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. 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.