Complex Interpolations for Bergman Spaces and Their Applications

GUO Yunyun, SHEN Fangning, LYU Xiaofen

(School of Science, Huzhou University, Huzhou 313000, China)

Abstract：In this paper, we discuss the complex interpolations for Bergman space with exponential weights using the knowledge of Complex Functions and Functional Analysis. As an application, the boundedness of Berezin transforms on Lebesgue space Lp is obtained.

Keywords：Bergman space with exponential weight; complex interpolation; Berezin transform

0 Introduction

LetDbe the unit disk on complex planeCand let dAbe the normalized area measure onD. For a subharmonic functionφonD, 0

and

whereH(D) is the set of all holomorphic functions onD.

|ρ(z)-ρ(w)|≤ε|z-w|,

wheneverz,w∈DE. We let

where Δφis Laplace operator acting onφ. Throughout this paper, we always suppose thatφis a subharmonic function onDwithφ∈W0. This weight is closed to that discussed by Borichev, Dhuez and Kellay in [1] and by Oleinik and Perel’man in [2].

whenever 1≤p≤, which can be seen in [3].

Supposegis a function onDsuch that

We then define Berezin transform ofgto be

It is easy to check that Berezin transforms can be well defined on the spaceLp.

In this paper, we study the complex interpolation for Bergman spaces by the boundedness of Bergman projections. As an application, we also consider the action of the Berezin transform onLpspaces.

In what follows, the symbolcstands for a positive constant, which may change from line to line, but does not depend on the functions being considered. Two quantitiesAandBare called equivalent, denoted byA?B, if there exists somecsuch thatc-1A≤B≤cA.

1 Complex interpolation

Definition1SupposeX1andX2are Banach spaces. A spaceXin Banach space will be called an intermediate space betweenX1andX2ifX1∩X2?X?X1+X2with continuous inclusion. The spaceXis also called an interpolation space betweenX1andX2, denote as [X1,X2]=X.

Proposition1[5]Suppose 1≤p0≤p1≤and 0≤θ≤1. Then we have

where

Theorem1Suppose 1≤p0≤p1≤∞ and 0≤θ≤1. Then

where

which follows from the definition of complex interpolation.

there then exist a functionF(z,ζ)(z∈D, 0≤Reζ≤1) and a positive constantMsuch that

(a)F(z,θ)=f(z) for allz∈D.

(b)‖F(xiàn)(·，ζ)‖p0,φ≤Mfor all Reζ=0.

(c)‖F(xiàn)(·，ζ)‖p1,φ≤Mfor all Reζ=1.

Define a functionG(z,ζ) to be

We get

G(z,θ)=Pf(z)=f(z).

If Reζ=0, we have

2 Berezin transforms

Berezin transform plays an important role in Bergman spaces, such as Carleson measures, Toeplitz operators, Hankel operators. In this section, we discuss the boundedness of Berezin transforms acting onLpspaces. We need the following lemmas which can be seen in [3] and [6]. The distancedρ(z,w) is defined as

where the infimum is taken over all piecewiseC1curvesγ：[0,1]→Dwithγ(0)=zandγ(1)=w.

(1) There exist positive constantsσ,csuch that

(2) There exist some constantsα>0,c>0 such that

In particularly,

K(z,z)?e2φ(z)ρ(z)-2,z∈D.

Lemma2[3]Supposeρ∈0,k>-2,σ>0,-0, we have

Theorem2For 1≤p≤, Berezin transforms are bounded onLp.

ProofUsing the interpolation, we only need to show that Berezin transforms are bounded onL1andL, respectively. In fact, by Fubini’s theorem, we have

By Lemma 1 and Lemma 2, we obtain

Similarly,

The proof is completed.

3 Conclusion

The idea of complex interpolations appears everywhere in mathematics, especially in PDE, harmonic analysis, appriximation theory and operator theory. The method is most successful in dealing with operators which either act on a scale of Banach spaces or belongs to a scale of Banach spaces.