=ζ∈Cn:∑j=1n|ζj|2=1 is its boundary. In this paper, X denotes the unit polydisk or the unit ball for n≥1 and ∂X denotes Tn for X=Un or Sn for X=Bn. The normalized (in the sense that σ(∂X)=1) Lebesgue measure on ∂X is denoted by dσ.The Hardy space onX is denoted by Hq(X)(0<q≤∞). The Nevanlinna class N(X) on X is defined as the set of all holomorphic functions f on X such that(1)sup0≤r<1⁡∫∂Xlog+⁡frζdσζ<∞holds. It is known that f∈N(X) has a finite nontangential limit, denoted by f∗, almost everywhere on ∂X.The Smirnov classN∗(X) is defined as the set of all f∈NX which satisfy the equality(2)sup0≤r<1∫∂Xlog+⁡frζdσζ=∫∂Xlog+⁡f∗ζdσζ.Define a metric(3)dN∗Xf,g=∫∂Xlog⁡1+f∗ζ-g∗ζdσζfor f,g∈N∗(X). With the metric dN∗X·,·N∗(X) is an F-algebra. Recall that an F-algebra is a topological algebra in which the topology arises from a complete metric.The Privalov classNpX, 1<p<∞, is defined as the set of all holomorphic functions f on X such that(4)sup0≤r<1⁡∫∂Xlog+⁡frζpdσζ<∞holds. It is well-known that Np(X) is a subalgebra of N∗(X); hence every f∈Np(X) has a finite nontangential limit almost everywhere on ∂X. Under the metric defined by(5)dNpXf,g=∫∂Xlog⁡1+f∗ζ-g∗ζpdσζ1/pfor f,g∈Np(X), Np(X) becomes an F-algebra (cf. [1]).Now we define the classMp(X). For 0<p<∞, the class Mp(X) is defined as the set of all holomorphic functions f on X such that(6)∫∂Xlog+⁡Mfζpdσζ<∞,where Mf(ζ)≔sup0≤r<1⁡frζ is the maximal function. The class Mp(X) with p=1 in the case n=1 was introduced by Kim in [2]. As for p>0 and n≥1, the class was considered in [3, 4]. For f,g∈MpX, define a metric(7)dMpXf,g=∫∂Xlog⁡1+Mf-gζpdσζαp/p,where αp=min⁡1,p. With this metric Mp(X) is also an F-algebra (see [5]).It is well-known that the following inclusion relations hold:(8)HqX⊊NpX⊊M1X⊊N∗X0<q≤∞,p>1.Moreover, it is known that N(X)⊊Mp(X)0<p<1 [6].A subsetL of a linear topological space A is said to bebounded if for any neighborhood U of zero in A there exists a real number α,0<α<1, such that αL=αf;f∈L⊂U. Yanagihara characterized bounded subsets of N∗X in the case n=1 [7]. As for Mp(X) with p=1 in the case n=1, Kim described some characterizations of boundedness (see [2]). For p>1 and n=1, these characterizations were considered by Meštrović [8]. As for Np(X) with p>1 in the case n≥1, Subbotin investigated the properties of boundedness [1].In this paper, we consider some characterizations of boundedness inMp(X) with 0<p<∞ in the case n≥1.


