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

Let $\mathcal{L}$ be the 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 ${\mathrm{\partial }}_t^{2}u-\mathcal{L}u=0$ with initial data $(u,{\mathrm{\partial }}_{t}u){\mathrm{\mid }}_{t=0}=(f,g)$. 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}|}|($\mathrm{I}\mathrm{d}$+$\mathcal{L}$)^{\frac{α_0}{2}}!f|{L^p(S ,$\mathrm{d}$ρ)}+(1+|t|),|($\mathrm{I}\mathrm{d}$+$\mathcal{L}$)^{\frac{α_1}{2}}!g|_{L^p(S,$\mathrm{d}$ρ)} \end{align*} for all $t\in {\mathbb{R}}^{\ast }$ and $1<p<\mathrm{\infty }$, where the exponents ${\alpha }_0=n\mid 1\mathrm{/}p-1\mathrm{/}2\mid$ and ${\alpha }_1=n\mid 1\mathrm{/}p-1\mathrm{/}2\mid -1$ attain their critical values. This result settles, in full generality, the conjecture raised by Müller, Thiele, and Vallarino.