Average decay of the Fourier transform of measures with applications

Abstract

We consider spherical averages of the Fourier transform of fractal measures and improve the lower bound on the rate of decay by taking advantage of multilinear estimates. 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 ${\dot{H}}^1({\mathbb{R}}^d)\times L^2({\mathbb{R}}^d)$.