Maximal operators associated with Fourier multipliers and applications

TL;DR

This paper establishes a criterion for the boundedness of maximal operators linked to Fourier multipliers on L^p spaces, with applications to fractional wave equations and surface averages.

Contribution

It introduces a new criterion for maximal operator boundedness associated with Fourier multipliers, including non-radial multipliers with limited decay, using square function estimates and bilinear interpolation.

Findings

Boundedness of maximal operators for Mikhlin-type multipliers.

Convergence results for fractional half-wave equations.

L^p boundedness for maximal operators in various contexts.

Abstract

In this paper, we introduce a criterion for maximal operators associated with Fourier multipliers to be bounded on $L^p(\mathbb{R}^d)$. Noteworthy examples satisfying the criterion are multipliers of the Mikhlin type or limited decay which are not necessarily radial. To do so, we make use of modified square function estimates and bilinear interpolation. In result, we obtain convergence results for fractional half-wave equations and surface averages as well as the $L^p$ boundedness for the maximal operators.