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Necessary conditions for weighted estimates of Multilinear Multipliers and Pseudo-Differential Operators

Bae Jun Park, Naohito Tomita

TL;DR

The paper addresses the problem of identifying sharp weight conditions for weighted $L^p$ bounds of multilinear Fourier multipliers and multilinear pseudo-differential operators. The authors develop explicit constructions of multiple weights and prototype model symbols, and employ Littlewood-Paley reductions and Bessel potential kernels to prove that the condition $w\in A_{(p_1/q,\dots,p_l/q)}$ with $q\ge\frac{nl}{s}$ (for multipliers) and $r\le q$ (for pseudo-differential operators) are necessary, establishing sharpness of previously known sufficient conditions. They further deduce optimal sharp maximal-function estimates for multilinear pseudo-differential operators, extending the linear theory to the multilinear setting. These results deepen the understanding of how multilinear weight interactions govern boundedness and provide precise limitations on weight classes in weighted harmonic analysis.

Abstract

We study optimal multiple weight assumptions in the weighted theory of multilinear Fourier multipliers and multilinear pseudo-differential operators. For multilinear Fourier multipliers, we revisit the weighted Hörmander-type theorem of Li and Sun, as a multilinear version of Kurtz and Wheeden, and show that their multiple weight condition is sharp. This provides the sharp necessary condition in the multilinear setting and simultaneously improves the classical linear necessity established by Kurtz and Wheeden. In the pseudo-differential setting, we consider recent weighted estimates of the authors for symbols in the multilinear Hörmander class and prove that their multiple weight hypothesis is also best possible. As a corollary, we can obtain the optimality of sharp maximal function estimates for multilinear pseudo-differential operators in the papers of the authors which originated from the results of Chanillo and Torchinsky.

Necessary conditions for weighted estimates of Multilinear Multipliers and Pseudo-Differential Operators

TL;DR

The paper addresses the problem of identifying sharp weight conditions for weighted bounds of multilinear Fourier multipliers and multilinear pseudo-differential operators. The authors develop explicit constructions of multiple weights and prototype model symbols, and employ Littlewood-Paley reductions and Bessel potential kernels to prove that the condition with (for multipliers) and (for pseudo-differential operators) are necessary, establishing sharpness of previously known sufficient conditions. They further deduce optimal sharp maximal-function estimates for multilinear pseudo-differential operators, extending the linear theory to the multilinear setting. These results deepen the understanding of how multilinear weight interactions govern boundedness and provide precise limitations on weight classes in weighted harmonic analysis.

Abstract

We study optimal multiple weight assumptions in the weighted theory of multilinear Fourier multipliers and multilinear pseudo-differential operators. For multilinear Fourier multipliers, we revisit the weighted Hörmander-type theorem of Li and Sun, as a multilinear version of Kurtz and Wheeden, and show that their multiple weight condition is sharp. This provides the sharp necessary condition in the multilinear setting and simultaneously improves the classical linear necessity established by Kurtz and Wheeden. In the pseudo-differential setting, we consider recent weighted estimates of the authors for symbols in the multilinear Hörmander class and prove that their multiple weight hypothesis is also best possible. As a corollary, we can obtain the optimality of sharp maximal function estimates for multilinear pseudo-differential operators in the papers of the authors which originated from the results of Chanillo and Torchinsky.
Paper Structure (14 sections, 6 theorems, 125 equations)

This paper contains 14 sections, 6 theorems, 125 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Muckenhoupt's weight space $A_p(\mathbb{R}^n)$
  4. Multiple weight space $A_{(p_1,\dots,p_l)}$
  5. Bessel potential
  6. Key Estimates
  7. Contruction of (multiple) weights
  8. Construction of (multilinear) multipliers
  9. Main weighted estimates
  10. Proof of Theorem \ref{['sharpnessmultilinear']}
  11. Proof of Theorem \ref{['multipseudonega']} and Corollary \ref{['sharp-maximal-function-estimate']}
  12. Reduction to symbols of type $(0,0)$
  13. Proof of Theorem \ref{['multipseudonega']} for symbols of type $(0,0)$.
  14. Proof of Corollary \ref{['sharp-maximal-function-estimate']}

Key Result

Theorem 1

Let $l\in \mathbb{N}$, $\frac{nl}{2}<s\le nl$, $0<q<\infty$, $\max\{q,\frac{nl}{s}\} \le p_1,\dots,p_l<\infty$, and $\frac{1}{p_1}+\cdots+\frac{1}{p_l}=\frac{1}{p}$. Assume that multiweibound holds for all multilinear symbols $\sigma$ and for all $\vec{w}\in A_{(\frac{p_1}{q},\dots,\frac{p_l}{q})}(\

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Remark
  • Corollary 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • proof