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.

National Natural Science Foundation of China10771121113012201140138711661052Natural Science Foundation of Zhejiang ProvinceLQ 14A010007Natural Science Foundation of Shandong ProvinceZR2012AQ020Fund of Doctoral Program Research of Shaoxing College of Art and Science201350181. Introduction

Let E be a complex Hilbert space of infinite dimension. Given a subdomain Ω of E, a function h:Ω→C is said to be holomorphic if it is Fréchet differentiable at each point z∈Ω or, equivalently, if h(z)=∑n=1∞Pn(z) for all z∈Ω, where Pn is an n-homogeneous polynomial.

A continuous complex-valued function f defined on Ω is said to be pluriharmonic if there are two holomorphic functions h and g on Ω such that f=h+g¯. We denote the class of all pluriharmonic mappings on Ω by P(Ω). Suppose that (ek)k∈Γ is an orthonormal basis of E. Then every z∈E can be written as z=∑k∈Γzkek and z¯=∑k∈Γzk¯ek. For a pluriharmonic mapping f∈P(Ω), we introduce the notion (1)∇f=∂f∂z1,…,∂f∂zk,…,k∈Γ,∇¯f=∂f∂z1¯,…,∂f∂zk¯,…,k∈Γ.

Let ω:[0,+∞)→[0,+∞) be a continuous increasing function with ω(0)=0. We say that ω is a majorant if ω(t)/t is nonincreasing for t>0. A function f:Ω→C is said to belong to Lipschitz space Λω(Ω) if there is a positive constant C such that(2)fz-fw≤Cωz-wfor all z,w∈Ω (cf. [1]).

Given a proper subdomain G of E and a majorant ω, we say that G is ω-extension if for each pair of points z,w∈G can be joined by a rectifiable curve γ⊂G satisfying(3)∫γωdGzdGzdsz<Cωz-wwith some fixed positive constant C=C(G,ω), where ds stands for the arc length measure on γ and dG(z) denotes the distance from z to the boundary ∂G of G (cf. [2]).

In [1], Dyakonov characterized the holomorphic functions in Λω(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 [4–6]. 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 ω be a majorant and Ω be a simply connected ω-extension subdomain of E. If f=h+g¯∈P(Ω), then the following statements are equivalent:

f∈Λω(Ω).

h∈Λω(Ω) and g∈Λω(Ω).

|h|∈Λω(Ω) and |g|∈Λω(Ω).

|h|∈Λω(Ω,∂Ω) and |g|∈Λω(Ω,∂Ω).

Here Λω(G,∂G) denotes the class of continuous functions on G∪∂G which satisfy (2) with some positive constant C, whenever z∈G and w∈∂G.

Let BE be the unit ball of E. For each f∈P(BE), we denote (4)Λfz=maxθ∈∂BE∇fz·θ+∇¯fz·θ¯.Following [4], the ω-α-Bloch space Bωα of P(BE) consists of all functions f∈P(BE) such that (5)fω,α=supz∈BEω1-z2αΛfz<∞,and the little ω-α-Bloch space Bω,0α consists of the functions f∈P(BE) such that (6)limz→1-supz∈BEω1-z2αΛfz=0.In particular, when f is holomorphic and ω(t)=t, the space Bωα is Bα of H(BE) which has been studied in [7].

Let h:BE→C be continuous. If there exists a constant C>0 such that (7)1-z21/21-w21/2hz-hwz-w≤C,for any z,w∈BE,z≠w, 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, α-Bloch, little α-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 Cn, Li and Wulan [10] and Zhao [11] characterized holomorphic α-Bloch space in terms of (1-|z|2)β(1-|w|2)α-β|f(z)-f(w)|/|z-w|. 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 E. 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 C, and they are positive and may differ from one occurrence to the other. The notation A≍B means that there exists a positive constant C such that B/C≤A≤CB.

2. Preliminaries

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

For a∈BE, the involution φa:BE→BE is defined as (8)φaz=saQa+Pamaz,where sa=1-|a|2 and ma:BE→BE is the analytic map (9)maz=a-z1-z,a,Pa:E→E is the orthogonal projection along the one-dimensional subspace spanned by a, more precisely, (10)Paz=z,aa,aaand Qa is the orthogonal complement, and Qa=Id-Pa. The automorphisms of the unit ball BE turn to be compositions of such analogous involutions with unitary transformations of E.

As in the finite dimensional case, the pseudohyperbolic and hyperbolic metrics on BE are, respectively, defined by (11)ρEz,w=φzw=φwz,βEz,w=12ln1+ρEz,w1-ρEz,w,z,w∈BE.It is known (see [7]) that(12)φzw2=1-1-z21-w21-z,w2(13)=-2Rz,w+z,w2+z2+w2-z2w21-z,w2(14)≤z-w21-z,w2.For each a∈BE and r∈(0,1), we define the pseudohyperbolic ball with center a and radius r as(15)Ea,r=z∈BE:ρEa,z<r.A simple computation gives that E(a,r) is a Euclidean ball with center and radius given by (16)1-r21-r2z2z,1-z21-r2z2r.

