Regularity of the $\bar\partial$-Neumann problem by means of superlogarithmic multipliers

TL;DR

This thesis explores regularity of the $ar ext{d}$-Neumann problem using superlogarithmic multipliers, combining classical energy estimates with geometric analysis of Levi form singularities, and introduces a new weighted formula for domain regularity.

Contribution

It introduces a novel weighted Kohn-Hörmander-Morrey formula twisted by a pseudodifferential operator, expanding the class of domains with locally regular $ar ext{d}$-Neumann problem solutions.

Findings

Established local regularity for new classes of domains.

Connected geometric properties of Levi form to regularity.

Developed a generalized weighted formula for analysis.

Abstract

This thesis starts from a review on current research on the local hypoellipticity of the $\bar\partial$-Neumann problem. It presents the classical method of regularity from estimates of the energy: subelliptic as well as superlogarithmic. More recent material is included in which the regularity of the solution is obtained from the geometry of the singularities of the Levi form. The new contribution to this discussion consists in a general weighted Kohn-H\"ormander-Morrey formula twisted by a pseudodifferential operator. As an application, a new class of domains for which the $\bar\partial$-Neumann problem is locally regular is exhibited.