Toeplitz operators on the unit ball with locally integrable symbols

TL;DR

This paper investigates the boundedness and compactness of Toeplitz operators with locally integrable symbols on weighted harmonic Bergman spaces over the unit ball, extending results to analytic function spaces.

Contribution

It provides new sufficient conditions for boundedness and compactness of Toeplitz operators with locally integrable symbols, generalizing previous analytic space results.

Findings

Derived a general sufficient condition for boundedness of Toeplitz operators.

Established a vanishing condition characterizing compactness.

Extended results to weighted Bergman spaces of analytic functions.

Abstract

We study the boundedness of Toeplitz operators $T_\psi$ with locally integrable symbols on weighted harmonic Bergman spaces over the unit ball of $\mathbb{R}^n$. Generalizing earlier results for analytic function spaces, we derive a general sufficient condition for the boundedness of $T_\psi$ in terms of suitable averages of its symbol. We also obtain a similar "vanishing" condition for compactness. Finally, we show how these results can be transferred to the setting of the standard weighted Bergman spaces of analytic functions.