Characterization of Forward, Vanishing, and Reverse Bergman Carleson Measures using Sparse Domination

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. In the case of forward and vanishing Bergman Carleson measures, our results extend the results of [J Funct Anal 280(6):26, 2021] from \(1\leqslant p\leqslant q< 2p\) to all \(0<p\leqslant q<\infty \). 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.