# Toeplitz Operators and Skew Carleson measures for weighted Bergman spaces on strongly pseudoconvex domains

## TL;DR

This paper characterizes the boundedness of Toeplitz operators on weighted Bergman spaces over strongly pseudoconvex domains using skew Carleson measures, extending previous results from the unit ball to more general domains.

## Contribution

It generalizes and refines the characterization of Toeplitz operator boundedness via skew Carleson measures for weighted Bergman spaces on strongly pseudoconvex domains.

## Key findings

- Toeplitz operators map between weighted Bergman spaces if and only if the measure is skew Carleson.
- The results extend previous work from the unit ball to strongly pseudoconvex domains.
- Provides explicit conditions relating weights, exponents, and measures for operator boundedness.

## Abstract

In this paper we study mapping properties of Toeplitz-like operators on weighted Bergman spaces of bounded strongly pseudconvex domains in $\mathbb{C}^n$. In particular we prove that a Toeplitz operator built using as kernel a weighted Bergman kernel of weight $\beta$ and integrating against a measure $\mu$ maps continuously (when $\beta$ is large enough) a weighted Bergman space $A^{p_1}_{\alpha_1}(D)$ into a weighted Bergman space $A^{p_2}_{\alpha_2}(D)$ if and only if $\mu$ is a $(\lambda,\gamma)$-skew Carleson measure, where $\lambda=1+\frac{1}{p_1}-\frac{1}{p_2}$ and $\gamma=\frac{1}{\lambda}\left(\beta+\frac{\alpha_1}{p_1}-\frac{\alpha_2}{p_2}\right)$. This theorem generalizes results obtained by Pau and Zhao on the unit ball, and extends and makes more precise results obtained by Abate, Raissy and Saracco on a smaller class of Toeplitz operators on bounded strongly pseudoconvex domains.