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

• Published in: American Journal of Mathematics
• Main Authors: Frank, Rupert L., Sabin, Julien
• Format: Article in Journal/Newspaper
• Language: unknown
• Published: Johns Hopkins University Press 2017
• Subjects: Hartree; Sogge; laptev
• Online Access: https://doi.org/10.1353/ajm.2017.0041

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 Schrödinger 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 Schrödinger operators with complex potentials, and to Schatten properties of the scattering matrix. © 2017 Johns Hopkins University Press. Manuscript received May 18, 2015; revised March 17, 2016. The authors are grateful to A. Laptev, M. Lewin and A. Pushnitski for useful discussions. J. S. thanks the Mathematics Department of Caltech for the Research Stay during which this work has been done. Financial support from the U.S. National Science Foundation through grant PHY-1347399 (R. F.), from the ERC MNIQS-258023 and from the ANR "NoNAP" (ANR-10-BLAN 0101) of the French ministry of research (J. S.) are acknowledged. Submitted - 1404.2817.pdf