Toeplitz operators on the unit ball with locally integrable symbols

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.

24 pages; added an example