The Bergman projection and weighted $C^k$ estimates for the canonical solution to \dbar on non-smooth domains

TL;DR

This paper develops weighted $C^k$ estimates for the canonical solution to the $ar{ ext{d}}$-problem on non-smooth strictly pseudoconvex domains using integral representations, extending regularity results to more complex geometries.

Contribution

It introduces new weighted $C^k$ estimates for the $ar{ ext{d}}$-problem on Henkin-Leiterer domains, a class of non-smooth strictly pseudoconvex domains.

Findings

Derived weighted $C^k$ estimates for the $ar{ ext{d}}$-problem on non-smooth domains.

Extended regularity results to Henkin-Leiterer domains with non-smooth boundaries.

Provided integral representation techniques for these estimates.

Abstract

We apply integral representations for functions on non-smooth strictly pseudoconvex domains, the Henkin-Leiterer domains, to derive weighted $C^k$ estimates for the component of a given function, $f$, which is orthogonal to holomorphic functions in terms of $C^k$ norms of $\mdbar f$. The weights are powers of the gradient of the defining function of the domain.