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

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.

💡 Research Summary

This paper investigates the optimal multiple‑weight hypotheses required for weighted norm inequalities of multilinear Fourier multipliers and multilinear pseudo‑differential operators. The authors focus on two central questions: (i) whether the multiple‑weight class introduced by Li and Sun in their Hörmander‑type theorem for multilinear multipliers is sharp, and (ii) whether the multiple‑weight condition used in recent weighted estimates for multilinear pseudo‑differential operators (by the authors themselves) can be weakened.

Background. In the linear setting, Muckenhoupt’s (A_{p}) class characterizes the weights for which the Hardy–Littlewood maximal operator and Calderón–Zygmund operators are bounded on (L^{p}(w)). For multilinear operators, Lerner et al. defined the class (A_{\vec p}) (with (\vec p=(p_{1},\dots ,p_{l}))) as the natural analogue; however, its internal structure is more intricate, lacking the simple monotonicity of the linear case.

Multilinear Fourier multipliers. Let (\sigma) be a bounded symbol satisfying a Sobolev‑type regularity
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