Inequalities for radial solutions to the wave equation on the hyperbolic space

Abstract

Let $\mathbb H^n$ be the $n$-dimensional hyperbolic space, and $\Delta_{\mathbb H^n}$ be the Laplace--Beltrami operator on $\mathbb H^n$. Let $u=u(t,\cdot)$ be the solution to the wave equation $\partial^2_{t}u-\big(\Delta_{\mathbb H^n}+\frac{1}{4}(n-1)^2\operatorname{Id}\big)u=0$ with initial conditions $u(0,x)=f(x)$ and $\partial_tu(0,x)=0$. In this article we show that for a radial function $f$, $2n/(n-1)\leq p<\infty$, $\gamma<1/p$ and $T\geq 0$, there exists a constant $C_p$ independent of $f$ and $T$ such that $$ \left(\int_T^{T+1}\|u(t,\cdot)\|^p_p\mathrm d t\right)^{\frac1p}\leq C_{p,\gamma}\,\mathrm e^{(n-1)\left(\frac12-\frac1p\right)T}\left\|(-\Delta_{\mathbb H^n})^{\frac{n-1}2\left(\frac12-\frac1p\right)-\frac\gamma2}f\right\|_p. $$ We also show that the exponential part $\mathrm e^{(n-1)(1/2-1/p)T}$ of the bound is optimal.