On the $p$-essential normality of principal submodules of the Bergman module on strongly pseudoconvex domains

Abstract

In this paper, we show that under a mild condition, a principal submodule of the Bergman module on a bounded strongly pseudoconvex domain with smooth boundary in $\mathbb{C}^n$ is $p$-essentially normal for all $p>n$. This improves a previous result by the first author and K. Wang, in which it was shown that any polynomial-generated principal submodule of the Bergman module on the unit ball $\mathbb{B}_n$ is $p$-essentially normal for all $p>n$. As a consequence, we show that the submodule of $L_a^2(\mathbb{B}_n)$ consisting of functions vanishing on an analytic subset of pure codimension $1$ is $p$-essentially normal for all $p>n$.