Strichartz estimates and Fourier restriction theorems on the Heisenberg group

TL;DR

This paper establishes Strichartz estimates for linear Schrödinger and wave equations on the Heisenberg group using Fourier restriction techniques, overcoming challenges posed by the non-dispersive nature of these equations.

Contribution

It introduces a novel approach based on Fourier restriction theorems on the Heisenberg group, extending Strichartz estimates to broader index ranges for wave equations.

Findings

Derived Strichartz estimates for Schrödinger equations on $\

Extended anisotropic Strichartz estimates for wave equations on $\

Abstract

This paper is dedicated to the proof of Strichartz estimates on the Heisenberg group $\mathbb{H}^d$ for the linear Schr\"odinger and wave equations involving the sublaplacian. The Schr\"odinger equation on $\mathbb{H}^d$ 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 $\mathbb{H}^d$, 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.