A new hypoelliptic operator on almost CR manifolds

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 $\mathbb{H}^{5}$.