Weighted $C^k$ estimates for a class of integral operators on non-smooth domains

TL;DR

This paper develops weighted $C^k$ estimates for $(0,q)$-forms on non-smooth strictly pseudoconvex domains using integral representations, linking the regularity of forms to their $ar{d}$ derivatives with specific weights.

Contribution

It introduces new weighted $C^k$ estimates for forms on Henkin-Leiterer domains, extending regularity results to non-smooth pseudoconvex domains.

Findings

Weighted $C^k$ estimates derived for $(0,q)$-forms

Estimates relate form regularity to $ar{d}$ and $ar{d}^*$ norms

Applicable to non-smooth strictly pseudoconvex domains

Abstract

We apply integral representations for $(0,q)$-forms, $q\ge1$, on non-smooth strictly pseudoconvex domains, the Henkin-Leiterer domains, to derive weighted $C^k$ estimates for a given $(0,q)$-form, $f$, in terms of $C^k$ norms of $\mdbar f$, and $\mdbar^{\ast} f$. The weights are powers of the gradient of the defining function of the domain.