Average decay of the Fourier transform of measures with applications

TL;DR

This paper improves bounds on the decay rates of Fourier transforms of fractal measures, with applications to maximal estimates for Schrödinger and wave equations, refining convergence and divergence properties of solutions.

Contribution

It provides new bounds on Fourier decay of fractal measures and applies these to enhance understanding of PDE solutions' behavior.

Findings

Improved upper and lower bounds on Fourier decay rates.

Enhanced maximal estimates for Schrödinger and wave equations.

Refined convergence results for solutions as time approaches zero.

Abstract

We consider spherical averages of the Fourier transform of fractal measures and improve both the upper and lower bounds on the rate of decay. Maximal estimates with respect to fractal measures are deduced for the Schr\"odinger and wave equations. This refines the almost everywhere convergence of the solution to its initial datum as time tends to zero. A consequence is that the solution to the wave equation cannot diverge on a $(d-1)$-dimensional manifold if the data belongs to the energy space $\dot{H}^1(\mathbb{R}^d)\times L^2(\mathbb{R}^d)$.