On Extremal Problems in Certain New Bergman Type Spaces in Some Bounded Domains inCn

Romi F Shamoyan,Olivera Mihić

doi:10.1155/2014/975434

Romi F Shamoyan, Olivera Mihić

Open Access

https://doi.org/10.1155/2014/975434

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Abstract

Based on recent results on boundedness of Bergman projection with positive Bergman kernel in analytic spaces in various types of domains inCn, we extend our previous sharp results on distances obtained for analytic Bergman type spaces in unit disk to some new Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains, and minimal bounded homogeneous domains.

Highlights

The goal of this paper is to add several new results for distances in analytic Bergman type spaces of functions of several variables
Based on recent results on boundedness of Bergman projection with positive Bergman kernel in analytic spaces in various types of domains in Cn, we extend our previous sharp results on distances obtained for analytic Bergman type spaces in unit disk to some new Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains, and minimal bounded homogeneous domains
We look at analytic Bergman type spaces in Lie ball, bounded symmetric domains of tube type, Siegel domains of second type, and minimal bounded homogeneous domains

Summary

Introduction

The goal of this paper is to add several new results for distances in analytic Bergman type spaces of functions of several variables. We use the boundedness of Bergman type projections with |K(z, w)| positive kernel acting from X to X together with ForelliRudin type sharp estimates of Bergman kernel These three tools were used in general Siegel domain of second type, polydisk, and unit ball in [1,2,3,4] (see various references there). In [13] the base of all our proofs in complex domains in higher dimension was the Bergman reproducing formula, while here all our assertions are based on some recent results on boundedness of Bergman projections with positive Bergman kernel in Bergman spaces in such type domains. We will need various definitions and assertions for formulations of main results These are assertions on various types of domains we consider in this paper and analytic functions on them. Assume that 1 ≤ p < +∞, α > −1 and that the complex number β satisfies the condition

Re β

PD of D is the orthogonal projection of the Hilbert space

Bergman projection

The transference principle also applies to the positive operators

The Bergman along with the projection associated

Note then we used that for kj