$L^p$ regularity of the Bergman projection on generalizations of the Hartogs triangle in ${\mathbb{C}}^{n+1}$

Abstract

In this paper we investigate a class of domains ${\Omega }_k^{n+1}=\{(z,w)\in {\mathbb{C}}^n\times \mathbb{C}:|z{|}^k<|w|<1\}$ for $k\in {\mathbb{Z}}^{+}$ which generalizes the Hartogs triangle. we first obtain the new explicit formulas for the Bergman kernel function on these domains and further give a range of $p$ values for which the $L^p$ boundedness of the Bergman projection holds. This range of $p$ is shown to be sharp.

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