Subelliptic Operators and Harmonic Analysis

Several problems and results of harmonic analysis are related to the Laplace operator, its spectral theory and functional calculus. In fact, it is often possible to replace the Laplace operator with a more general elliptic operator. However, in various contexts, particularly on noncommutative Lie groups and sub-Riemannian manifolds, the natural substitute for the Laplace operator need not be elliptic, and it may be just subelliptic. Here new and interesting phenomena arise, many basic questions are far from being completely understood, and the known results typically involve a combination of tools from different areas of mathematics, such as evolution equations, representation theory, differential geometry.