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

TL;DR

This paper establishes conditions under which the Kohn-Laplace equation on abstract CR manifolds has solutions with good regularity properties, extending previous results to more general settings and introducing the concept of plurisubharmonic CR manifolds.

Contribution

It proves the existence and regularity of the complex Green operator on abstract CR manifolds under new geometric conditions, and introduces the notion of plurisubharmonic CR manifolds.

Findings

Existence of the Green operator under CR-plurisubharmonic conditions.

Global regularity of solutions on CR manifolds with weak compactness property.

Introduction of plurisubharmonic CR manifolds generalizing classical notions.

Abstract

Let $M$ be a compact, pseudoconvex-oriented, $(2n+1)$-dimensional, abstract CR manifold of hypersurface type, $n\geq 2$. We prove the following: (i) If $M$ admits a strictly CR-plurisubharmonic function on $(0,q_0)$-forms, then the complex Green operator $G_q$ exists and is continuous on $L^2_{0,q}(M)$ for degrees $q_0\le q\le n-q_0$. In the case that $q_0=1$, we also establish continuity for $G_0$ and $G_n$. Additionally, the $\bar\partial_b$-equation on $M$ can be solved in $C^\infty(M)$. (ii) If $M$ satisfies "a weak compactness property" on $(0,q_0)$-forms, then $G_q$ is a continuous operator on $H^s_{0,q}(M)$ and is therefore globally regular on $M$ for degrees $q_0\le q\le n-q_0$; and also for the top degrees $q=0$ and $q=n$ in the case $q_0=1$. We also introduce the notion of a "plurisubharmonic CR manifold" and show that it generalizes the notion of "plurisubharmonic defining function" for a a domain in $\mathbb C^N$ and implies that $M$ satisfies the weak compactness property.