A New Hypoelliptic Operator on Almost CR Manifolds

Details A New Hypoelliptic Operator on Almost CR Manifolds A New Hypoelliptic Operator on Almost CR Manifolds

2008-08-27

arxiv.org/abs/0808.3796v3

Rev. Mat. Iberoamericana 27 (2011), no. 2, 393-414

Mathematics

Complex Variables

v3: final version (16 pages)

Scientific paper

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.

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