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

TL;DR

This paper extends restriction theorems and Strichartz inequalities to systems of orthonormal functions, providing new bounds and applications in quantum mechanics, PDEs, and spectral theory, including solving an open problem and establishing uniform Sobolev estimates.

Contribution

It generalizes classical Fourier restriction and Strichartz results to orthonormal systems with optimal dependence, and extends uniform Sobolev estimates to Schatten spaces, with multiple applications.

Findings

Established optimal bounds for orthonormal systems in restriction theorems.

Solved an open problem on Strichartz bounds for Schrödinger solutions.

Extended uniform Sobolev estimates to Schatten spaces with applications.

Abstract

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.