Scale-free Hankel factorization and explicit one-dimensional plunge bounds for time-frequency localization operators

Abstract

The plunge region records the spectral transition of a time-frequency localization operator. Let $A_0,B_0\subset\mathbb{R}$ be bounded measurable sets of positive measure with finite boundaries, and let $S_{cA_0,B_0}=P_{cA_0}Q_{B_0}P_{cA_0}$. We prove that, for every $c>0$ and $0<\varepsilon<1/2$, $Λ_\varepsilon\le C(A_0,B_0)\widetilde L[1+\ln_+(ca/\widetilde L)]$, where $\widetilde L=\ln(1/[\varepsilon(1-\varepsilon)])$, and $\ln_+x=\max(0,\ln x)$. If $M$ and $K$ count the interval components and $a$ and $b$ are their maximal lengths, one may take $C(A_0,B_0)=63MK(1+b)$. The estimate is uniform in $c$ and $\varepsilon$ and becomes $O(\widetilde L)$ when $\widetilde L\ge ca$. On the range $c\ge2$ and $α^{-c}<\varepsilon<1/2$ covered by the parallelepiped theorem of Kulikov and Dam Larsen (2026), this recovers its $d=1$ order; their complementary very-small-threshold result is sharper. We give an independent direct proof with explicit geometry dependence and a single formulation for all $c$ and $\varepsilon$. It acts on the off-diagonal factor of the localization operator: an exact one-dimensional oscillation factorization reduces each far-field piece to two modulated copies of the scale-free Hankel kernel $1/[2π(s+r)]$, Bernstein-ellipse approximation gives uniform singular-value decay, and a Taylor-rank estimate controls the boundary layer. Choosing this width at the spectral depth absorbs finer scales and leaves only $\ln_+(ca/\widetilde L)$ dyadic scales, yielding the self-improving logarithm. It also identifies tangential phase dependence as an obstruction to this factorization in nonproduct higher-dimensional geometry.