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 $\beta$ and integrating against a measure $\mu$ maps continuously (when $\beta$ is large enough) a weighted Bergman space $A_{{\alpha }_1}^{p_1}(D)$ into a weighted Bergman space $A_{{\alpha }_2}^{p_2}(D)$ 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.

21 pages. arXiv admin note: text overlap with arXiv:1710.01631