Lipschitz-Type and Bloch-Type Spaces of Pluriharmonic Mappings in a Hilbert Space

Abstract

We investigate some properties of pluriharmonic mappings in an infinite dimensional complex Hilbert space. Several characterizations for pluriharmonic mappings to be in Lipschitz-type and Bloch-type spaces are given, which are generalizations of the corresponding known ones for holomorphic functions with several complex variables.

1. Introduction

Let be a complex Hilbert space of infinite dimension. Given a subdomain of , a function is said to be holomorphic if it is Fréchet differentiable at each point or, equivalently, if for all , where is an -homogeneous polynomial.

A continuous complex-valued function defined on is said to be pluriharmonic if there are two holomorphic functions and on such that . We denote the class of all pluriharmonic mappings on by . Suppose that is an orthonormal basis of . Then every can be written as and . For a pluriharmonic mapping , we introduce the notion

Let be a continuous increasing function with . We say that is a majorant if is nonincreasing for . A function is said to belong to Lipschitz space if there is a positive constant such thatfor all (cf. [1]).

Given a proper subdomain of and a majorant , we say that is -extension if for each pair of points can be joined by a rectifiable curve satisfyingwith some fixed positive constant , where stands for the arc length measure on and denotes the distance from to the boundary of (cf. [2]).

In [1], Dyakonov characterized the holomorphic functions in in terms of their modulus. Later, Pavlović [3] came up with a relatively simple proof of the results of Dyakonov. For the generalizations of this topic, we refer to [4–6]. In this paper, we consider the corresponding problem in the case of . Our first result is the following theorem which can be viewed as an extension of [6, Theorem ] to the infinite dimensional setting.

Theorem 1. Let be a majorant and be a simply connected -extension subdomain of . If , then the following statements are equivalent: (a).(b) and .(c) and .(d) and .Here denotes the class of continuous functions on which satisfy (2) with some positive constant , whenever and .

Let be the unit ball of . For each , we denote Following [4], the --Bloch space of consists of all functions such that and the little --Bloch space consists of the functions such that In particular, when is holomorphic and , the space is of which has been studied in [7].

Let be continuous. If there exists a constant such that for any , then we say that satisfies weighted Lipschitz condition (cf. [8]).

In the theory of function spaces, the relationship between Bloch spaces and weighted Lipschitz functions has attracted much attention. In 1986, Holland and Walsh established a standard criterion for analytic Bloch space in the unit disc in terms of weighted Lipschitz functions. Since then, a series of work has been carried out to characterize Bloch, -Bloch, little -Bloch, and Besov spaces of holomorphic and harmonic functions along this line. For instance, Ren and Tu [9] extended Holland and Walsh’s criterion to the Bloch space in the unit ball of , Li and Wulan [10] and Zhao [11] characterized holomorphic -Bloch space in terms of . For the related results of harmonic functions, we refer to [4, 12–14] and the references therein.

The second purpose of this paper is to consider the corresponding problems for pluriharmonic mappings in an infinite dimensional complex Hilbert space . In Section 2, we collect some known results that will be needed in the sequel. Our main results and their proofs are presented in Sections 3 and 4.

Throughout this paper, constants are denoted by , and they are positive and may differ from one occurrence to the other. The notation means that there exists a positive constant such that .

2. Preliminaries

We need the following preliminary material (see [7, 8] for the details).

For , the involution is defined as where and is the analytic map is the orthogonal projection along the one-dimensional subspace spanned by , more precisely, and is the orthogonal complement, and . The automorphisms of the unit ball turn to be compositions of such analogous involutions with unitary transformations of .

As in the finite dimensional case, the pseudohyperbolic and hyperbolic metrics on are, respectively, defined by It is known (see [7]) thatFor each and , we define the pseudohyperbolic ball with center and radius asA simple computation gives that is a Euclidean ball with center and radius given by

The following lemma will be needed in the sequel. See [15] for the analogue of this result in several complex variables.

Lemma 2. Let and . Then .

Proof. From (15), we have . It follows from (12) that Similarly, we can obtain that . Combining these two inequalities with (12), we have

The following lemma comes from [4].

Lemma 3. Let be a majorant and and . Then, for ,

A combination of Lemmas 2 and 3 yields the following.

Lemma 4. Let and . Then .

3. Lipschitz Spaces

We begin this section with some lemmas which will be used in the proof of Theorem 1.

3.1. Several Lemmas

Lemma 5. Let be a real pluriharmonic function of with . Then, for each ,

Proof. For a fixed , let in the unit disc . Then is a harmonic function on with . It follows from [16] that which implies that where . This completes the proof.

Lemma 6. Let be a majorant and be a -extension domain of . If be a holomorphic function on and , then .

Proof. Fixing a point and considering the function , defined on by here Since is pluriharmonic in and , by Lemma 5, we have that, for each , which in turn gives Hence By the assumption , we have, for each , which implies that Thus, for any , we haveFor a pair of points , we let be a rectifiable curve which joins and satisfying (2). Integrating (30) along leads toCombining (31) with (3), we have which completes the proof.

As an application of Lemma 6, we can obtain the following.

Lemma 7. Let be a majorant and let be a pluriharmonic mapping on a simply connected -extension domain . Then if and only if both .

Proof. We only need to prove necessity since the sufficiency is obvious. Let , where are real. As and by Lemma 6, we see that .

The following lemma is an analogue of [5, Theorem ] for holomorphic functions in an infinite dimensional Hilbert space . Since the proof is almost the same as the one in [5], we leave it to the readers.

Lemma 8. Let be a majorant and be a simply connected -extension domain of . If is a holomorphic function on , then the following statements are equivalent:(i).(ii).(iii),where denotes the class of continuous functions on which satisfy (2) with some positive constant , whenever and .

3.2. The Proof of Theorem 1

Proof. The proof follows from Lemmas 5–8.

4. Bloch Spaces

In this section, we show some characterizations of the spaces and in terms of on the unit ball of . We first extend [4, Theorem ] to the setting of as follows.

Theorem 9. Let , , and . Then if and only if

Proof. For a fixed point in (34), we can find which satisfies that and where , and . Consequently, we have By letting , we deduce that Conversely, we assume that . For , Since, for and ,we get where the last integral converges since . Thus This completes the proof of Theorem 9.

Theorem 10. Let , , and ku. Then if and only if

Proof.  
Sufficiency. Assume that (42) holds. Then, for any , there exists such that whenever . It follows by an argument similar to that in the proof of Theorem 9 that we have whenever . Hence Necessity. For , let . By the proof of Theorem 9, we have for all . By the triangle inequality, we have In the above inequality, first by letting and then letting , we obtain the desired result.

In the following, by adding the restriction , we generalize [10, Theorems and ] to the following forms.

Theorem 11. Let , , and . Then if and only if

Proof. We only need to prove the necessity since the sufficiency easily follows from the proof of Theorem 9. Assume that . Then for any , we have Since, for ,by Lemma 2, we obtain that Thus, This completes the proof of Theorem 11.

Similarly, we can obtain the following.

Theorem 12. Let , , and . Then if and only if

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work was supported by the NNSF of China (nos. 10771121, 11301220, 11401387, and 11661052), the NSF of Zhejiang Province, China (no. LQ 14A010007), the NSF of Shan-dong Province, China (no. ZR2012AQ020), and the Fund of Doctoral Program Research of Shaoxing College of Art and Science (20135018).