Embedding theorems for Bergman spaces via harmonic analysis

TL;DR

This paper characterizes measures for which differentiation operators are bounded between weighted Bergman spaces and L^q spaces, using harmonic analysis and developing a theory of tent spaces for these spaces.

Contribution

It introduces geometric characterizations of measures ensuring bounded differentiation operators on weighted Bergman spaces with doubling weights.

Findings

Characterization of measures for bounded differentiation operators

Development of tent space theory for weighted Bergman spaces

Extension of harmonic analysis techniques to Bergman space embeddings

Abstract

Let $A^p_\omega$ denote the Bergman space in the unit disc induced by a radial weight~$\omega$ with the doubling property $\int_{r}^1\omega(s)\,ds\le C\int_{\frac{1+r}{2}}^1\omega(s)\,ds$. The positive Borel measures such that the differentiation operator of order $n\in\mathbb{N}\cup\{0\}$ is bounded from $A^p_\omega$ into $L^q(\mu)$ are characterized in terms of geometric conditions when $0<p,q<\infty$. En route to the proof a theory of tent spaces for weighted Bergman spaces is built.