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

Lemma 2.

Let r∈(0,1) and w∈E(z,r). Then 1-|z|2≍1-|w|2≍|1-〈z,w〉|.

Proof.

From (15), we have |φz(w)|<r. It follows from (12) that (17)1-r24<1-z21-w241-z,w2≤1-z21-w241-z2≤1-w21-z2.Similarly, we can obtain that 1-r2/4<(1-|z|2)/(1-|w|2). Combining these two inequalities with (12), we have(18)1-z2≍1-w2≍1-z,w.

The following lemma comes from [4].

Lemma 3.

Let ω(t) be a majorant and u∈(0,1] and v∈(1,∞). Then, for t∈(0,∞), (19)ωut≥uωt,ωvt≤vωt.

A combination of Lemmas 2 and 3 yields the following.

Lemma 4.

Let r∈(0,1) and u,v∈E(z,r). Then ω(1-|u|2)≍ω(1-|v|2).

3. Lipschitz Spaces

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

3.1. Several LemmasLemma 5.

Let f be a real pluriharmonic function of BE with |f|<1. Then, for each θ∈∂BE, (20)∇fz·θ≤4π1-fz21-z2.

Proof.

For a fixed θ∈∂BE, let F(ζ)=f(θζ) in the unit disc D. Then F is a harmonic function on D with |F(ζ)|<1. It follows from [16] that (21)Fζ≤4π1-Fζ21-ζ2,which implies that (22)∇fz·θ≤4π1-fz21-z2,where z=θζ. This completes the proof.

Lemma 6.

Let ω be a majorant and Ω be a ω-extension domain of E. If g=u+iv be a holomorphic function on Ω and u∈Λω(Ω), then g∈Λω(Ω).

Proof.

Fixing a point z∈Ω and considering the function U, defined on BE by (23)Uw=uz+dzwMz,w∈BE,here (24)dz=dΩz,Mz=supuζ:ζ-z<dz.Since U is pluriharmonic in BE and |U(w)|<1, by Lemma 5, we have that, for each θ∈∂BE, (25)∇U0·θ≤4π1-U02≤8π1-U0,which in turn gives (26)dz∇uz·θ≤8πMz-uz.Hence (27)dz∇gz·θ≤16πMz-uz.

By the assumption u∈Λω(Ω), we have, for each ζ∈B(z,d(z)), (28)uζ-uz≤uζ-uz≤Cωdz,which implies that (29)Mz-uz≤Cωdz.Thus, for any θ∈∂BE, we have(30)dz∇gz·θ≤Cωdz,z∈Ω.

For a pair of points z1,z2∈Ω, we let γ be a rectifiable curve which joins z1 and z2 satisfying (2). Integrating (30) along γ leads to(31)gz1-gz2≤C∫γωdzdzdsz.Combining (31) with (3), we have (32)gz1-gz2≤Cωz1-z2,which completes the proof.

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

Lemma 7.

Let ω be a majorant and let f=h+g¯ be a pluriharmonic mapping on a simply connected ω-extension domain Ω. Then f∈Λω(Ω) if and only if both g,h∈Λω(Ω).

Proof.

We only need to prove necessity since the sufficiency is obvious. Let f=h+g¯=u+iv, where u,v are real. As f,f¯∈Λω(Ω) and (33)u=f+f¯2=h+g+h+g¯2,v=f-f¯2i=h-g-h-g¯2i,by Lemma 6, we see that g,h∈Λω(Ω).

The following lemma is an analogue of [5, Theorem 1] for holomorphic functions in an infinite dimensional Hilbert space E. Since the proof is almost the same as the one in [5], we leave it to the readers.

Lemma 8.

Let ω be a majorant and Ω be a simply connected ω-extension domain of E. If h is a holomorphic function on Ω, then the following statements are equivalent:

h∈Λω(Ω).

|h|∈Λω(Ω).

|h|∈Λω(Ω,∂Ω),

where Λω(Ω,∂Ω) denotes the class of continuous functions on Ω∪∂Ω which satisfy (2) with some positive constant C, whenever z∈Ω and w∈∂Ω.
3.2. The Proof of Theorem <xref ref-type="statement" rid="thm1.1">1</xref>Proof.

The proof follows from Lemmas 5–8.

4. Bloch Spaces

In this section, we show some characterizations of the spaces Bωα and Bω,0α in terms of |f(z)-f(w)|/|z-w| on the unit ball of BE. We first extend [4, Theorem 3] to the setting of P(BE) as follows.

Theorem 9.

