Multi-parameter maximal Fourier restriction

TL;DR

This paper enhances Fourier restriction estimates to multi-parameter maximal estimates, explores Lebesgue point properties with ellipsoids, proves a multi-parameter Menshov--Paley--Zygmund theorem, and strengthens Strichartz estimates for dispersive PDEs.

Contribution

It introduces a multi-parameter maximal Fourier restriction estimate and extends classical theorems to higher dimensions and more general settings.

Findings

Strengthened Fourier restriction to multi-parameter maximal estimates.

Established a multi-parameter Lebesgue point property with ellipsoids.

Proved a multi-parameter Menshov--Paley--Zygmund theorem for  Fourier transforms.

Abstract

The main result of this note is the strengthening of a quite arbitrary a priori Fourier restriction estimate to a multi-parameter maximal estimate of the same type. This allows us to discuss a certain multi-parameter Lebesgue point property of Fourier transforms, which replaces Euclidean balls by ellipsoids. Along the lines of the same proof, we also establish a $d$-parameter Menshov--Paley--Zygmund-type theorem for the Fourier transform on $\mathbb{R}^d$. Such a result is interesting for $d\geq2$ because, in a sharp contrast with the one-dimensional case, the corresponding endpoint $L^2$ estimate (i.e., a Carleson-type theorem) is known to fail since the work of C. Fefferman in 1970. Finally, we show that a Strichartz estimate for a given homogeneous constant-coefficient linear dispersive PDE can sometimes be strengthened to a certain pseudo-differential version.