Bounded Subsets of the Zygmund F -Algebra

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1 International Journal of Mathematical Analysis Vol. 12, 2018, no. 9, HIKARI Ltd, Bounded Subsets of the Zygmund F -Algebra Yasuo Iida Department of Mathematics, Kanazawa Medical University, 1-1, Daigaku, Uchinada, Ishikawa , Japan Copyright c 2018 Yasuo Iida. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We will consider some characterizations of boundedness in the Zygmund F -algebra N log α NX) α > 0) of holomorphic functions f on the unit polydisk or the unit ball that satisfy ϕ α log + frζ) ) dσζ) <, where ϕ α t) = t{logc α + t)} α for t 0 and c α = max{e, e α }. Mathematics Subject Classification: 30H15, 32A38, 46B04 Keywords: bounded subsets, Zygmund F -algebra, Privalov class, Smirnov class 1 Introduction Let n N and C n = {z = z 1,..., z n ) z j C, 1 j n} be the space of n-complex variables. The unit polydisk {z C n z j < 1, 1 j n} is denoted by U n and the distinguished boundary T n is {ζ C n ζ j = 1, 1 j n}. The unit ball {z C n n j=1 z j 2 < 1} is denoted by B n and S n = {ζ C n n j=1 ζ j 2 = 1} is its boundary. In this paper, X denotes the unit polydisk or the unit ball and denotes T n for X = U n or S n for X = B n. The normalized Lebesgue measure on is denoted by dσ.

2 426 Yasuo Iida H q X) 0 < q ) denotes the Hardy space on X. We denote the Nevanlinna class on X by NX), which consists of all holomorphic functions f on X such that log + frζ) dσζ) < holds. It is well-known that each function f NX) has the nontangential limit f ζ) = lim frζ) a.e. ζ ). r 1 The Smirnov class N X) is defined as the set of all f NX) which satisfy log + frζ) dσζ) = log + f ζ) dσζ). The metric in the class N X) is introduced by d N X)f, g) = log1 + f ζ) g ζ) ) dσζ) f, g N X)). With the metric d N X), ) N X) is an F -algebra, that is, a complete linear metric space with multiplication continuous. The Privalov class N p X), 1 < p <, is the set of all holomorphic functions f on X satisfying log + frζ) ) p dσζ) <. It is known that N p X) is a subalgebra of N X), hence each function f N p X) has the nontangential limit almost everywhere on. Under the metric defined by d N p X)f, g) = ) 1 log1 + f ζ) g ζ) )) p p dσζ) f, g N p X)), N p X) is also an F -algebra cf. [7]). For 0 < p <, the class M p X) consists of holomorphic functions f on X for which log + Mfζ) ) p dσζ) <, where Mfζ) := frζ). Define a metric { d M p X)f, g) = } αp log1 + Mf g)ζ))) p p dσζ) f, g M p X)), where α p = min1, p). With this metric M p X) is also an F -algebra see [2]).

3 Bounded subsets of the Zygmund F -algebra 427 It is well-known that it holds the following inclusion relations: H q X) N p X) M 1 X) N X) NX) 0 < q, p > 1). Moreover, it is known that N p X) = M p X) p > 1) and NX) M p X) 0 < p < 1). We shall define the Zygmund F -algebra N log α NX) α > 0). The class N log α NX) is the set of all holomorphic functions f on X such that ϕ α log + frζ) ) dσζ) <, 1) where ϕ α t) = t{logc α + t)} α for t 0 and c α = max{e, e α }. It is verified that 1) is equivalent to the condition f α := ϕ α log1 + frζ) )) dσζ) <. 2) This class was considered by Zygmund first [10]. Further, the topological properties of this class were studied in [1, 3, 8]. It is known that the following relations hold: H q X) N log α NX) N X) α > 0). q>0 This implies that every f N log α NX) has a finite nontangential limit almost everywhere on. Therefore the characteristic defined by 2) satisfies the relation f α = ϕ α log1 + f ζ) )) dσζ). For f, g N log α NX), we can define a metric d N log α NX)f, g) := f g α = ϕ α log1 + f ζ) g ζ) )) dσζ). With this metric N log α NX) becomes an F -algebra see [1, 3]). 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 λ, 0 < λ < 1, such that λl = {λf ; f L} U. Yanagihara investigated the properties of boundedness in N X) in the case n = 1 [9]. As for N p X) with p > 1 in the case n 1, Subbotin described some characterizations of boundedness [7]. As for M p X) with p = 1 in the case n = 1, Kim characterized bounded subsets of the class see [5]). For p > 1 and n = 1, these characterizations were described by Meštrović [6]. In recent paper [4], the author considered bounded subsets of M p X) with 0 < p < in the case n 1. In this paper, some characterizations of boundedness in the Zygmund F - algebra N log α NX) with α > 0 in the case n 1 will be described.

