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

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. Slavíková, 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.