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Atomic decomposition of real-variable type for Bergman spaces in the unit ball of $\mathbb{C}^n$

Zeqian Chen and Wei Ouyang

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Abstract

In this paper we show that, for any $0 < p \le 1$ and $\alpha > -1$, every (weighted) Bergman space $\mathcal{A}^p_{\alpha} (\mathbb{B}_n)$ admits an atomic decomposition of real-variable type. More precisely, for each $f \in \mathcal{A}^p_{\alpha} (\mathbb{B}_n)$ there exist a sequence of $(p, \infty)_{\alpha}$-atoms $a_k$ with compact support and a scalar sequence $\{\lambda_k \}$ such that $f = \sum_k \lambda_k a_k$ in the sense of distribution and $\sum_k | \lambda_k |^p \lesssim \| f \|^p_{p, \alpha};$ and moreover, $f = \sum_k \lambda_k P_{\alpha} ( a_k)$ in $\mathcal{A}^p_{\alpha} (\mathbb{B}_n),$ where $P_{\alpha}$ is the orthogonal projection from $L^2_{\alpha} (\mathbb{B}_n)$ onto $\mathcal{A}^2_{\alpha} (\mathbb{B}_n).$ The proof is constructive and our construction is based on analysis inside the unit ball $\mathbb{B}_n$ associated with a quasimetric.

Article information

Source
Publ. Mat., Volume 58, Number 2 (2014), 353-377.

Dates
First available in Project Euclid: 21 July 2014

Permanent link to this document
https://projecteuclid.org/euclid.pm/1405949323

Mathematical Reviews number (MathSciNet)
MR3264502

Zentralblatt MATH identifier
1300.34162

Subjects
Primary: 32A36: Bergman spaces 32A50: Harmonic analysis of several complex variables [See mainly 43-XX]

Keywords
Bergman space atomic decomposition Bergman kernel homogeneous space maximal function

Citation

Chen, Zeqian; Ouyang, Wei. Atomic decomposition of real-variable type for Bergman spaces in the unit ball of $\mathbb{C}^n$. Publ. Mat. 58 (2014), no. 2, 353--377. https://projecteuclid.org/euclid.pm/1405949323


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