Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

On maximally hypoelliptic differential operators

The class of maximally hypoelliptic differential operators is a large class of differential operators which contains elliptic operators as well as Hörmander's sum of squares. I will present our work where we define a principal symbol generalising the classical principal symbol for elliptic operators which should be thought of as the analogue of the principal symbol in sub-Riemannian geometry. Our main theorem is that maximal hypoellipticity is equivalent to invertibility of our principal symbol, thus generalising the main regularity theorem for elliptic operators and confirming a conjecture of Helffer and Nourrigat. While defining our principal symbol, we will answer the question: What is the tangent space in sub-Riemman geometry in the sense of Gromov?