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

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