Hypoellipticity of the $\bar\partial$-Neumann problem by means of subelliptic multipliers

TL;DR

This paper establishes local hypoellipticity of the complex Laplacian and Kohn Laplacian on pseudoconvex boundaries using subelliptic multipliers, advancing understanding of regularity in several complex variables.

Contribution

It introduces a new approach to hypoellipticity by employing subelliptic multipliers related to the boundary's Levi form and gradient, extending previous regularity results.

Findings

Proves local hypoellipticity of $ox$ and $ox_b$ under subelliptic multiplier conditions.

Connects boundary geometry with regularity properties of complex Laplacians.

Provides a framework for analyzing hypoellipticity using subelliptic multipliers.

Abstract

We prove local hypoellipticity of the complex Laplacian $\Box$ and of the Kohn Laplacian $\Box_b$ in a pseudoconvex boundary when, for a system of cut-off $\eta$, the gradient $\partial_b\eta$ and the Levi form $\partial_b\bar\partial_b\eta^2$ are subelliptic multipliers in the sense of Kohn.