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

Abstract Motivated by problems in control theory concerning decay rates for the damped wave equation $$\begin{aligned} w_{tt}(x,t) + \gamma (x) w_t(x,t) + (-\Delta + 1)^{s/2} w(x,t) = 0, \end{aligned}$$ w tt ( x , t ) + γ ( x ) w t ( x , t ) + ( - Δ + 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 $$E \subset \mathbb {R}^+$$ E ⊂ R + is $$\mu _\alpha $$ μ α -relatively dense (where $$d\mu _\alpha (x) \approx x^{2\alpha +1}\, dx$$ d μ α ( x ) ≈ x 2 α + 1 d x ) for $$\alpha > -1/2$$ α > - 1 / 2 , and $$\operatorname {supp} \mathcal {F}_\alpha (f) \subset [R,R+1]$$ supp F α ( f ) ⊂ [ R , R + 1 ] , then we show $$\begin{aligned} \Vert f\Vert _{L^2_\alpha (\mathbb {R}^+)} \lesssim \Vert f\Vert _{L^2_\alpha (E)}, \end{aligned}$$