The Bergman kernel and projection on a class of bounded Hartogs domains
Studia Mathematica
MSC: Primary 47G10; Secondary 32A36, 32A25
DOI: 10.4064/sm250214-27-2
Published online: 3 July 2026
Abstract
We study the Bergman kernel and projection on the class of bounded Hartogs domains defined by $$ H^{\mathbf {p}}_{k+m}:=\{z=(\widetilde {z},z_{k+1},\ldots ,z_{k+m})\in \mathbb {C}^{k+m}: \|\widetilde {z}\|^{p_1} \lt |z_{k+1}|^{p_{k+1}} \lt \cdots \lt |z_{k+m}|^{p_{k+m}} \lt 1\},$$ where $k,m\in \mathbb {Z}^+$, $\mathbf {p}:=(p_1,p_{k+1},\ldots ,p_{k+m})\in (\mathbb {R}^+)^{m+1}$ and $\widetilde {z}:=(z_1,\ldots ,z_k)\in \mathbb {C}^k$. We obtain an explicit formula for the Bergman kernel of $H^{\mathbf {p}}_{k+m}$ and use it to establish an optimal estimate. We then study the $L^p$ regularity and irregularity of the Bergman projection and further investigate the weak-type estimates at the endpoints of the $L^p$ boundedness range.
