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

TL;DR

This paper establishes the sharp necessary conditions for weighted estimates in multilinear Fourier multipliers and pseudo-differential operators, improving existing linear and multilinear theories.

Contribution

It proves the optimality of multiple weight conditions in the weighted theory of multilinear Fourier multipliers and pseudo-differential operators, refining previous results and establishing sharpness.

Findings

Multiple weight condition is sharp for multilinear Fourier multipliers.

Multiple weight hypothesis is optimal for multilinear pseudo-differential operators.

Results confirm the sharpness of maximal function estimates in the multilinear setting.

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 H\"ormander-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 H\"ormander 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.