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Maximal inequalities and the decay of Fourier transforms of measures

Terence L. J. Harris

Abstract

It is shown that Schrödinger 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 Lucà and Rogers, which used a quantitative ergodic lemma instead of the Marstrand projection theorem. Lucà and Rogers gave a second proof of Bourgain's necessary condition $s\geq \frac{n}{2(n+1)}$ for Schrödinger 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.

Maximal inequalities and the decay of Fourier transforms of measures

Abstract

It is shown that Schrödinger maximal inequalities over fractals are equivalent to the 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 Lucà and Rogers, which used a quantitative ergodic lemma instead of the Marstrand projection theorem. Lucà and Rogers gave a second proof of Bourgain's necessary condition for Schrödinger solutions in 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.
Paper Structure (3 sections, 2 theorems, 67 equations)

This paper contains 3 sections, 2 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Proof for the Schrödinger equation
  3. Proof for the wave equation

Key Result

Theorem 1.1

Let $n \geq 1$. Let $\alpha>0$ and let $s \geq 0$. There is a constant $K$ depending only on $s$ and $n$ such that following holds.

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1
  • proof : Proof of \ref{['localversion']} of Theorem \ref{['equivalence']}
  • Remark 2
  • Remark 3
  • proof : Re-proof of Bourgain's necessary condition $s \geq \frac{n}{2(n+1)}$
  • Theorem 3.1
  • proof
  • Remark 4