A solution operator for $\bar\partial$ on the Hartogs triangle and $L^p$ estimates

TL;DR

This paper constructs an integral solution operator for the ar on product domains including the punctured bidisc, achieving broad $L^p$ estimates that extend beyond traditional bounds, with applications to the Hartogs triangle.

Contribution

It introduces a novel ar solution operator on product domains that satisfies $L^p$ estimates with non-standard bounds, extending the known range of $p$ for the Hartogs triangle.

Findings

The operator satisfies $L^p$ estimates for all $1 \u2264 p < \u221e$.

It extends $L^p$ estimates for ar on the Hartogs triangle beyond the canonical solution.

The estimates involve non-standard bounding terms compared to strongly pseudoconvex domains.

Abstract

An integral solution operator for $\bar\partial$ is constructed on product domains that include the punctured bidisc. This operator is shown to satisfy $L^p$ estimates for all $1\leq p <\infty$, though with non-standard -- relative to strongly pseudoconvex domains -- bounding term. These estimates imply $L^p$ estimates for $\bar\partial$ on the Hartogs triangle, with greater range of $p$ than the canonical solution satisfies.