#### Abstract

Some characterizations of boundedness in will be described, where are -algebras which consist of holomorphic functions defined by maximal functions.

#### 1. Introduction

Let be a positive integer. The space of -complex variables is denoted by . The unit polydisk is denoted by and the distinguished boundary is . The unit ball is denoted by and is its boundary. In this paper, denotes the unit polydisk or the unit ball for and denotes for or for . The normalized (in the sense that ) Lebesgue measure on is denoted by .

The Hardy space on is denoted by . The Nevanlinna class on is defined as the set of all holomorphic functions on such thatholds. It is known that has a finite nontangential limit, denoted by , almost everywhere on .

The Smirnov class is defined as the set of all which satisfy the equalityDefine a metricfor . With the metric is an -algebra. Recall that an -algebra is a topological algebra in which the topology arises from a complete metric.

The Privalov class , , is defined as the set of all holomorphic functions on such thatholds. It is well-known that is a subalgebra of ; hence every has a finite nontangential limit almost everywhere on . Under the metric defined byfor , becomes an -algebra (cf. [1]).

Now we define the class . For , the class is defined as the set of all holomorphic functions on such thatwhere is the maximal function. The class with in the case was introduced by Kim in [2]. As for and , the class was considered in [3, 4]. For , define a metricwhere . With this metric is also an -algebra (see [5]).

It is well-known that the following inclusion relations hold:Moreover, it is known that [6].

A subset of a linear topological space is said to be* bounded* if for any neighborhood of zero in there exists a real number , such that . Yanagihara characterized bounded subsets of in the case [7]. As for with in the case , Kim described some characterizations of boundedness (see [2]). For and , these characterizations were considered by Meštrović [8]. As for with in the case , Subbotin investigated the properties of boundedness [1].

In this paper, we consider some characterizations of boundedness in with in the case .

#### 2. The Results

Theorem 1. *Let . is bounded if and only if**(i) there exists such thatfor all ;**(ii) for each there exists such thatfor any measurable set with the Lebesgue measure .*

*Proof. ****Necessity.* Let be a bounded subset of . We put .

(i) For any , there is a number such thatfor all and . It follows thatfor all and . Sinceusing the elementary inequalitywe haveThus (i) is satisfied.

(ii) For given , we take as and as above. Next take such thatThen, for each set with and for every , we obtainTherefore, the condition (ii) is satisfied.*Sufficiency*. Letbe a neighborhood of in . Take such thatThen, there is such that (ii) is satisfied. For , we can find so thatby Chebyshev’s inequality. We haveChoose such that . Then, using inequality (14) andwe obtain, for every ,Therefore we get , which shows is a bounded subset of .

The proof of the theorem is complete.

*Remark 2. *We note that the characterization of boundedness in has the same conditions as the characterization of boundedness in the Smirnov class in the case ([7], Theorem ), the class with in the case ([2], Theorem ), and the Privalov class ([1], Theorem ). On the other hand, we see that (ii) implies (i) in Theorem 1. Suppose that (ii) holds. Then there is a positive integer such thatfor any measurable set with . There are measurable sets such that , for , and for every . Thenholds (cf. [8] (Theorem and Remark )).

Next we show a standard example of a bounded set of . The following theorem is easily proved in the same way of [1] (p.236) and [2] (Theorem ); therefore, we do not prove it here.

Theorem 3. *Let . If , then form a bounded set in .*

Let and we set . Subbotin proved an equivalent condition that a subset is bounded. The following is a theorem by Subbotin.

Theorem 4 (see [1]). *Let . A subset is bounded if and only if the following two conditions are satisfied:**(i) There exists such that for all .**(ii) For each there exists such thatfor any measurable set with the Lebesgue measure .*

As shown in [1, 3], for any the class coincides with the class and the metrics and are equivalent. Therefore the topologies induced by these metrics are identical on the set .

The following theorem is clear; therefore the proof may be omitted.

Theorem 5. *Let . A subset is bounded if and only if the following two conditions are satisfied:**(i) There exists such thatfor all .**(ii) For each there exists such thatfor any measurable set with the Lebesgue measure .*

#### 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).