# Decay estimates for the linear damped wave equation on the Heisenberg group

## TL;DR

This paper establishes decay estimates for solutions to the linear damped wave equation on the Heisenberg group, utilizing Fourier analysis and Hermite functions to improve understanding of solution behavior over time.

## Contribution

It provides new $L^2$ decay estimates for the damped wave equation on the Heisenberg group, including enhancements under additional $L^1$ regularity of initial data.

## Key findings

- Derived $L^2$ decay estimates for solutions and derivatives.
- Improved decay estimates with $L^1$ regularity of initial data.
- Utilized group Fourier transform and Hermite functions in analysis.

## Abstract

This paper is devoted to the derivation of $L^2$ - $L^2$ decay estimates for the solution of the homogeneous linear damped wave equation on the Heisenberg group $\mathbf{H}_n$, for its time derivative and for its horizontal gradient. Moreover, we consider the improvement of these estimates when further $L^1(\mathbf{H}_n)$ regularity is required for the Cauchy data. Our approach will rely strongly on the group Fourier transform of $\mathbf{H}_n$ and on the properties of the Hermite functions that form a maximal orthonormal system for $L^2(\mathbb{R}^n)$ of eigenfunctions of the harmonic oscillator.