The Interplay of Shifted Square and Maximal Function Estimates in the Context of Multilinear Fourier Multipliers

TL;DR

This paper extends boundedness criteria for multilinear Fourier multipliers using shifted square and maximal functions, addressing sharpness issues and applying results to singular integrals with rough kernels in Orlicz spaces.

Contribution

It generalizes a recent bilinear theorem to multilinear operators and refines the use of shifted functions to improve sharpness in boundedness results.

Findings

Established boundedness of singular integrals with rough kernels in Orlicz spaces

Removed shift from square functions, simplifying the analysis

Extended bilinear results to multilinear Fourier multipliers

Abstract

Following their appearance in 2014, so-called shifted square and maximal functions have seen an eruption of use in the study of singular integral operators. In this paper, we will generalize a recent theorem of G. Dosidis, B. Park, and L. Slav\'ikov\'a, which gave a sharp boundedness criterion for certain bilinear Fourier multipliers, to the general multilinear setting. In so doing, we will witness how the combined use of shifted square and maximal functions causes a loss of sharpness; we, then, repair this through a trick, which allows us to remove the shift from the square functions, placing it purely on the maximal functions. As an application to our main theorem, we establish the boundedness of certain singular integrals with rough homogeneous kernels lying in the Orlicz space $L(\log L)^\alpha$ when restricted to the unit sphere. This represents an edge case to what was previously known in the literature.