Projected Composition Operators on Pseudoconvex Domains

Abstract

Let \(\Omega \subset {\mathbb {C}}^n\) be a smooth bounded pseudoconvex domain and \(A^2 (\Omega )\) denote its Bergman space. Let \(P:L^2(\Omega )\longrightarrow A^2(\Omega )\) be the Bergman projection. For a measurable \(\varphi :\Omega \longrightarrow \Omega \), the projected composition operator is defined by \((K_\varphi f)(z) = P(f \circ \varphi )(z), z \in \Omega , f\in A^2 (\Omega ).\) In 1994, Rochberg studied boundedness of \(K_\varphi \) on the Hardy space of the unit disk and obtained different necessary or sufficient conditions for boundedness of \(K_\varphi \). In this paper we are interested in projected composition operators on Bergman spaces on pseudoconvex domains. We study boundedness of this operator under the smoothness assumptions on the symbol \(\varphi \) on \({{\overline{\Omega }}}\).