Growth Estimates for Solutions to the Wave Equation on Damek--Ricci Spaces

TL;DR

This work proves sharp in-regularity $L^p$ bounds for solutions to the wave equation $\partial_t^2 u-\mathcal{L}u=0$ on Damek--Ricci spaces, where $\mathcal{L}$ is the left-invariant distinguished Laplacian. The authors derive time-growth estimates $\|u(t,\cdot)\|_{L^p(S)} \lesssim_p (1+|t|)^{2|\frac{1}{p}-\frac{1}{2}|} \| (\mathrm{Id}+\mathcal{L})^{\frac{\alpha_0}{2}} f \|_{L^p} + (1+|t|) \| (\mathrm{Id}+\mathcal{L})^{\frac{\alpha_1}{2}} g \|_{L^p}$ for $1<p<\infty$, with endpoint regularities $\alpha_0=n|\frac{1}{p}-\frac{1}{2}|$ and $\alpha_1=n|\frac{1}{p}-\frac{1}{2}|-1$, proving the conjecture of Müller–Thiele–Vallarino in full generality. The method combines spherical harmonic analysis on Damek--Ricci spaces, local Hardy space techniques, and precise asymptotics of spherical functions via Harish-Chandra expansions, yielding endpoint $L^1$ bounds for the wave propagator and subsequent interpolation to $L^p$ bounds. The results highlight that long-time growth is governed by the dimension at infinity rather than the topological dimension, and they extend endpoint-Wang–Yan type results from the ax+b group to the full Damek–Ricci class. Overall, the paper advances the understanding of dispersive properties of waves on noncompact, non-symmetric spaces and settles a key regularity question for this geometric setting.

Abstract

Let $\mathcal{L}$ be the left-invariant distinguished Laplacian, and let $\mathrm{d}ρ$ denote the right Haar measure on a Damek--Ricci space $S$. Let $u(t,x)$ denote the solution to the wave equation $\partial_t^2 u-\mathcal{L} u=0$ with initial data $(u,\partial_t u)|_{t=0}=(f,g)$. In this paper, we establish the sharp-in-regularity $L^p$ bounds \begin{align*} \|u(t,\cdot)\|_{L^p(S ,\mathrm{d}ρ)} \lesssim_p (1+|t|)^{2|\frac{1}{p}-\frac{1}{2}|}\|(\mathrm{Id}+\mathcal{L})^{\frac{α_0}{2}}\!f\|_{L^p(S ,\mathrm{d}ρ)}+(1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\frac{α_1}{2}}\!g\|_{L^p(S,\mathrm{d}ρ)} \end{align*} for all $t\in\mathbb{R}^*$ and $1<p<\infty$, where the exponents $α_0 = n\left|1/p-1/2\right|$ and $α_1 = n\left|1/p-1/2\right| -1$ attain their critical values. This result settles, in full generality, the conjecture raised by Müller, Thiele, and Vallarino.

Theorems & Definitions (23)

• Theorem 1.1: Müller&Vallarino MuVa
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof