Maximal inequalities and the decay of Fourier transforms of measures

TL;DR

This paper establishes a connection between maximal inequalities for Schrödinger and wave equations over fractals and the decay rates of Fourier transforms of fractal measures, providing new proofs of known conditions without ergodic or number theory methods.

Contribution

It demonstrates the equivalence between Schrödinger maximal inequalities and Fourier decay rates, introducing a novel proof of Bourgain's necessary condition without relying on ergodic or number theoretic techniques.

Findings

Schrödinger maximal inequalities are equivalent to Fourier decay rates of fractal measures.

A new proof of Bourgain's necessary condition is provided, avoiding ergodic theory.

The connection between wave equation averages and Fourier decay is established.

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.