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 $\partial_t^2 u-\mathcal{L} u=0$ 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}\rho)} \lesssim_p (1+|t|)^{2|\frac{1}{p}-\frac{1}{2}|}\|(\mathrm{Id}+\mathcal{L})^{\frac{\alpha_0}{2}}\!f\|_{L^p(S ,\mathrm{d}\rho)}+(1+|t|)\,\|(\mathrm{Id}+\mathcal{L})^{\frac{\alpha_1}{2}}\!g\|_{L^p(S,\mathrm{d}\rho)} \end{align*} for all $t\in\mathbb{R}^*$ and $1<p<\infty$, where the exponents $\alpha_0 = n\left|1/p-1/2\right|$ and $\alpha_1 = n\left|1/p-1/2\right| -1$ attain their critical values. This result settles, in full generality, the conjecture raised by M\"{u}ller, Thiele, and Vallarino.