Regularity of a $\bar\partial$-solution operator for strongly $\mathbf C$-linearly convex domains with minimal smoothness

TL;DR

This paper establishes regularity results for the $ar ext{-}\partial$-problem solutions in certain convex domains with minimal smoothness, using new domain characterizations and advanced analytical techniques.

Contribution

It introduces a novel analytic characterization of strongly $\mathbf C$-linearly convex domains with $C^{1,1}$ smoothness, enabling regularity proofs for the $ar\partial$-problem.

Findings

Proves regularity of solutions in Hölder-Zygmund spaces.

Develops a new characterization of the domain class.

Extends techniques from strongly pseudoconvex domains to less smooth domains.

Abstract

We prove regularity of solutions of the $\bar\partial$-problem in the H\"older-Zygmund spaces of bounded, strongly $\mathbf C$-linearly convex domains of class $C^{1,1}$. The proofs rely on a new, analytic characterization of said domains which is of independent interest, and on techniques that were recently developed by the first-named author to prove estimates for the $\bar\partial$-problem on strongly pseudoconvex domains of class $C^2$.