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

In this paper we study mapping properties of Toeplitz-like operators on weighted Bergman spaces of bounded strongly pseudoconvex domains in several complex variables. 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 into another weighted Bergman space if and only if μ is a (λ,γ)-skew Carleson measure, where λ and γ can be computed explicitly. 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 for a smaller class of Toeplitz operators on bounded strongly pseudoconvex domains.