Estimates for maximal Fourier multiplier operators on $\Bbb R^2$ via square functions

TL;DR

This paper establishes sharp $L^p$ bounds for maximal Fourier multiplier operators on $R^2$ using Littlewood-Paley square functions, extending classical results and providing new insights into Bochner-Riesz type multipliers.

Contribution

It generalizes Carbery's 1983 result by deriving sharp estimates for maximal Fourier multipliers on $R^2$ through advanced square function techniques.

Findings

Proved sharp $L^p$ bounds for certain maximal Fourier multipliers.

Extended classical results to a broader class of Bochner-Riesz type operators.

Demonstrated the effectiveness of Littlewood-Paley square functions in this context.

Abstract

We consider certain Littlewood-Paley square functions on $\Bbb R^2$ and prove sharp estimates for them, from which we can deduce $L^p$ boundedness of maximal functions defined by Fourier multipliers of Bochner-Riesz type on $\Bbb R^2$. This is a generalization of a result due to A. Carbery 1983.