Two Problems in Bergman Spaces with Non-radial Weights

Abstract

This paper investigates two problems unified by the study of the uniform boundedness of the dilation operators (UBD) T_r f(z)=f(rz), 0<r<1, acting on weighted Bergman spaces A^p_omega with not necessarily radial weights. We first characterize the random symbol space for A^p_omega under a mild admissible condition (Theorem 1.2). This extends the main result of [7] from radial weights to non-radial weights. We then introduce two new notions, namely non-radial mixed norm spaces M(p,q;omega) and analytic tent spaces A(p,q;omega), and we characterize their corresponding symbol spaces as well (Theorem 1.8, Theorem 1.12). The novelty here is to employ a measure-disintegration framework to prove a general Littlewood-type theorem (M(p,q;omega))* = H(2,q;omega_r), from which the Bergman space result (A^p_omega)* = H(2,p;omega_r) follows as the special case p=q. Among other things, UBD plays a pivotal role in the proofs of the preceding theorems. The second main problem addressed in this paper is to establish a local-to-global criterion for UBD, which remains largely unexplored for non-radial weights. Our principle result in this part (Theorem 1.16) asserts that UBD is guaranteed by two local geometric conditions: bounded hyperbolic oscillation (BHO) and a reverse-Carleson tail condition (RC). This is the technical heart of the paper. In the course of our investigation, three new types of problems arise naturally, each of independent interest: a two-weight top-maximal operator (Theorem 1.17); a truncated maximal operator over hyperbolic balls (Theorem 1.18); and a single-testing Carleson embedding problem (Theorem 1.19).