Maximal inequalities and the decay of Fourier transforms of measures

Abstract

It is shown that Schr\"odinger maximal inequalities over fractals are equivalent to the $L^2$ decay rates of Fourier transforms of fractal measures over the paraboloid. A similar connection is shown between the wave equation and cone averages. One implication is well-known and follows from the Kolmogorov-Seliverstov-Plessner method, but the other implication is nontrivial and relies on a variant of the Marstrand projection theorem. The idea of the proof is to insert an extra averaging parameter into a proof of Luc\`a and Rogers, which used a quantitative ergodic lemma instead of the Marstrand projection theorem. Luc\`a and Rogers gave a second proof of Bourgain's necessary condition $s\geq \frac{n}{2(n+1)}$ for Schr\"odinger solutions in $\mathbb{R}^{n+1}$ to converge pointwise a.e. back to the initial data as time tends to zero. One application of the main theorem in this article is a proof of Bourgain's necessary condition which does not use ergodic theory or number theory.