Characterization of Schatten class Hankel operators on weighted Bergman spaces

TL;DR

This paper provides a complete characterization of when Hankel operators on weighted Bergman spaces belong to Schatten classes, resolving a long-standing conjecture by Kehe Zhu from 1991.

Contribution

It introduces a new criterion based on local mean oscillation functions for Schatten class membership of Hankel operators, confirming Zhu's conjecture.

Findings

Characterization of Schatten class membership via local mean oscillation.

Resolution of Kehe Zhu's 1991 conjecture.

Unified framework for Hankel operators on weighted Bergman spaces.

Abstract

We completely characterize the simultaneous membership in the Schatten ideals $S_ p$, $0<p<\infty$ of the Hankel operators $H_ f$ and $H_{\bar{f}}$ on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.