Hankel operators induced by radial Bekollé-Bonami weights on Bergman spaces

Abstract

We study big Hankel operators $H_f^{\nu }:A_{\omega }^p\to L_{\nu }^q$ generated by radial Bekollé-Bonami weights $\nu$, when $1<p\le q<\infty$. Here the radial weight $\omega$ is assumed to satisfy a two-sided doubling condition, and $A_{\omega }^p$ 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.

25 pages