Maximal and Area Integral Characterizations of Bergman Spaces in the Unit Ball ofℂn

Zeqian Chen,Wei Ouyang

doi:10.1155/2013/167514

Abstract

We present maximal and area integral characterizations of Bergman spaces in the unit ball ofℂn. The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the radial derivative, the complex gradient, and the invariant gradient. As an application, we obtain new maximal and area integral characterizations of Besov spaces. Moreover, we give an atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces.

Highlights

Introduction and Main ResultsLet C denote the set of complex numbers
For α > −1 and p > 0, the Bergman space Apα consists of holomorphic functions f in Bn with
It is known that the family of the generalized Bergman spaces Apα covers tions in the unit most of the spaces ball of Cn, such as of holomorphic functhe classical diagonal

Summary

Introduction and Main Results

Let C denote the set of complex numbers. Throughout the paper we fix a positive integer n, and let. Call it the invariant measure on Bn, since dτ = dτ ∘ φ for any automorphism φ of Bn. For α > −1 and p > 0, the (weighted) Bergman space Apα consists of holomorphic functions f in Bn with. We first establish a maximal-function characterization for the Bergman spaces. To this end, we define for each γ > 0 and f ∈ H(Bn):. It is known that the family of the generalized Bergman spaces Apα covers tions in the unit most of the spaces ball of Cn, such as of holomorphic functhe classical diagonal. Any notation and terminology not otherwise explained are as used in [3] for spaces of holomorphic functions in the unit ball of Cn

Proofs of Theorems 1 and 2

Atomic Decomposition for Bergman Spaces

Area Integral Inequalities