Weak BMO and Toeplitz operators on Bergman spaces

TL;DR

This paper introduces weak BMO and VMO conditions for functions on the unit disc, analyzes their properties, and applies these to generalize results on Toeplitz operators' spectra and norms.

Contribution

It defines weak BMO and VMO conditions, explores their properties, and extends known results on Toeplitz operators using these new function classes.

Findings

Average functions of BWMO are boundedly oscillating

Results extend to spectra and norms of Toeplitz operators

Examples of VWMO functions not in classical VMO or BMO

Abstract

Inspired by our previous work on the boundedness of Toeplitz operators, we introduce weak BMO and VMO type conditions, denoted by BWMO and VWMO, respectively, for functions on the open unit disc of the complex plane. We show that the average function of a function $f$ in BWMO is boundedly oscillating, and the analogous result holds for $f$ in VWMO. The result is applied for generalizations of known results on the essential spectra and norms of Toeplitz operators. Finally, we provide examples of functions satisfying the VWMO condition which are not in the classical VMO or even in BMO.