Fourier restriction, wave equations, and applications to spectral multipliers in commutative and non-commutative settings

• Applicant: Professor Dr. Detlef Müller
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 462433077

It is well-known by now that there are close and fundamental connections between Fourier restriction estimates (a concept that had been introduced by E. M. Stein in the seventies), estimates for linear and non-linear wave equations (and more general dissipative equations), and spectral multiplier problems for elliptic and even sub-elliptic linear differential operators. The goal of this research project is to push forward important aspects of these theories and their connections. This includes applications of the powerful polynomial partitioning method, which has been successfully applied for the first time to problems in Fourier restriction theory in recent seminal papers by L. Guth, to Fourier restriction for smooth 2-hypersurfaces with points of vanishing Gaussian curvature (the case of non-vanishing curvature has very recently been settled in two independent articles, one of them by the applicant and his co-authors). Also extensions of Guth's work on higher-dimensional paraboloids to more general higher-dimensional quadrics are aimed at. Moreover, in the non-commutative setting, we plan to establish as sharp as possible L^p - spectral multiplier estimates for subelliptic differential operators through the study of associated wave equations, including local smoothing estimates. The substantial problems arising in the study of such operators are related to the underlying, highly complex sub-Riemannian geometries. The results on groups of Heisenberg type which have been obtained so far in joint works with E.M. Stein, and A. Seeger, are based on methods which mostly avoid the usage of Fourier integral operators (FIOs), in sharp contrast to the approaches applied in the study of elliptic operators. A very recent joint article with A. Martini und S. Nicolussi-Golo, however, gives rise to some hope that suitable FIOs techniques might still be applicable to a wider extent.

• DFG Programme: Research Grants