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 ${\partial }_t^{2}u-({\Delta }_{{\mathbb{H}}^n}+\frac{1}{4}(n-1{)}^2\mathrm{Id})u=0$ with initial conditions $u(0,x)=f(x)$ and ${\partial }_{t}u(0,x)=0$. In this article we show that for a radial function $f$, $2n/(n-1)\le p<\infty$, $\gamma <1/p$ and $T\ge 0$, there exists a constant $C_p$ independent of $f$ and $T$ such that $${\left({\int }_T^{T+1}\Vert u(t,\cdot ){\Vert }_p^p\mathrm{d}t\right)}^{\frac{1}{p}}\le C_{p,\gamma }{\mathrm{e}}^{(n-1)\left(\frac{1}{2}-\frac{1}{p}\right)T}{\Vert (-{\Delta }_{{\mathbb{H}}^n}{)}^{\frac{n-1}{2}\left(\frac{1}{2}-\frac{1}{p}\right)-\frac{\gamma }{2}}f\Vert }_p.$$ We also show that the exponential part ${\mathrm{e}}^{(n-1)(1/2-1/p)T}$ of the bound is optimal.