On the Fourier asymptotics of absolutely continuous measures with power-law singularities

TL;DR

This paper establishes precise estimates for the long-term average behavior of Fourier transforms of absolutely continuous measures with power-law singularities, with applications to spectral measures of perturbed Laplacians.

Contribution

It provides sharp asymptotic estimates for Fourier transforms of measures with power-law singularities, extending understanding of their spectral properties.

Findings

Sharp bounds on Fourier transform averages for singular measures

Application to spectral analysis of finite-rank Laplacian perturbations

Enhanced understanding of measure behavior near singularities

Abstract

We prove sharp estimates on the time-average behavior of the squared absolute value of the Fourier transform of some absolutely continuous measures that may have power-law singularities, in the sense that their Radon-Nikodym derivatives diverge with a power-law order. We also discuss an application to spectral measures of finite-rank perturbations of the discrete Laplacian.