Strichartz estimates and Fourier restriction theorems on the Heisenberg group

Hajer Bahouri (1, 2) , Davide Barilari (3) , Isabelle Gallagher (4)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
4 DMA - Département de Mathématiques et Applications - ENS Paris

This paper is dedicated to the proof of Strichartz estimates on the Heisen-berg group H^d for the linear Schrödinger and wave equations involving the sublaplacian. The Schrödinger equation on H^d is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available. Our approach, inspired by the Fourier transform restriction method initiated by Tomas and Stein, is based on Fourier restriction theorems on H^d, using the non-commutative Fourier transform on the Heisenberg group. It enables us to obtain also an anisotropic Strichartz estimate for the wave equation, for a larger range of indices than was previously known.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse fonctionnelle [math.FA]

Hajer Bahouri, Davide Barilari, Isabelle Gallagher. Strichartz estimates and Fourier restriction theorems on the Heisenberg group. Journal of Fourier Analysis and Applications, 2021, 27, pp.21. ⟨10.1007/s00041-021-09822-5⟩. ⟨hal-02357244v2⟩