A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform

Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t)+\gamma (x)w_t(x,t)+(-\mathrm{\Delta }+1{)}^{s\mathrm{/}2}w(x,t)=0,$$ 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 $\alpha >-1\mathrm{/}2$, and $\mathrm{supp}{\mathcal{F}}_{\alpha }(f)\subset [R,R+1]$, then we show $$\mathrm{\parallel }f{\mathrm{\parallel }}_{L_{\alpha }^2({\mathbb{R}}^{+})}\lesssim \mathrm{\parallel }f{\mathrm{\parallel }}_{L_{\alpha }^2(E)},$$ for all $f\in L_{\alpha }^2({\mathbb{R}}^{+})$, 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.