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

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.


Introduction
Let  be a complex Hilbert space of infinite dimension.Given a subdomain Ω of , a function ℎ : Ω → C is said to be holomorphic if it is Fréchet differentiable at each point  ∈ Ω or, equivalently, if ℎ() = ∑ ∞ =1   () for all  ∈ Ω, where   is an -homogeneous polynomial.
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  ⊂  satisfying with 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 Λ  (D) 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][5][6].In this paper, we consider the corresponding problem in the case of P(Ω).Our first result is the following theorem which can be viewed as an extension of [6, Theorem 1] to the infinite dimensional setting.

Theorem 1.
Let  be a majorant and Ω be a simply connected -extension subdomain of .If  = ℎ +  ∈ P(Ω), then the following statements are equivalent:

Journal of Function Spaces
Here Λ  (, ) denotes the class of continuous functions on  ∪  which satisfy (2) with some positive constant , whenever  ∈  and  ∈ .
Let B  be the unit ball of .For each  ∈ P(B  ), we denote Following [4], the --Bloch space B   of P(B  ) consists of all functions  ∈ P(B  ) such that         , = sup and the little --Bloch space B  ,0 consists of the functions In particular, when  is holomorphic and () = , the space B   is B  of (B  ) which has been studied in [7].Let ℎ : B  → C be continuous.If there exists a constant  > 0 such that for any ,  ∈ B  ,  ̸ = , then we say that ℎ 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 D 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 C  , 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][13][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 / ≤  ≤ .

Preliminaries
We need the following preliminary material (see [7,8] for the details).
For  ∈ B  , the involution   : B  → B  is defined as where   = √ 1 − || 2 and   : B  → B  is the analytic map :  →  is the orthogonal projection along the onedimensional subspace spanned by , more precisely, and   is the orthogonal complement, and   =  −   .The automorphisms of the unit ball B  turn to be compositions of such analogous involutions with unitary transformations of .
As in the finite dimensional case, the pseudohyperbolic and hyperbolic metrics on B  are, respectively, defined by It is known (see [7]) that For each  ∈ B  and  ∈ (0, 1), we define the pseudohyperbolic ball with center  and radius  as A 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  ∈ (0, 1) and  ∈ (, ).

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

Several
which completes the proof.
As an application of Lemma 6, we can obtain the following.
The following lemma is an analogue of [5, Theorem 1] 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 Ω be a simply connected -extension domain of .If ℎ is a holomorphic function on Ω, then the following statements are equivalent: where Λ  (Ω, Ω) denotes the class of continuous functions on Ω ∪ Ω which satisfy (2) with some positive constant , whenever  ∈ Ω and  ∈ Ω.

The Proof of Theorem 1
Proof.The proof follows from Lemmas 5-8.

Bloch Spaces
In this section, we show some characterizations of the spaces B   and B  ,0 in terms of |()−()|/|−| on the unit ball of B  .We first extend [4,Theorem 3] to the setting of P(B  ) as follows.
Theorem 9. Let  ∈ P(B  ), 0 ≤  < 1, and  ≤  < 1 + .Then  ∈ B   if and only if Proof.For a fixed point  in (34), we can find  ∈ B  which satisfies that  =  +  0 and where  > 0, and  0 ∈ B  .Consequently, we have By letting  → 0, we deduce that Conversely, we assume that  ∈ B   .For ,  ∈ B  , Since, for ,  ∈ B  and  ∈ [0, 1], where the last integral converges since  < 1 + .Thus for all ,  ∈ B  .By the triangle inequality, we have sup In the above inequality, first by letting || → 1 − and then letting  → 1 − , we obtain the desired result.
In the following, by adding the restriction  ∈ (, ), we generalize [10, Theorems 1 and 2] to the following forms.(52) This completes the proof of Theorem 11.
Proof.For a fixed  ∈ B  , let () = () in the unit disc D. Then  is a harmonic function on D with |()| < 1.Proof.Fixing a point  ∈ Ω and considering the function , defined on B  by LemmasLemma 5. Let  be a real pluriharmonic function of B  with || < 1.Then, for each  ∈ B  , where  = .This completes the proof.Lemma 6.Let  be a majorant and Ω be a -extension domain of .If  =  + V be a holomorphic function on Ω and  ∈ Λ  (Ω), then  ∈ Λ  (Ω).