Strongly Pseudoconvex Domains with Finite Dimensional $p$-Bergman Spaces

Abstract

We show that for every $n\ge2$, $p>2$, and positive integer $m$, there exists a finite-volume pseudoconvex domain $D\subset\mathbb C^n$ with real-analytic strongly pseudoconvex boundary whose $p$-Bergman space $A^p(D)$ has dimension $m$. These domains admit smooth strictly plurisubharmonic exhaustions and satisfy $H^\infty(D)=\mathbb C$ and $\dim A^s(D)=\infty$ for every $0<s\le2$. Moreover, for every finite $s\ge p$, we have $\dim A^s(D)=\lceil pm/s\rceil$. As an application, we show that the Bergman projection fails to map $\mathcal{C}_c^\infty(D)$ into $A^p(D)$ and admits no bounded extension to $L^s(D)$ or $L^{s/(s-1)}(D)$ for any finite $s\ge p$.