Decay estimates for a class of wave equations on the Heisenberg group

TL;DR

This paper establishes decay estimates for dispersive wave equations on the Heisenberg group using harmonic analysis tools, and applies these results to derive Strichartz inequalities for various fractional and higher-order Schrödinger and wave equations.

Contribution

It introduces decay estimates for a class of dispersive semigroups on the Heisenberg group and derives Strichartz inequalities for related equations, extending analysis on non-commutative groups.

Findings

Decay estimates for dispersive semigroups on H^n

Strichartz inequalities for fractional and higher-order equations

Application to specific wave and Schrödinger equations

Abstract

In this paper, we study a class of dispersive wave equations on the Heisenberg group $H^n$. Based on the group Fourier transform on $H^n$, the properties of the Laguerre functions and the stationary phase lemma, we establish the decay estimates for a class of dispersive semigroup on $H^n$ given by $e^{it\phi(\mathcal{L})}$, where $\phi: \mathbb{R}^+ \to \mathbb{R}$ is smooth, and $\mathcal{L}$ is the sub-Laplacian on $H^n$. Finally, using the duality arguments, we apply the obtained results to derive the Strichartz inequalities for the solutions of some specific equations, such as the fractional Schr\"{o}dinger equation, the fractional wave equation and the fourth-order Schr\"{o}dinger equation.