# Bounded Subsets of Smirnov and Privalov Classes on the Upper Half Plane

Authors: Yasuo Iida
Journal: International Journal of Analysis
 (2017)

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## Abstract

Some characterizations of boundedness inN⁎(D) and Np(D)  (1<p<∞) will be described, where N⁎(D) denote the Smirnov class and Np(D) the Privalov class on the upper half plane D={z∈C∣Im⁡z>0}, respectively.


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## Body

## 1. Introduction

LetU and T denote the unit disk and the unit circle in C, respectively. The Privalov class Np(U), 1<p<∞, is defined as the set of all holomorphic functions f on U, satisfying(1)sup0<r<1∫Tlog⁡1+frζpdσζ<+∞, where dσ denotes normalized Lebesgue measure on T. The notion of Np(U) was introduced by Privalov [1] and has been explored by several authors (see [2–4]). Letting p=1, we have the Nevanlinna class N(U). It is well-known that each function f in N(U) has the nontangential limit f∗(ζ)=limr→1-f(rζ)  (a.e. ζ∈T) and that log⁡1+f (and, hence, (log⁡1+f)p for p>1) is subharmonic if f is holomorphic. Define a metric(2)dNpUf,g=∫Tlog⁡1+f∗ζ-g∗ζpdσζ1/p for f,g∈Np(U). With the metric dNp(U)(·,·)Np(U) becomes an F-algebra [2]. Recall that an F-algebra is a topological algebra in which the topology arises from a complete metric.We denote the Smirnov class byN∗(U), which consists of all holomorphic functions f on U such that log⁡1+fz≤Q[ϕ](z)  (z∈U) for some ϕ∈L1(T),  ϕ≥0, where the right side denotes the Poisson integral of ϕ on U. It is known that if f∈N(U), f belongs to N∗(U) if and only if(3)limr→1-∫Tlog⁡1+frζdσζ=∫Tlog⁡1+f∗ζdσζ.Under the metric(4)dN∗Uf,g=∫Tlog⁡1+f∗ζ-g∗ζ⁡dσζ for f,g∈N∗(U), the class is also an F-algebra (see [5]).For0<p<∞, the class Mp(U) is defined as the set of all holomorphic functions f on U such that(5)∫Tlog⁡1+Mfζpdσζ<+∞,where Mf(ζ)=sup0≤r<1frζ is the maximal function. The class M1(U) was introduced by Kim in [6]. As for p>0, the class was considered in [7, 8]. For f,g∈Mp(U), define a metric(6)dMpUf,g=∫Tlog⁡1+Mf-gζpdσζαp/p, where αp=min⁡(1,p). With this metric Mp(U) is also an F-algebra (see [9]).It is well-known thatHq(U)⊊Np(U)⊊M1(U)⊊N∗(U)⊊N(U)  (0<q≤+∞,p>1), where Hq(U) denotes the Hardy space on U. Moreover, it is known that N(U)⊊Mp(U)  (0<p<1) [6].Mochizuki [10] introduced the Nevanlinna class N0(D) and the Smirnov class N∗(D) on the upper half plane D≔{z∈C∣Im⁡z>0}: the class N0(D) is the set of all holomorphic functions f on D satisfying(7)supy>0∫Rlog⁡1+fx+iy⁡dx<+∞ and N∗(D) the set of all holomorphic functions f on D satisfying log⁡1+fz≤P[ϕ](z)  (z∈D) for some ϕ∈L1(R), ϕ≥0, where the right side denotes the Poisson integral of ϕ on D. It is well-known that each function f in N0(D) has the nontangential limit f∗(x)=limy→0+f(x+iy) (a.e. x∈R). Let f∈N0(D). Then f∈N∗(D) if and only if (8)limy→0+∫Rlog⁡1+fx+iydx=∫Rlog⁡1+f∗xdx(see [10]). Moreover, under the metric(9)dN∗Df,g=∫Rlog⁡1+f∗x-g∗x⁡dx,the class N∗(D) becomes an F-algebra [10].The classMp(D)  (0<p<∞) is defined as the set of all holomorphic functions f on D such that(10)∫Rlog⁡1+Mfxpdx<+∞,where Mf(x)=supy>0fx+iy. The class Mp(X) with p=1 was introduced by Ganzhula in [11]. As for p>0, Efimov and Subbotin investigated this class [12]. For f,g∈Mp(D), define a metric(11)dMpDf,g=∫Rlog⁡1+Mf-gxpdxαp/p, where αp=min⁡(1,p). With this metric Mp(D) is also an F-algebra (see [11, 12]).In [13], the class Np(D) was introduced, analogous to Np(U); that is, we denote by Np(D)(p>1) the set of all holomorphic functions f on D such that(12)supy>0∫Rlog⁡1+fx+iypdx<+∞. Each f∈Np(D) has the nontangential limit f∗(x) for a.e. x∈R, and under the metric,(13)dNpDf,g=∫Rlog⁡1+f∗x-g∗xpdx1/p,the class Np(D) becomes an F-algebra [13].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∗(U) [14]. As for Mp(U) with p=1, Kim described some characterizations of boundedness (see [6]). For p>1, these characterizations were considered by Meštrović [15]. As for Mp(D) with p=1, Ganzhula investigated the properties of boundedness [11] and Efimov characterized bounded subsets of Mp(D) in the case 0<p<∞ [16]. In recent paper [17], the author described bounded subsets of Mp(U) in the case 0<p<1.In this paper, we consider some characterizations of boundedness inN∗(D) and Np(D)(p>1).


