Zero sequences, factorization and sampling measures for weighted Bergman spaces

Taneli Korhonen,Jouni Rättyä

doi:10.1007/s00209-019-02243-7

Abstract

The zero sets of the Bergman space A^p_omega induced by either a radial weight omega admitting a certain doubling property or a non-radial Bekollé-Bonami type weight are characterized in the spirit of Luecking’s results from 1996. Accurate results obtained en route to this characterization are used to generalize Horowitz’s factorization result from 1977 for functions in A^p_omega . The utility of the obtained factorization is illustrated by applications to integration and composition operators as well as to small Hankel operator induced by a conjugate analytic symbol. Dominating sets and sampling measures for the weighted Bergman space A^p_omega induced by a doubling weight are also studied. Several open problems related to the scheme of the paper are posed.

Highlights

Introduction and main resultsLet H(D) denote the space of analytic functions in the unit disc D = {z ∈ C : |z| < 1} of the complex plane C
A function ω : D → [0, ∞), integrable over D, is called a weight. It is radial if ω(z) = ω(|z|) for all z ∈ D
For 0 < p < ∞ and a weight ω, the weighted Bergman space Aωp consists of f ∈ H(D) such that

Summary

Introduction and main results

1 − |w|2 α d A(w), z ∈ D, for α = 2 on certain Lq -spaces, and this step follows at once by using the classical characterization of the one-weight inequality for the Bergman projection by Bekollé and Bonami [3,4] This yields the hypothesis ω ∈ B∞ in Theorems 1 and 4, the proofs of which are presented in Sect. The studies on Carleson measures [29,30,31,36] strongly support the use of Carleson squares instead of pseudohyperbolic discs as testing sets, at least when ω induces a very small weighted Bergman space. The measures μ satisfying the inequality are the p-Carleson measures for Aωp These measures in the case ω ∈ D have been studied in [30,31,36], and can be characterized in terms of the weighted maximal function.

Dominating sets

Sampling measures