Optimal control of singular Fourier multipliers by maximal operators

TL;DR

This paper establishes a method to control a wide class of singular Fourier multipliers using geometrically-defined maximal operators, with applications to oscillatory integrals, dispersive PDE, and higher-dimensional analysis.

Contribution

It introduces a novel approach to controlling singular Fourier multipliers via maximal operators involving fractional averages and extends the theory to higher dimensions.

Findings

Controlled broad class of singular Fourier multipliers.

Developed new maximal operators involving fractional averages.

Applied results to oscillatory integrals and dispersive PDEs.

Abstract

We control a broad class of singular (or "rough") Fourier multipliers by geometrically-defined maximal operators via general weighted $L^2(\mathbb{R})$ norm inequalities. The multipliers involved are related to those of Coifman--Rubio de Francia--Semmes, satisfying certain weak Marcinkiewicz-type conditions that permit highly oscillatory factors of the form $e^{i|\xi|^\alpha}$ for both $\alpha$ positive and negative. The maximal functions that arise are of some independent interest, involving fractional averages associated with tangential approach regions (related to those of Nagel and Stein), and more novel "improper fractional averages" associated with "escape" regions. Some applications are given to the theory of $L^p-L^q$ multipliers, oscillatory integrals and dispersive PDE, along with natural extensions to higher dimensions.