Weighted estimates for the Bergman projection on the Hartogs triangle

Abstract

We establish a weighted inequality for the Bergman projection with matrix weights for a class of pseudoconvex domains. We extend a result of Aleman-Constantin and obtain the following estimate for the weighted norm of P : P L 2 (Ω,W) ≤ C(B 2 (W)) 2. Here B 2 (W) is the Bekollé-Bonami constant for the matrix weight W and C is a constant that is independent of the weight W but depends upon the dimension and the domain.

Dyadic Carleson embedding and sparse domination of weighted composition operators on strictly pseudoconvex domains

“…Dyadic decomposition on Ω. For this part, we mainly follow the framework built in [19,20,22]. Heuristically, the construction of a dyadic decomposition on Ω consists of two steps: I. bΩ is a space of homogeneous type, and hence it admits a dyadic decomposition; II.…”

“…To our best knowledge, the first appearance of such an estimate appears in the work of Aleman, Pott and Reguera [4], in which, the Sarason conjecture on the Bergman spaces was studied via a pointwise sparse domination estimate of the Bergman projection. Later, by using similar idea, Rahm Tchoundja and Wick [38] were able to prove a sharp weighted estimate for the Berezin transform and Bergman projection (see, also [20,21,22,14] for some extension of this work). In a recent paper [18], the authors extended this line of research by studying the behavior of weighted composition operators on the upper half plane and the unit ball via the sparse domination technique.…”

Weak‐type estimates for the Bergman projection on the polydisc and the Hartogs triangle

“…Remark It can be shown that if the Bergman projection on a weighted space is of weak‐type , then is bounded on . The idea of the proof can be found in [29, Theorem 1] and [17, Theorem 1.2]. Theorem 4.2, on the other hand, shows a different phenomenon in the Hartogs triangle case: the Bergman projection on is of weak‐type (4,4) but not ‐bounded.…”

Bekollé-Bonami estimates on some pseudoconvex domains

“…Later, Rahm, Tchoundja, and Wick [RTW17] generalized the results of Pott and Reguera to the unit ball case, and also obtained estimates for the Berezin transform. Weighted norm estimates of the Bergman projection have also been obtained [HW19] on the Hartogs triangle.…”