Average decay of the Fourier transform of measures with applications

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 spaceḢ 1 (R d ) × L 2 (R d ).

Pointwise Convergence of the Schrödinger Flow

“…As mentioned in Section 2, it is standard that this estimate implies pointwise convergence e it∆ f (x) → f (x) as t → 0 for almost every x ∈ T d . The problem of identifying the minimal regularity s for which (27) holds is still open. The following result has been proved in [32] when d = 1 and in [47] when d ≥ 2.…”

Pointwise Convergence of Schrödinger Solutions and Multilinear Refined Strichartz Estimates

“…We say that µ is α-dimensional if it is a probability measure supported in the unit ball B n (0, 1) and satisfies that (1.4) µ(B(x, r)) ≤ C µ r α , ∀r > 0, ∀x ∈ R n . Lemma 1.5 (Lucà-Rogers, Lemma 7.1 in [14]). Let α > α 0 ≥ n − 2s and suppose that sup…”

A note on pointwise convergence for the Schrödinger equation

“…This is a consequence of the Dahlberg-Kenig example combined with [1], where it was proven that α n (s) ≤ n − 2s in the range n/4 ≤ s ≤ n/2. For the best known upper bounds with lower regularity, see [14,Theorem 1.2].…”