Compactness of the Complex Green Operator on CR-Manifolds of Hypersurface Type

TL;DR

This paper establishes conditions under which the complex Green operator on CR manifolds of hypersurface type is compact, extending potential theoretic criteria and employing microlocal analysis techniques.

Contribution

It introduces the (CR-P_q) condition, generalizing Catlin's property to CR manifolds of arbitrary codimension, and proves compactness of the Green operator under these conditions.

Findings

Compactness of the Green operator is achieved under (CR-P_q) and (CR-P_{n-1-q}) conditions.

The (CR-P_q) condition generalizes Catlin's property (P_q) to higher codimension.

Microlocal norms using CR-plurisubharmonic functions effectively control tangent bundle directions.

Abstract

The purpose of this article is to study compactness of the complex Green operator on CR manifolds of hypersurface type. We introduce (CR-P_q), a potential theoretic condition on $(0,q)$-forms that generalizes Catlin's property (P_q) to CR manifolds of arbitrary codimension. We prove that if an embedded CR-manifold of hypersurface type satisfies (CR-P_q) and (CR-P_{n-1-q}) and is of real dimension at least five, then the complex Green operator is a compact operator on the Sobolev spaces $H^s_{0,q}(M)$, if $1\leq q \leq n-2$ and $s\geq 0$. We use CR-plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle.