Hankel operators induced by radial Bekoll\'e-Bonami weights on Bergman spaces

TL;DR

This paper characterizes the boundedness of big Hankel operators induced by radial Bekollé-Bonami weights on weighted Bergman spaces, revealing stronger weight dependencies than classical cases and analyzing anti-analytic symbols.

Contribution

It provides a new characterization of Hankel operator boundedness for a broader class of weights, extending previous results to more general weighted Bergman spaces.

Findings

Characterization of boundedness in terms of weighted mean oscillation

Stronger dependence on weights compared to classical weights

Analysis of anti-analytic symbols

Abstract

We study big Hankel operators $H_f^\nu:A^p_\omega \to L^q_\nu$ generated by radial Bekoll\'e-Bonami weights $\nu$, when $1<p\leq q<\infty$. Here the radial weight $\omega$ is assumed to satisfy a two-sided doubling condition, and $A^p_\omega$ denotes the corresponding weighted Bergman space. A characterization for simultaneous boundedness of $H_f^\nu$ and $H_{\overline{f}}^\nu$ is provided in terms of a general weighted mean oscillation. Compared to the case of standard weights that was recently obtained by Pau, Zhao and Zhu (Indiana Univ. Math. J. 2016), the respective spaces depend on the weights $\omega$ and $\nu$ in an essentially stronger sense. This makes our analysis deviate from the blueprint of this more classical setting. As a consequence of our main result, we also study the case of anti-analytic symbols.