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

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 is a significant improvement of the results of the first author and K. Wang, where the same result is shown to hold for polynomial-generated principal submodules of the Bergman module on the unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$. As a consequence of our main result, we prove that the submodule of $L_a^2(\mathbb{B}_n)$ consisting of functions vanishing on a pure analytic subsets of codimension $1$ is $p$-essentially normal for all $p>n$.