Maximal operators given by Fourier multipliers with dilation of fractional dimensions

TL;DR

This paper establishes criteria for $L^p$ bounds of maximal Fourier multiplier operators with fractional dilation, covering various classes of multipliers and advancing understanding of their boundedness properties.

Contribution

It introduces a new criterion based on dilation set dimensions that ensures $L^p$ bounds for a broad class of Fourier multipliers.

Findings

Criterion covers Mikhlin-type multipliers

Includes multipliers with limited decay

Applicable to multipliers with slow decay

Abstract

In this paper, we investigate $L^p$ bounds of maximal Fourier multiplier operators with dilation of fractional dimensions. For Fourier multipliers, we suggest a criterion related to dimensions of dilation sets which guarantees $L^p$ bounds of the maximal operators for each $p$. Our criterion covers Mikhlin-type multipliers, multipliers with limited decay, and multipliers with slow decay.