Toeplitz operators on the unit ball with locally integrable symbols (Q2140259)

The authors study the boundedness of Toeplitz operators \(T_{\psi}\) with locally integrable symbols on weighted harmonic Bergman spaces over the unit ball in \(\mathbb{R}^n\). The main result of the paper is a general sufficient condition for the boundedness of \(T_{\psi}\) in terms of suitable averages of its symbol \(\psi\) on certain sets. Note that the authors propose a new kind of these sets, certain spherical boxes, and furthermore, the modulus in their condition is located outside of the integral, which makes the condition stronger. A~similar ``vanishing'' condition for the compactness of Toeplitz operators is also given. Finally, the authors show how the above results can be transferred to the setting of the standard weighted Bergman spaces of analytic functions.