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 ${{\overline{\Omega }}}$.