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 $S_{c{A}_0,B_0}=P_{c{A}_0}{Q}_{B_0}{P}_{c{A}_0}$. We prove that, for every $c>0$ and $0<\epsilon <1/2$, ${\Lambda }_{\epsilon }\le C(A_0,B_0)\tilde{L}[1+{\ln }_{+}(ca/\tilde{L})]$, where $\tilde{L}=\ln (1/[\epsilon (1-\epsilon )])$, 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 $\epsilon$ and becomes $O(\tilde{L})$ when $\tilde{L}\ge ca$. On the range $c\ge 2$ and ${\alpha }^{-c}<\epsilon <1/2$ 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 $\epsilon$. 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 $1/[2\pi (s+r)]$, 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 ${\ln }_{+}(ca/\tilde{L})$ dyadic scales, yielding the self-improving logarithm. It also identifies tangential phase dependence as an obstruction to this factorization in nonproduct higher-dimensional geometry.

15 pages. Revised title and discussion sharpen the comparison with the Kulikov-Dam Larsen theorem; all mathematical statements and proofs are unchanged