Weighted decoupling with lower-dimensional frequency localization

TL;DR

This paper establishes new weighted decoupling estimates for functions with Fourier support near the sphere or paraboloid, incorporating lower-dimensional frequency localization, with applications to fractal restriction and Falconer distance problems.

Contribution

It introduces weighted decoupling estimates that incorporate lower-dimensional frequency localization, improving bounds for fractal restriction and Falconer distance set problems.

Findings

Recovered fractal $L^2$ restriction estimate with sharper weight dependence.

Derived weighted refined decoupling estimates for Falconer distance problem.

Improved earlier results under stronger $ ext{α}$-dimensional weight assumptions.

Abstract

We prove weighted $L^2$ and refined $L^p$ decoupling estimates for functions whose Fourier transforms are supported in a small neighborhood of the unit sphere or the truncated paraboloid with an additional lower-dimensional frequency localization property. As a special case, we recover the fractal $L^2$ restriction estimate of Du and Zhang, with a sharper dependence on the density of the weight. We also derive weighted refined decoupling estimates related to the Falconer distance set problem, improving earlier results under the stronger assumption that the underlying weight is $\alpha$-dimensional at every scale.