Hankel operators on Fock spaces and related Bergman kernel estimates

TL;DR

This paper characterizes bounded Hankel operators with anti-holomorphic symbols on weighted Fock spaces, linking their boundedness to BMOA and Bloch spaces via Bergman kernel estimates, and extends to compact and Schatten class operators.

Contribution

It provides a comprehensive characterization of Hankel operators on weighted Fock spaces, connecting boundedness to BMOA and Bloch spaces through Bergman kernel and metric estimates, including compactness and Schatten class criteria.

Findings

Bounded Hankel operators are characterized by symbols in BMOA/Bloch spaces.

The paper establishes estimates for Bergman kernels and metrics.

Results include criteria for compact and Schatten class Hankel operators.

Abstract

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on $\cn$. The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with H\"{o}rmander's $L^2$ estimates for the $\bar{\partial}$ operator are key ingredients in the proof.