The Kohn-Laplace equation on abstract CR manifolds: Local regularity

TL;DR

This paper proves local regularity results for the complex Green operator solving the Kohn-Laplace equation on abstract CR manifolds, introducing a new potential condition called the sigma-superlogarithmic property.

Contribution

It establishes local regularity of solutions under the sigma-superlogarithmic condition and analyzes the smoothness of the integral kernel of the Green operator.

Findings

Solutions are smooth where the data is smooth and the sigma condition holds.

The integral kernel of the Green operator is also shown to be smooth under certain conditions.

Abstract

The purpose of this paper is to establish local regularity of the solution operator to the Kohn-Laplace equation, called the complex Green operator, on abstract CR manifolds of hypersurface type. For a cut-off function $\sigma$, we introduce the $\sigma$-superlogarithmic property, a potential theoretical condition on CR manifolds. We prove that if the given datum is smooth on an open set containing the support of $\sigma$ then the solution is smooth on the interior of $\{x\in M:\sigma(x)=1\}$. Furthermore, we also study the smoothness of the integral kernel of the complex Green operator.