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

Andrew Haar

TL;DR

This work extends shifted square and maximal function techniques to multilinear Fourier multipliers, delivering a sharp Lp-boundedness criterion tied to a kernel moment D_λ(K) and an optimal logarithmic exponent λ. A central innovation is a change-of-variables trick that relocates shifts from square functions to maximal functions, restoring sharpness beyond the bilinear case. The results apply to rough homogeneous kernels on spheres, residing in Orlicz spaces L(log L)^α, thereby pushing the boundary of known kernel-based multiplier theorems. The combination of decomposition, shifted operator control, and multilinear interpolation yields a comprehensive sharp theorem with a matching counterexample.

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. 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 $L(\log L)^α$ when restricted to the unit sphere. This represents an edge case to what was previously known in the literature.

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

TL;DR

This work extends shifted square and maximal function techniques to multilinear Fourier multipliers, delivering a sharp Lp-boundedness criterion tied to a kernel moment D_λ(K) and an optimal logarithmic exponent λ. A central innovation is a change-of-variables trick that relocates shifts from square functions to maximal functions, restoring sharpness beyond the bilinear case. The results apply to rough homogeneous kernels on spheres, residing in Orlicz spaces L(log L)^α, thereby pushing the boundary of known kernel-based multiplier theorems. The combination of decomposition, shifted operator control, and multilinear interpolation yields a comprehensive sharp theorem with a matching counterexample.

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. 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 when restricted to the unit sphere. This represents an edge case to what was previously known in the literature.

Paper Structure

This paper contains 13 sections, 9 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminary Material
  3. Shifted Operators
  4. The Peetre Maximal Function
  5. A Trick with Change of Variables
  6. Proof of the Positive Result
  7. Reducing the Proof to One Estimate
  8. The Vertex Points
  9. The Non-Edge Cases
  10. Some Edge Cases
  11. Interpolating for the Remaining Edge Cases
  12. Proof of Sharpness
  13. The Counterexample

Key Result

Theorem 1

Suppose $1 \leqslant p_1,p_2,\ldots,p_n,p \leqslant \infty$ satisfy Further, take $K$ to be an integrable function in $\mathbb{R}^{nd}$ with Define and assume that this quantity is finite for $K$. The $n$-linear operator, $T$, is given through $K$ as in eq:multopcon. If then there is a constant $C = C(p_1,\ldots,p_n,d,n) > 0$ such that for all $f_1,\ldots,f_n \in \mathcal{S}_0(\mathbb{R}^d)$.

Theorems & Definitions (25)

  • Remark 1.1
  • Theorem 1
  • Remark 1.2
  • Theorem 2
  • Remark 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.5
  • ...and 15 more