A New Hypoelliptic Operator on Almost CR Manifolds

TL;DR

This paper introduces a new hypoelliptic operator on almost CR manifolds derived from the Kohn-Rossi complex, analyzing its properties, hypoellipticity conditions, and Fredholm index, with examples illustrating its limitations.

Contribution

It constructs a novel hypoelliptic operator on almost CR manifolds and examines its hypoelliptic and Fredholm properties, expanding understanding of differential operators in this geometric setting.

Findings

Operator acts on all (p,q)-forms but is hypoelliptic only on specific forms.

Hypoellipticity depends on a finite type condition for certain forms.

Fredholm index of the operator is zero.

Abstract

The aim of this paper is to present the construction, out of the Kohn-Rossi complex, of a new hypoelliptic operator $Q_{L}$ on almost CR manifolds equipped with a real structure. The operator acts on all (p,q)-forms, but when restricted to (p,0)-forms and (p,n)-forms it is a sum of squares up to sign factor and lower order terms. Therefore, only a finite type condition condition is needed to have hypoellipticity on those forms. However, outside these forms $Q_{L}$ may fail to be hypoelliptic, as it is shown in the example of the Heisenberg group. We also look at the Fredholm properties of $Q_{L}$ and show that the corresponding Fredholm index is zero.