Appears in collection : 2025 - T1 - WS3 - Analysis on homogeneous spaces and operator algebras

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? If time permits, I will also talk about the heat kernel of maximally hypoelliptic differential operators. This is partly joint work with Androulidakis and Yuncken.