Decay estimates and Strichartz inequalities for a class of dispersive equations on H-type groups

TL;DR

This paper establishes decay estimates and Strichartz inequalities for dispersive equations on H-type groups, extending Euclidean and Heisenberg results to a broader non-commutative setting with sharp decay rates.

Contribution

It introduces a frequency localization approach to handle non-homogeneous functions of the sub-Laplacian on H-type groups, deriving new decay estimates and Strichartz inequalities.

Findings

Derived decay estimates for dispersive semigroups on H-type groups.

Established sharp time decay rates for specific dispersive equations.

Proved new Strichartz inequalities for fractional and higher-order Schrödinger equations.

Abstract

Let $\mathcal{L}$ be the sub-Laplacian on H-type groups and $\phi: \mathbb{R}^+ \to \mathbb{R}$ be a smooth function. The primary objective of the paper is to study the decay estimate for a class of dispersive semigroup given by $e^{it\phi(\mathcal{L})}$. Inspired by earlier work of Guo-Peng-Wang \cite{GPW2008} in the Euclidean space and Song-Yang \cite{SY2023} on the Heisenberg group, we overcome the difficulty arising from the non-homogeneousness of $\phi$ by frequency localization, which is based on the non-commutative Fourier transform on H-type groups, the properties of the Laguerre functions and Bessel functions, and the stationary phase theorem. Finally, as applications, we derive the new Strichartz inequalities for the solutions of some specific equations, such as the fractional Schr\"{o}dinger equation, the fourth-order Schr\"odinger equation, the beam equation and the Klein-Gordon equation, which corresponds to $\phi(r)=r^\alpha$, $r^2+r,\sqrt{1+r^2},\sqrt{1+r}$, respectively. Moreover, we also prove that the time decay is sharp in these cases.