IDA function and asymptotic behavior of singular values of Hankel operators on weighted Bergman spaces

TL;DR

This paper characterizes the asymptotic behavior of singular values of Hankel operators on various weighted Bergman spaces using the IDA function, linking it to mean oscillation and Schatten class membership.

Contribution

It introduces a novel approach using the IDA function to analyze singular value asymptotics of Hankel operators on weighted Bergman spaces, including Fock spaces.

Findings

Asymptotic behavior of singular values characterized by IDA function.

Simultaneous asymptotics of $H_f$ and $H_{ar{f}}$ linked to mean oscillation.

Hankel operators' membership in weak Schatten classes demonstrated.

Abstract

In this paper, we use the non-increasing rearrangement of ${\rm IDA}$ function with respect to a suitable measure to characterize the asymptotic behavior of the singular values sequence $\{s_n(H_f)\}_n$ of Hankel operators $H_f$ acting on a large class of weighted Bergman spaces, including standard Bergman spaces on the unit disc, standard Fock spaces and weighted Fock spaces. As a corollary, we show that the simultaneous asymptotic behavior of $\{s_n(H_f)\}$ and $\{s_n(H_{\bar{f}})\}$ can be characterized in terms of the asymptotic behavior of non-increasing rearrangement of mean oscillation function. Moreover, in the context of weighted Fock spaces, we demonstrate the Berger-Coburn phenomenon concerning the membership of Hankel operators in the weak Schatten $p$-class.