Carleson measures and Toeplitz operators on small Bergman spaces on the ball

TL;DR

This paper extends the characterization of Carleson measures and analyzes Toeplitz operators on small weighted Bergman spaces in higher dimensions, providing criteria for boundedness, compactness, and Schatten class membership.

Contribution

It generalizes Seip's results from the unit disc to the unit ball in complex space, offering new criteria for Toeplitz operators on these spaces.

Findings

Characterization of Carleson measures on small Bergman spaces in higher dimensions

Necessary and sufficient conditions for Toeplitz operator boundedness and compactness

Criteria for Toeplitz operators to belong to Schatten p classes

Abstract

We study the Carleson measures and the Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip's results from the unit disc of $\mathbb C$ to the unit ball of $\mathbb C^n$. We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten $p$ classes membership of Toeplitz operators for $1<p<\infty$.