| Résume | Résumé :
The theory of elliptic PDEs is distinguished by the availability of sharp results for remarkably general classes of operators, and by its deep connections with harmonic analysis, the Fourier transform, Riemannian geometry, and the calculus of pseudodifferential operators.
Beginning with the work of Hörmander, Kohn, Folland, Stein, and Rothschild in the 1960s and 1970s, a broad generalization of ellipticity emerged, now known as maximal subellipticity or maximal hypoellipticity. In this setting, Riemannian geometry is replaced by Carnot–Carathéodory geometry, and the role of pseudodifferential operators can be taken over by algebras of singular integral operators.
In this talk I will discuss results on general maximally subelliptic PDEs from this point of view. The focus will be on sharp regularity theorems for linear and fully nonlinear equations in adapted function spaces, including adapted Sobolev, Zygmund–Hölder, Besov, and Triebel–Lizorkin spaces, and on the central role of algebras of singular integral operators. |
