On Conic Fourier Multipliers

TL;DR

This paper establishes a weighted inequality linking conic Fourier multipliers to lacunary directional maximal operators, leading to boundedness results for these multipliers on L^p spaces in three dimensions.

Contribution

It introduces a novel inequality that connects conic Fourier multipliers with lacunary maximal operators, enabling new boundedness results.

Findings

Proves a weighted inequality controlling conic Fourier multipliers.

Shows boundedness of these multipliers on L^p(R^3) for 1<p< 0.

Links maximal operator bounds to Fourier multiplier boundedness.

Abstract

We prove a weighted inequality which controls conic Fourier multiplier operators in terms of lacunary directional maximal operators. By bounding the maximal operators, this enables us to conclude that the multiplier operators are bounded on $L^p(\mathbb{R}^3)$ with $1<p<\infty$.