JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi 10.1155/2017/7260602 7260602 Research Article Bounded Subsets of Classes MpX of Holomorphic Functions http://orcid.org/0000-0003-1113-2669 Iida Yasuo 1 Zhu Kehe Department of Mathematics Kanazawa Medical University Uchinada Ishikawa 920-0293 Japan kanazawa-med.ac.jp 2017 8102017 2017 19 07 2017 05 09 2017 8102017 2017 Copyright © 2017 Yasuo Iida. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

Kanazawa Medical University K2017-6
1. Introduction

Let n be a positive integer. The space of n-complex variables z=(z1,,zn) is denoted by Cn. The unit polydisk zCn:zj<1,1jn is denoted by Un and the distinguished boundary Tn is ζCn:ζj=1,1jn. The unit ball zCn:j=1n|zj|2<1 is denoted by Bn and Sn=ζCn:j=1n|ζj|2=1 is its boundary. In this paper, X denotes the unit polydisk or the unit ball for n1 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 on X 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)sup0r<1Xlog+frζdσζ<holds. It is known that fN(X) has a finite nontangential limit, denoted by f, almost everywhere on X.

The Smirnov class N(X) is defined as the set of all fNX which satisfy the equality(2)sup0r<1Xlog+frζdσζ=Xlog+fζdσζ.Define a metric(3)dNXf,g=Xlog1+fζ-gζdσζfor f,gN(X). With the metric dNX·,·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 class NpX, 1<p<, is defined as the set of all holomorphic functions f on X such that(4)sup0r<1Xlog+frζpdσζ<holds. It is well-known that Np(X) is a subalgebra of N(X); hence every fNp(X) has a finite nontangential limit almost everywhere on X. Under the metric defined by(5)dNpXf,g=Xlog1+fζ-gζpdσζ1/pfor f,gNp(X), Np(X) becomes an F-algebra (cf. ).

Now we define the class Mp(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)sup0r<1frζ is the maximal function. The class Mp(X) with p=1 in the case n=1 was introduced by Kim in . As for p>0 and n1, the class was considered in [3, 4]. For f,gMpX, define a metric(7)dMpXf,g=Xlog1+Mf-gζpdσζαp/p,where αp=min1,p. With this metric Mp(X) is also an F-algebra (see ).

It is well-known that the following inclusion relations hold:(8)HqXNpXM1XNX0<q,p>1.Moreover, it is known that N(X)Mp(X)0<p<1 .

A subset L of a linear topological space A is said to be bounded if for any neighborhood U of zero in A there exists a real number α,0<α<1, such that αL=αf;fLU. Yanagihara characterized bounded subsets of NX in the case n=1 . As for Mp(X) with p=1 in the case n=1, Kim described some characterizations of boundedness (see ). For p>1 and n=1, these characterizations were considered by Meštrović . As for Np(X) with p>1 in the case n1, Subbotin investigated the properties of boundedness .

In this paper, we consider some characterizations of boundedness in Mp(X) with 0<p< in the case n1.

2. The Results Theorem 1.

Let 0<p<. LMp(X) is bounded if and only if

(i) there exists K< such that(9)Xlog+Mfζpdσζ<Kfor all fL;

(ii) for each ε>0 there exists δ>0 such that(10)Elog+Mfζpdσζ<ε,fL,for any measurable set EX with the Lebesgue measure E<δ.

Proof.

Necessity. Let L be a bounded subset of Mp(X). We put βp=max1,p=p/αp.

(i) For any η>0, there is a number α0=α0η0<α0<1 such that(11)dMpXαf,0βp=Xlog1+αMfζpdσζ<ηβpfor all fL and |α|α0. It follows that(12)Xlog+αMfζpdσζ<ηβpfor all fL and |α|α0. Since(13)log+Mflog+α0Mf+log1α0,using the elementary inequality(14)a+bp2pap+bpa0,b0,p>0,we have(15)Xlog+Mfζpdσζ2pXlog+α0Mfζpdσζ+Xlog1α0pdσζ=2pηβp+log1α0p=K=constant.Thus (i) is satisfied.

(ii) For given ε>0, we take η as η<ε/2p+11/βp and α0=α0(η) as above. Next take δ>0 such that(16)δlog1α0p<ε2p+1.Then, for each set EX with |E|<δ and for every fL, we obtain(17)Elog+Mfζpdσζ2pElog+α0Mfζpdσζ+Elog1α0pdσζ2pηβp+2pElog1α0p<ε2+ε2<ε.Therefore, the condition (ii) is satisfied.

Sufficiency. Let(18)V=gMpX;dMpXg,0<ηbe a neighborhood of 0 in Mp(X). Take ε>0 such that(19)log1+εp+2plog2pε+2pε<ηβp.Then, there is δ(0<δ<ε) such that (ii) is satisfied. For fL, we can find EfX so that(20)XEf<δ,log+MfζpKδonEfby Chebyshev’s inequality. We have(21)MfζexpKδ1/p=Aδ=AonEf.Choose α such that 0<α<ε/A. Then, using inequality (14) and(22)log1+xlog2+log+xx>0,we obtain, for every fL,(23)dMpXαf,0βp=Xlog1+αMfζpdσζ=Ef+XEfEflog1+εpdσζ+2pXEflog2pdσζ+XEflog+Mfζpdσζlog1+εp+2plog2pδ+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 in Mp(X)0<p< has the same conditions as the characterization of boundedness in the Smirnov class N(X) in the case n=1 (, Theorem 1), the class Mp(X) with p=1 in the case n=1 (, Theorem 4.1), and the Privalov class Np(X)(1<p<) (, 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,fL,for any measurable set EX with |E|1/K. There are measurable sets E1,,EKX such that j=1KEj=X, EiEj= for ij, and |Ej|=1/K for every j. Then(25)Xlog+Mfζpdσζ=j=1KEjlog+Mfζpdσζ<Kholds (cf.  (Theorem 19 and Remark 20)).

Next we show a standard example of a bounded set of Mp(X). The following theorem is easily proved in the same way of  (p.236) and  (Theorem 4.6); therefore, we do not prove it here.

Theorem 3.

Let 0<p<. If fMp(X), then fρ(z)=fρzzX,0ρ<1 form a bounded set in Mp(X).

Let p>1 and we set fNpXdNp(X)(f,0). Subbotin proved an equivalent condition that a subset LNp(X)1<p< is bounded. The following is a theorem by Subbotin.

Theorem 4 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let p>1. A subset LNp(X) is bounded if and only if the following two conditions are satisfied:

(i) There exists K< such that fNp(X)K for all fL.

(ii) For each ε>0 there exists δ>0 such that(26)Elog+fζpdσζ<ε,fL,for any measurable set EX 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.

Let p>1. A subset LMp(X) is bounded if and only if the following two conditions are satisfied:

(i) There exists K< such that(27)Xlog+fζpdσζ<Kfor all fL.

(ii) For each ε>0 there exists δ>0 such that(28)Elog+fζpdσζ<ε,fL,for any measurable set EX with the Lebesgue measure |E|<δ.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The author is partly supported by the Grant for Assist KAKEN from Kanazawa Medical University (K2017-6).

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