4 428 Yasuo Iida 2 The results Theorem 2.1. Let α > 0. L N log α NX) is bounded if and only if i) there exists a K < such that ϕ α log + f ζ) ) dσζ) < K for any f L; ii) for each ε > 0 there exists δ > 0 such that ϕ α log + f ζ) ) dσζ) < ε, E for any f L and for any measurable set E with the Lebesgue measure E < δ. Proof. We follow [4]. Necessity. Let L be a bounded subset of N log α NX). i) For any η > 0, we can find a number λ 0 = λ 0 η) 0 < λ 0 < 1) such that d N log α NX)λf, 0) = ϕ α log1 + λf ζ) )) dσζ) < η for all f L and λ λ 0. Since fg α 2 α+2 f α + g α ), 3) which is derived from the definition of ϕ α and the elementary inequality we obtain a + b) α 2 α a α + b α ) a 0, b 0, α > 0), ϕ α log + f ζ) ) dσζ) ϕ α log 1 + λ 1 0 λ 0 f ζ) )) dσζ) 2 ) α+2 λ 1 0 α + λ 0 f α = 2 α+2 ϕ α log1 + λ 1 0 ) ) + η ) = K = constant. Thus the condition i) is satisfied. ii) Given ε > 0, take η as η < ε/2 α+3 and λ 0 = λ 0 η) as above. Next we take δ > 0 such that δ ϕ α log1 + λ 1 0 ) ) < ε 2. α+3

5 Bounded subsets of the Zygmund F -algebra 429 If E < δ for each set E and for any f L, we have ϕ α log + f ζ) ) dσζ) E E ϕ α log1 + λ 1 λ0 f ζ) ) ) dσζ) 0 2 α+2 ϕ α log1 + λ 1 0 ) ) E + η ) < ε 2 + ε 2 < ε. Therefore the condition ii) holds. Sufficiency. Let V = {g N log α NX) ; d N log α NX)g, 0) < η} be a neighborhood of zero in N log α NX). We take ε > 0 such that ϕ α log1 + ε)) + 2 α+2 ϕ α log 2)ε + 2 α+2 ε < η. Then we can find a δ 0 < δ < ε) so that ii) is satisfied. For f L, there is a measurable set E f such that \E f < δ, ϕ α log + f ζ) ) K δ on E f by Chebyshev s inequality. Thus we obtain )) K f ζ) exp ϕ 1 α = Aδ) = A on E f. δ Take λ such that 0 < λ < ε/a. Then, using inequalities and log1 + x) log 2 + log + x x > 0) ϕ α x + y) 2 α+2 ϕ α x) + ϕ α y)) x, y 0), which is derived in the same way of 3), we have, for any f L, d N log α NX)λf, 0) = ϕ α log1 + λf ζ) )) dσζ) = E f + \E f ϕ α log1 + ε)) dσζ) + E f ϕ α log1 + f ζ) )) dσζ) \E f

6 430 Yasuo Iida ϕ α log1 + ε)) +2 α+2 ϕ α log 2) dσζ) + ϕ α log + f ζ) ) ) dσζ) \E f \E f ϕ α log1 + ε)) + 2 α+2 ϕ α log 2)δ + 2 α+2 ε < η. Hence we get d N log α NX)λf, 0) < η. N log α NX). This completes the proof. Therefore L is a bounded subset of Remark. Note that the characterization of boundedness in N log α NX) α > 0) has the same conditions as the characterization of boundedness in the Smirnov class N X) in the case n = 1 [9, Theorem 1], the Privalov class N p X) 1 < p < ) [7, Theorem 5], and the class M p X) 0 < p < ) [4, Theorem 1] [5, Theorem 4.1]. Next we will show a standard example of a bounded set of N log α NX). Theorem 2.2. Let α > 0. If f N log α NX), then f r z) = frz) X, 0 r < 1) form a bounded set in N log α NX). z To prove Theorem 2.2, we utilize the following result by Eminyan. Theorem 2.3. see [1, Theorem 3]) Let α > 0. If f N log α NX) and f r z) = frz) for z X and 0 r < 1, then f r f as r 1 in the metric d N log α NX). Proof of Theorem 2.2. Let V = {g N log α NX) ; d N log α NX)g, 0) < η} be a neighborhood of zero in N log α NX). Choose λ, 0 < λ < 1, so that d N log α NX)λ f, 0) < η/2. Let r 0 be sufficiently near to 1 such that d N log α NX)f, f r ) < η/2 for r 0 r < 1 by Theorem 2.3. Then we have d N log α NX)λ f r, λ f) < η/2. For r r 0 we obtain d N log α NX)λ f r, 0) d N log α NX)λ f r, λ f) + d N log α NX)λ f, 0) < η. For 0 r r 0 we can find λ so small that d N log α NX)λ f r, 0) < η. Therefore, if λ = minλ, λ ), we have {λf r } V.

7 Bounded subsets of the Zygmund F -algebra 431 References [1] O. M. Eminyan, Zygmund F -algebras of holomorphic functions in the ball and in the polydisk, Doklady Math., ), [2] V. I. Gavrilov and A. V. Subbotin, F -algebras of holomorphic functions in a ball containing the Nevanlinna class, Math. Montisnigri, ), Russian) [3] V. I. Gavrilov and A. V. Subbotin, Linear and metric properties of F - algebras of N log N holomorphic functions of several complex variables, Math. Montisnigri, ), Russian) [4] Y. Iida, Bounded subsets of classes M p X) of holomorphic functions, J. Funct. Sp., ), [5] H. O. Kim, On an F -algebra of holomorphic functions, Can. J. Math., ), [6] R. Meštrović, On F -algebras M p 1 < p < ) of holomorphic functions, The Scientific World Journal, ), [7] A. V. Subbotin, Functional properties of Privalov spaces of holomorphic functions in several variables, Math. Notes., ), [8] S. Ueki, Isometries of the Zygmund F -algebra, Proc. Amer. Math. Soc., ), [9] N. Yanagihara, Bounded subsets of some spaces of holomorphic functions, Sci. Pap. Coll. Gen. Ed., Univ. Tokyo, ), [10] A. Zygmund, Trigonometric Series, Vol. 2, Cambridge University Press, Received: August 3, 2018; Published: September 3, 2018