On New Decomposition Theorems in some Analytic Function Spaces in Bounded Pseudoconvex Domains

Romi F Shamoyan,Elena B Tomashevskaya

doi:10.17516/1997-1397-2020-13-4-503-514

Abstract

We provide new sharp decomposition theorems for multifunctional Bergman spaces in the unit ball and bounded pseudoconvex domains with smooth boundary expanding known results from the unit ball. Namely we prove that mΠ j=1 jjfj jjXj ≍ jjf1 : : : fmjj Ap for various (Xj) spaces of analytic functions in bounded pseudoconvex domains with smooth boundary where f; fj ; j = 1; : : : ;m are analytic functions and where Ap ; 0 < p < 1; > ��1 is a Bergman space. This in particular also extend in various directions a known theorem on atomic decomposition of Bergman Ap spaces.

Highlights

Introduction and preliminariesThe problem we consider is well-known for functional spaces in Rn.Let X, (Xj) be a function space in a fixed domain D in Cn we wish to find equivalent expression for ||f1 . . . fm||X ; m ∈ N
We provide new sharp decomposition theorems for multifunctional Bergman spaces in the unit ball and bounded pseudoconvex domains with smooth boundary expanding known results from the unit ball
∏m Namely we prove that ||fj||Xj ≍ ||f1 . . . fm||Apα for various (Xj) spaces of analytic functions in j=1 bounded pseudoconvex domains with smooth boundary where f, fj, j = 1, . . . , m are analytic functions and where Apα, 0 < p < ∞, α > −1 is a Bergman space

Summary

Introduction and preliminaries

The problem we consider is well-known for functional spaces in Rn (the problem of equivalent norms) (see, for example, [1]). For m = 1 Hardy space case (see, for example, [2,3,4]) To study such group of functions it is natural, for example, to ask about structure of each {fj}m j=1 of this group. This idea was used for Bergman spaces in the unit ball in bounded pseudoconvex domains with smooth boundary in recent papers [5]. We refer to [5, 6] for a complete and not difficult proof of a basic known "purely Apα" case in this paper show in details how to modify it to get new results. The old known proof is simple and very flexible as it turns out and we can get, as we can see below, various new interesting results from it directly. The case of Apα Bergman space in more general bounded pseudoconvex domain can be seen in our recent paper [6]

We define Apα space as usual

Apαii z

Apαii m k p