Bergman projections on weighted Fock spaces in several complex variables

Xiaofen Lv

doi:10.1186/s13660-017-1560-3

Abstract

Let ϕ be a real-valued plurisubharmonic function on {mathbb {C}}^{n} whose complex Hessian has uniformly comparable eigenvalues, and let mathcal{F}^{p}(phi) be the Fock space induced by ϕ. In this paper, we conclude that the Bergman projection is bounded from the pth Lebesgue space L^{p}(phi ) to mathcal{F}^{p}(phi) for 1leq p leqinfty. As a remark, we claim that Bergman projections are also well defined and bounded on Fock spaces mathcal{F}^{p}(phi) with 0< p<1. We also obtain the estimates for the distance induced by ϕ and the L^{p}(phi)-norm of Bergman kernel for mathcal{F}^{2}(phi).

Highlights

We say that φ belongs to the weight class W if φ satisfies the following statements: (I) There exists c > such that for z ∈ Cn inf sup φ(w) > ; ( )
The symbol dv denotes the Lebesgue volume measure on Cn, andB(z, r) = w ∈ Cn : |w – z| < r for z ∈ Cn and r > .Suppose φ : Cn → R is a C plurisubharmonic function
In Section, we will discuss the boundedness of Bergman projections from Lp(φ) to F p(φ) with ≤ p ≤ ∞

Summary

Introduction

We say that φ belongs to the weight class W if φ satisfies the following statements: (I) There exists c > such that for z ∈ Cn inf sup φ(w) > ; ( ) L∞(φ) is the set of all Lebesgue measurable functions f on Cn with f ∞,φ = sup f (z) e–φ(z) < ∞. Let H(Cn) be the family of all holomorphic functions on Cn. The weighted Fock space is defined as Given φ ∈ W, let K(·, ·) be the weighted Bergman kernel for

Objectives

Results

Conclusion