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

Motivated by problems in control theory concerning decay rates for the damped wave equation wₓₓ (x, t) + γ (x) wₜ (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 R^+ is μ_α-relatively dense (where dμ_α (x) x^2α+1\, dx) for α> -1/2, and supp F_α (f) R, R+1, then we show \|f\|₋ℂ_⏐ (ₑ^+) \|f\|₋ℂ_⏐ (₄), for all f L²_α (R^+), where the constants in 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.