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

Abstract

Let $\mathcal{L}$ be the positive definite left-invariant distinguished Laplacian, and let $\mathrm{d}\rho$ 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 $t\in {\mathbb{R}}^{\ast }$ and $1<p<\infty$, where the exponents ${\alpha }_0=(n-1)\left|1/p-1/2\right|$ and ${\alpha }_1=(n-1)\left|1/p-1/2\right|-1$ attain their critical values. This result settles, in full generality, the conjecture raised by Müller, Thiele, and Vallarino.