Maximal Estimates for the $\bar\partial$-Neumann Problem on Non-pseudoconvex domains

TL;DR

This paper investigates the conditions under which maximal estimates hold for the $ar ext ext{d}$-Neumann problem on non-pseudoconvex domains, extending known results from the pseudoconvex case and providing new necessary and sufficient criteria.

Contribution

It establishes necessary and sufficient conditions for maximal estimates in non-pseudoconvex domains, broadening the understanding beyond pseudoconvex cases and including new examples.

Findings

Derived necessary conditions for maximal estimates in non-pseudoconvex domains.

Identified sufficient conditions and when they coincide with necessary conditions.

Extended previous pseudoconvex results to more general non-pseudoconvex settings.

Abstract

It is well known that elliptic estimates fail for the $\bar\partial$-Neumann problem. Instead, the best that one can hope for is that derivatives in every direction but one can be estimated by the associated Dirichlet form, and when this happens, we say that the $\bar\partial$-Neumann problem satisfies maximal estimates. In the pseudoconvex case, a necessary and sufficient geometric condition for maximal estimates has been derived by Derridj (for $(0,1)$-forms) and Ben Moussa (for $(0,q)$-forms when $q\geq 1$). In this paper, we explore necessary conditions and sufficient conditions for maximal estimates in the non-pseudoconvex case. We also discuss when the necessary conditions and sufficient conditions agree and provide examples. Our results subsume the earlier known results from the pseudoconvex case.