## 2. The Results

Theorem 1.
Let0<p<∞. L⊂Mp(X) is bounded if and only if
(i) there existsK<∞ such that(9)∫∂Xlog+⁡Mfζpdσζ<Kfor all f∈L;
(ii) for eachε>0 there exists δ>0 such that(10)∫Elog+⁡Mfζpdσζ<ε,∀f∈L,for any measurable set E⊂∂X with the Lebesgue measure E<δ.Proof.
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Necessity. Let L be a bounded subset of Mp(X). We put βp=max⁡1,p=p/αp.
(i) For anyη>0, there is a number α0=α0η0<α0<1 such that(11)dMpXαf,0βp=∫∂Xlog⁡1+αMfζpdσζ<ηβpfor all f∈L and |α|≤α0. It follows that(12)∫∂Xlog+⁡αMfζpdσζ<ηβpfor all f∈L and |α|≤α0. Since(13)log+⁡Mf≤log+⁡α0Mf+log⁡1α0,using the elementary inequality(14)a+bp≤2pap+bpa≥0,b≥0,p>0,we have(15)∫∂Xlog+⁡Mfζpdσζ≤2p∫∂Xlog+⁡α0Mfζpdσζ+∫∂Xlog⁡1α0pdσζ=2pηβp+log⁡1α0p=K=constant.Thus (i) is satisfied.
(ii) For givenε>0, we take η as η<ε/2p+11/βp and α0=α0(η) as above. Next take δ>0 such that(16)δlog⁡1α0p<ε2p+1.Then, for each set E⊂∂X with |E|<δ and for every f∈L, we obtain(17)∫Elog+⁡Mfζpdσζ≤2p∫Elog+⁡α0Mfζpdσζ+∫Elog⁡1α0pdσζ≤2pηβp+2pElog⁡1α0p<ε2+ε2<ε.Therefore, the condition (ii) is satisfied.
Sufficiency. Let(18)V=g∈MpX;dMpXg,0<ηbe a neighborhood of 0 in Mp(X). Take ε>0 such that(19)log⁡1+εp+2plog⁡2pε+2pε<ηβp.Then, there is δ(0<δ<ε) such that (ii) is satisfied. For f∈L, we can find Ef⊂∂X so that(20)∂X∖Ef<δ,log+⁡Mfζp≤KδonEfby Chebyshev’s inequality. We have(21)Mfζ≤exp⁡Kδ1/p=Aδ=AonEf.Choose α such that 0<α<ε/A. Then, using inequality (14) and(22)log⁡1+x≤log⁡2+log+⁡xx>0,we obtain, for every f∈L,(23)dMpXαf,0βp=∫∂Xlog⁡1+αMfζpdσζ=∫Ef​+∫∂X∖Ef​≤∫Eflog⁡1+εpdσζ+2p∫∂X∖Eflog⁡2pdσζ+∫∂X∖Eflog+⁡Mfζpdσζ≤log⁡1+εp+2plog⁡2pδ+2pε<ηβp.Therefore we get dMp(X)(αf,0)<η, which shows L is a bounded subset of Mp(X).
The proof of the theorem is complete.Remark 2.
We note that the characterization of boundedness inMp(X)0<p<∞ has the same conditions as the characterization of boundedness in the Smirnov class N∗(X) in the case n=1 ([7], Theorem 1), the class Mp(X) with p=1 in the case n=1 ([2], Theorem 4.1), and the Privalov class Np(X)(1<p<∞) ([1], Theorem 5). On the other hand, we see that (ii) implies (i) in Theorem 1. Suppose that (ii) holds. Then there is a positive integer K such that(24)∫Elog+⁡Mfζpdσζ<1,∀f∈L,for any measurable set E⊂∂X with |E|≤1/K. There are measurable sets E1,…,EK⊂∂X such that ⋃j=1KEj=∂X, Ei∩Ej=⌀ for i≠j, and |Ej|=1/K for every j. Then(25)∫∂Xlog+⁡Mfζpdσζ=∑j=1K∫Ejlog+⁡Mfζpdσζ<Kholds (cf. [8] (Theorem 19 and Remark 20)).Next we show a standard example of a bounded set ofMp(X). The following theorem is easily proved in the same way of [1] (p.236) and [2] (Theorem 4.6); therefore, we do not prove it here.Theorem 3.
Let0<p<∞. If f∈Mp(X), then fρ(z)=fρzz∈X,0≤ρ<1 form a bounded set in Mp(X).Letp>1 and we set fNpX≔dNp(X)(f,0). Subbotin proved an equivalent condition that a subset L⊂Np(X)1<p<∞ is bounded. The following is a theorem by Subbotin.Theorem 4 (see [1]).
Letp>1. A subset L⊂Np(X) is bounded if and only if the following two conditions are satisfied:
(i) There existsK<∞ such that fNp(X)≤K for all f∈L.
(ii) For eachε>0 there exists δ>0 such that(26)∫Elog+⁡f∗ζpdσζ<ε,∀f∈L,for any measurable set E⊂∂X with the Lebesgue measure E<δ.As shown in [1, 3], for any p>1 the class Mp(X) coincides with the class Np(X) and the metrics dMp(X) and dNp(X) are equivalent. Therefore the topologies induced by these metrics are identical on the set Mp(X)=Np(X).The following theorem is clear; therefore the proof may be omitted.Theorem 5.
Letp>1. A subset L⊂Mp(X) is bounded if and only if the following two conditions are satisfied:
(i) There existsK<∞ such that(27)∫∂Xlog+⁡f∗ζpdσζ<Kfor all f∈L.
(ii) For eachε>0 there exists δ>0 such that(28)∫Elog+⁡f∗ζpdσζ<ε,∀f∈L,for any measurable set E⊂∂X with the Lebesgue measure |E|<δ.



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*Source: 7260602-2017-10-08.xml*