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

TL;DR

This paper derives explicit formulas for the Bergman kernel on generalized Hartogs triangles in complex space and determines the exact range of p for which the Bergman projection is bounded on L^p spaces.

Contribution

It provides new explicit formulas for the Bergman kernel on a class of generalized Hartogs triangles and establishes the sharp range of p for L^p boundedness of the Bergman projection.

Findings

Explicit formulas for the Bergman kernel on  domains

Sharp range of p for L^p boundedness of the Bergman projection

Identification of conditions for boundedness on generalized Hartogs triangles

Abstract

In this paper we investigate a class of domains $\Omega^{n+1}_k =\{(z,w)\in \mathbb{C}^n\times \mathbb{C}: |z|^k < |w| < 1\}$ 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.