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Journal of Operator Theory

Volume 84, Issue 2, Fall 2020  pp. 339-364.

Toeplitz operators and skew Carleson measures for weighted Bergman spaces on strongly pseudoconvex domains

Authors:  Marco Abate (1), Samuele Mongodi (2), Jasmin Raissy (3)
Author institution: (1) Dipartimento di Matematica, Universita di Pisa, Largo Pontecorvo 5 Pisa, I-56127, Italy
(2) Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9 Milano, I-20133, Italy
(3) Institut de Mathematiques de Toulouse, UMR5219, Universite de Toulouse, CNRS, UPS, Toulouse, F-31062, France

Summary: In this paper we study properties of Toeplitz operators on weighted Bergman spaces of bounded strongly pseudoconvex domains. We prove that a Toeplitz operator built using a weighted Bergman kernel of weight $\beta$ and integrating against a measure $\mu$ maps continuously a weighted Bergman space $A^{p_1}_{\alpha_1}(D)$ into $A^{p_2}_{\alpha_2}(D)$ if and only if $\mu$ is a $(\lambda,\gamma)$-skew Carleson measure, where $\lambda=1+\frac{1}{p_1}-\frac{1}{p_2}$ and $\gamma=\frac{1}{\lambda}(\beta+\frac{\alpha_1}{p_1}-\frac{\alpha_2}{p_2})$. This generalizes results obtained by Pau and Zhao on the unit ball, and by Abate, Raissy and Saracco on a smaller class of Toeplitz operators on strongly pseudoconvex domains.

Keywords: Carleson measure, Toeplitz operator, strongly pseudoconvex domain, weighted Bergman spaces

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