Characterization of forward, vanishing, and reverse Bergman Carleson measures using sparse domination

TL;DR

This paper uses sparse domination from harmonic analysis to characterize various Bergman Carleson measures, including the previously unresolved reverse measures, extending existing results to broader parameter ranges and weighted spaces.

Contribution

It introduces a new technique to characterize all types of Bergman Carleson measures, including the open problem of reverse measures, and extends results to weighted spaces and broader parameters.

Findings

Characterized forward, vanishing, and reverse Bergman Carleson measures.

Extended results to all $0<p extless q extless\infty$ for forward and vanishing measures.

Provided new characterizations for measures related to the boundedness and compactness of radial differentiation operators.

Abstract

In this paper, using a new technique from harmonic analysis called sparse domination, we characterize the positive Borel measures including forward, vanishing, and reverse Bergman Carleson measures. The main novelty of this paper is determining the reverse Bergman Carleson measures which have remained open from the work of Luecking [Am. J. Math. 107 (1985) 85-111]. Moreover, in the case of forward and vanishing measures, our results extend the results of [J. Funct. Anal. 280 (2021), no. 6, 108897, 26 pp] from $1\leq p\leq q< 2p$ to all $0<p\leq q<\infty$. In a more general case, we characterize the positive Borel measures $\mu$ on $\mathbb{B}$ so that the radial differentiation operator $R^{k}:A_\omega^p(\mathbb{B})\rightarrow L^q(\mathbb{B},\mu)$ is bounded and compact. Although we consider the weighted Bergman spaces induced by two-side doubling weights, the results are new even on classical weighted Bergman spaces.