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

Abstract

We show that for every $n\ge 2$, $p>2$, 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 $0<s\le 2$. 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 $s\ge p$.

arXiv comment25 pages, comments are welcome!