Fractal Uncertainty and Quantitative Uniqueness for the Fourier Bessel Transform

Abstract

We establish quantitative uniqueness and a fractal uncertainty principle for the Fourier Bessel transform. In arbitrary dimension, we prove a quantitative uniqueness estimate on relatively dense sets for functions whose Fourier Bessel transforms decay according to a quasi-analytic weight. In dimension one, if $X\subset[0,1]$ and $Y\subset[a,a+h^{-1}]$ are$δ$-regular on the relevant scales, then, for $a\geq a_0h^{-1}$, \[ \operatorname{supp}\mathcal H_νf\subset Y \quad\Longrightarrow\quad \|\mathbf 1_Xf\|_{L^2_ν} \leq Ch^β\|f\|_{L^2_ν}. \] The lack of translation invariance prevents a direct application of the classical Fourier argument. We overcome this by constructing damping functions adapted to translated regular sets and combining Beurling Malliavin multipliers with large-argument Bessel asymptotics and a Bourgain Dyatlov multiscale iteration.