Projected composition operators on pseudoconvex domains

TL;DR

This paper investigates the boundedness of projected composition operators on Bergman spaces over smooth bounded pseudoconvex domains, extending prior work from Hardy spaces and providing conditions based on the smoothness of the symbol.

Contribution

It introduces new boundedness criteria for projected composition operators on Bergman spaces in pseudoconvex domains, considering smoothness assumptions on the symbol.

Findings

Boundedness conditions depend on the smoothness of the symbol.

Extension of Rochberg's results from Hardy to Bergman spaces.

Provides criteria for boundedness of projected composition operators.

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 $\overline\Omega$.