Average decay of the Fourier transform of measures with applications

2015/02/28 by Lucà, Renato, Rogers, Keith
#42B37 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1503.00105

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 Schrödinger 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 H¹(ℝ^d)× L²(ℝ^d).

Related

• Fourier Decay from L²-Flattening
• Higher-order Fourier dimension and frequency decompositions
• Sharp rate of average decay of the Fourier transform of a bounded set
• L^p Fourier asymptotics, Hardy type inequality and fractal measures
• Sharp Fourier decay estimates for measures supported on the well-approximable numbers