We prove several relations between spectrum and dynamics including wave trace expansion, sharp/improved Weyl laws, propagation of singularities and quantum ergodicity for the sub-Riemannian (sR) Laplacian in the four dimensional quasi-contact case. A key role in all results is played by the presence of abnormal geodesics and represents the first such appearance of these in sub-Riemannian spectral geometry.
In this talk I will describe joint work with D. Alonso-Orán and A. Córdoba where we extend a result, proved independently by Kiselev-Nazarov-Volberg and Caffarelli-Vasseur, for the critical dissipative SQG equation on a two dimensional sphere. The proof relies on De Giorgi technique following Caffarelli-Vasseur intermingled with a nonlinear maximum principle that appeared later in the approach of Constantin-Vicol. The final result can be paraphrased as follows: if the data is sufficiently smooth initially then it is smooth for all times.