Let f∈P(BE), 0≤β<1, and β≤α<1+β. Then f∈Bωα if and only if(34)L=supz,w∈BE,z≠wω1-z2β1-w2α-βfz-fwz-w<∞.

Proof.

For a fixed point z in (34), we can find w∈BE which satisfies that w=z+rθ0 and (35)Λfz=∇fz·θ0+∇¯fz·θ0¯,where r>0, and θ0∈∂BE. Consequently, we have (36)ω1-z2β1-w2α-βfz-fw=ω1-z2β1-w2α-β∫zw¯fζζdζ+fζ¯ζdζ¯.By letting r→0, we deduce that (37)ω1-z2αΛfz≤L.

Conversely, we assume that f∈Bωα. For z,w∈BE, (38)fz-fw=∫zw¯fζζdζ+fζ¯ζdζ¯≤z-w∫01Λfw+tz-wdt≤Cz-wfω,α∫01dtω1-tz+1-tw2α.Since, for z,w∈BE and t∈[0,1],(39)1-tz+1-tw2α≥1-tz+1-twα≥t1-z+1-t1-wα≥t1-z22+1-t1-w22α≥t2β1-t2α-β1-z2β1-w2α-β,we get (40)fz-fwz-w≤C∫01dtω1-tz+1-tw2α≤C∫01dtωt/2β1-t/2α-β1-z2β1-w2α-β≤Cω1-z2β1-w2α-β∫01dstβ1-tα-β≤Cω1-z2β1-w2α-β,where the last integral converges since α<1+β. Thus (41)ω1-z2β1-w2α-βfz-fwz-w<∞.This completes the proof of Theorem 9.

Theorem 10.

Let f∈P(BE), 0≤β<1, and kuβ≤α<1+β. Then f∈Bω,0α if and only if (42)limz→1-supw∈BE,z≠wω1-z2β1-w2α-βfz-fwz-w=0.

Proof.

Sufficiency. Assume that (42) holds. Then, for any ϵ>0, there exists δ∈(0,1) such that (43)supw∈BE,z≠wω1-z2β1-w2α-βfz-fwz-w<ϵwhenever δ<|z|<1. It follows by an argument similar to that in the proof of Theorem 9 that we have (44)ω1-z2αΛfz<Csupw∈BE,z≠wω1-z2β1-w2α-βfz-fwz-w≤Cϵ,whenever δ<|z|<1. Hence (45)limz→1-ω1-z2αΛfz=0.

Necessity. For t∈(0,1), let ft(z)=f(tz). By the proof of Theorem 9, we have (46)ω1-z2β1-w2α-βf-ftz-f-ftwz-w≤Cf-ftω,α,ω1-z2β1-w2α-βftz-ftwz-w<ω1-z2β1-w2α-βω1-tz2β1-tw2α-βω1-tz2β1-tw2α-βftz-ftwtz-w≤Cω1-z2β1-w2α-βω1-tz2β1-tw2α-βfω,αfor all z,w∈BE. By the triangle inequality, we have (47)supz,w∈BE,z≠wω1-z2β1-w2α-βfz-fwz-w≤Cf-ftω,α+Cω1-z2β1-w2α-βω1-tz2β1-tw2α-βfω,α.In the above inequality, first by letting |z|→1- and then letting t→1-, we obtain the desired result.

In the following, by adding the restriction w∈E(z,r), we generalize [10, Theorems 1 and 2] to the following forms.

Theorem 11.

Let r∈(0,1), f∈P(BE), and 0<β≤α. Then f∈Bωα if and only if(48)K=supw∈Ez,r,z≠wω1-z2β1-w2α-βfz-fwz-w<∞.

Proof.

We only need to prove the necessity since the sufficiency easily follows from the proof of Theorem 9. Assume that f∈Bωα. Then for any w∈E(z,r),z≠w, we have (49)fz-fw=∫zw¯fζζdζ+fζ¯ζdζ¯≤Cz-w∫01dsω1-sz+1-sw2α.Since, for s∈[0,1],(50)1-sz+1-sw2≥1-sz+1-sw≥s1-z+1-s1-w≥s1-z22+1-s1-w22≥121-z2s1-w21-s,by Lemma 2, we obtain that (51)fz-fwz-w≤C∫01dsω1/2α1-z2αs1-w2α-αs≤Cω1-z2α≤Cω1-z2β1-w2α-β.Thus, (52)supw∈Ez,r,z≠wω1-z2β1-w2α-βfz-fwz-w<∞.This completes the proof of Theorem 11.

Similarly, we can obtain the following.

Theorem 12.

Let r∈(0,1), f∈H(BE), and 0<β≤α. Then f∈Bω,0α if and only if (53)limz→1-supw∈Ez,r,z≠wω1-z2β1-w2α-βfz-fwz-w=0.

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).

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