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 \({{\overline{\Omega }}}\).

令\(\Omega \subset {\mathbb {C}}^n\)是一个光滑有界伪凸域，\(A^2 (\Omega )\)表示它的Bergman 空间。设\(P:L^2(\Omega )\longrightarrow A^2(\Omega )\)是伯格曼投影。对于可测量的\(\varphi :\Omega \longrightarrow \Omega \)，投影合成算子定义为\((K_\varphi f)(z) = P(f \circ \varphi )(z), z \在\欧米茄中，f \在A ^ 2（\欧米茄）。\）在1994年，Rochberg研究的有界\（K_ \ varphi \）上单位圆盘Hardy空间和获得的不同需要或充分条件的有界\ (K_\varphi\). 在本文中，我们对赝凸域上 Bergman 空间上的投影复合算子感兴趣。我们在符号\(\varphi \)上\({{\overline{\Omega }}}\)的平滑假设下研究了这个算子的有界性。