## 2. The Results

Theorem 1.
Letp>1. L⊂Np(D) is bounded if and only if
(i) there exists aK<∞ such that(14)∫Rlog⁡1+f∗xpdx<K for all f∈L;
(ii) for eachε>0 there exists δ>0 such that(15)∫Elog⁡1+f∗xpdx<ε,∀f∈L,for any measurable set E⊂R with the Lebesgue measure E<δ.Proof.
We follow [16, Theorem 1].
Necessity. Let L be a bounded subset of Np(D).
(i) For any numberη>0 there exists α=α(η), 0<α<1, such that(16)dNpDαf,0p=∫Rlog⁡1+αf∗xpdx<ηpfor all f∈L. Utilizing the inequality (1+x)α≤1+αx  (0<α<1,x≥0), it follows that, from (16),(17)∫Rlog⁡1+f∗xpdx≤∫Rlog⁡1+αf∗x1/αpdx=1αp∫Rlog⁡1+αf∗xpdx<ηαp=K=constantfor all f∈L. Therefore, condition (i) holds.
(ii) For any numberε>0, we take η as η<ε1/p/2. Choose a number α=α(ε), 0<α<1, such that equality (16) holds for all f∈L. Then for any measurable set E⊂R, using Minkowski’s inequality, we have the estimate(18)∫Elog⁡1+f∗xpdx<∫Elog⁡1α+f∗xpdx=∫Elog⁡1α+log⁡1+αf∗xpdx≤Elog⁡1αp1/p+∫Rlog⁡1+αf∗xp⁡dx1/pp<E1/plog⁡1α+ηp.If we take δ>0 as δ<ε/(2plog⁡(1/α)p), then(19)∫Elog⁡1+f∗xpdx<δ1/plog⁡1α+ε1/p2p<ε1/p2+ε1/p2p=εfor all f∈L and any measurable set E⊂R, |E|<δ. Thus condition (ii) holds.
Sufficiency. Let conditions (i) and (ii) hold for a subset L of Np(D), p>1. Consider a neighborhood(20)V=g∈NpD:dNpDg,0<η.Take ε>0 as ε<ηp/3. According to (ii), there exists a number δ>0 such that(21)∫Elog⁡1+f∗xpdx<ε<ηp3for all f∈L and any measurable set E⊂R, |E|<δ. Next there exists a finite constant K>0 such that condition (i) holds for all f∈L. Applying Chebyshev’s inequality to the Lebesgue measure of the set Ef=x∈R∣log⁡1+f∗xp>K/δ for f∈L, the following estimate is valid:(22)Ef≤δK∫Rlog1+f∗xpdx<δ.Then we may assume E=Ef and f∗x>exp⁡(K/δ)1/p-1=C in inequality (21); that is, f∗x/C<1 for all x∈R\Ef. Therefore, for any number α(0<α<1) and all f∈L, we have the following:(23)∫Rlog⁡1+αf∗xpdx=∫Eflog⁡1+αf∗xpdx+∫R\Eflog⁡1+αf∗xpdx<∫Eflog⁡1+f∗xpdx+∫E1log⁡1+αf∗xpdx+∫E2log⁡1+αf∗xpdx,where R\Ef=E1∪E2, E1=x∈R∣f∗x<1, and E2=x∈R∣1≤f∗x<C. By using the elementary inequality 1+βt≤(1+t)2β  (0≤t<1,0<β<1/2) to the second integral in (23), using (21) and taking(24)α=min⁡12,12ηp3K1/p,1C2ηp/3K1/p-1,we have the following estimate(25)∫Rlog⁡1+αf∗xpdx<ηp3+2αpK+ηp3K∫E2log⁡1+1pdx≤ηp3+ηp3+ηp3K∫Rlog⁡1+f∗xpdx<ηp.Therefore, αL⊂V and the set L is bounded in Np(D) by definition.
The proof of the theorem is complete.Next we consider some characterizations of boundedness inN∗(D). Proof of the following theorem can be obtained by taking p=1 in the whole proof of Theorem 1; therefore, this proof may be omitted.Theorem 2.
L
⊂
N
∗
(
D
) is bounded if and only if
(i) there existsK<∞ such that(26)∫Rlog⁡1+f∗xdx<K for all f∈L;
(ii) for eachε>0 there exists δ>0 such that(27)∫Elog⁡1+f∗xdx<ε,∀f∈L,for any measurable set E⊂R with the Lebesgue measure |E|<δ.



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