Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates

Understand

We generalize the theorems of Stein--Tomas and Strichartz about surface restrictions of Fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions.

• We deduce the corresponding Strichartz bounds for solutions to Schr\"odinger equations up to the endpoint, thereby solving an open problem of Frank, Lewin, Lieb and Seiringer.
• We also prove uniform Sobolev estimates in Schatten spaces, extending the results of Kenig, Ruiz, and Sogge.
• We finally provide applications of these results to a Limiting Absorption Principle in Schatten spaces, to the well-posedness of the Hartree equation in Schatten spaces, to Lieb--Thirring bounds for eigenvalues of Schr\"odinger operators with complex potentials, and to Schatten properties of the scattering matrix.