Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{ tt}(x,t) + \gamma(x) w_t(x,t) + (-\Delta + 1)^{ s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if $E \subset \mathbb{ R}^+$ is $\mu_\alpha$-relatively dense (where $d\mu_\alpha(x) \approx x^{ 2\alpha+1}\, dx$) for $\alpha > -1/2$, and $\operatorname{ supp} \mathcal{ F}_\alpha(f) \subset [R,R+1]$, then we show $$\|f\|_{ L^2_\alpha(\mathbb{ R}^+)} \lesssim \|f\|_{ L^2_\alpha(E)},$$ for all $f\in L^2_\alpha(\mathbb{ R}^+)$, where the constants in $\lesssim$ do not depend on $R > 0$. Previous results on PLS theorems for the Fourier-Bessel transform by Ghobber and Jaming (2012) provide bounds that depend on $R$. In contrast, our techniques yield bounds that are independent of $R$, offering a new perspective on such results. This result is applied to derive decay rates of radial solutions of the damped wave equation.
Categories: math.CA