A High-Frequency Uncertainty Principle for the Fourier-Bessel Transform
Benjamin Jaye, Rahul Sethi
TL;DR
The paper establishes an $R$-independent Paneah–Logvinenko–Sereda type inequality for the Fourier–Bessel transform: if $E\subset\mathbb{R}^+$ is $\mu_\alpha$-relatively dense and a function $f$ has $\operatorname{supp}\mathcal{F}_\alpha(f)\subset [R,R+1]$, then $\|f\|_{L^2_\alpha} \le C\|f\|_{L^2_\alpha(E)}$ with a constant $C$ independent of $R$ for large $R$. This improves previous $R$-dependent bounds and is achieved by combining the Bessel function asymptotics with Kovrizhkin’s multi-interval uncertainty principle, translating the problem into an oscillatory integral bound controlled by the density of $E$. The result has a direct application to decay rates of radial solutions to the damped wave equation, showing exponential or polynomial decay under radial GCC-type density assumptions. The authors also extend the approach to finite unions of unit intervals when $\alpha-\tfrac12\in\mathbb{N}$ and to radial functions with spherical harmonics, broadening the applicability to higher-dimensional radial problems and angular modes.
Abstract
Motivated by problems in control theory concerning decay rates for the damped wave equation $$w_{tt}(x,t) + γ(x) w_t(x,t) + (-Δ+ 1)^{s/2} w(x,t) = 0,$$ we consider an analogue of the classical Paneah-Logvinenko-Sereda theorem for the Fourier Bessel transform. In particular, if $E \subset \mathbb{R}^+$ is $μ_α$-relatively dense (where $dμ_α(x) \approx x^{2α+1}\, dx$) for $α> -1/2$, and $\operatorname{supp} \mathcal{F}_α(f) \subset [R,R+1]$, then we show $$\|f\|_{L^2_α(\mathbb{R}^+)} \lesssim \|f\|_{L^2_α(E)},$$ for all $f\in L^2_α(\mathbb{R}^+)$, 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.