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

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 $β$ and integrating against a measure $μ$ maps continuously (when $β$ is large enough) a weighted Bergman space $A^{p_1}_{α_1}(D)$ into a weighted Bergman space $A^{p_2}_{α_2}(D)$ if and only if $μ$ is a $(λ,γ)$-skew Carleson measure, where $λ=1+\frac{1}{p_1}-\frac{1}{p_2}$ and $γ=\frac{1}λ\left(β+\frac{α_1}{p_1}-\frac{α_2}{p_2}\right)$. 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.