On the failure of multilinear multiplier theorem with endpoint smoothness conditions

TL;DR

This paper investigates the limitations of a multilinear H"ormander multiplier theorem at endpoint smoothness conditions, establishing necessary and sufficient conditions for boundedness and demonstrating failure at critical endpoint cases.

Contribution

It identifies the precise endpoint smoothness conditions under which the multilinear multiplier theorem holds or fails, clarifying the theorem's limitations.

Findings

The estimate fails when ext{min}(s_1,...,s_n)=d/2.

The estimate fails when extstyle ext{sum}_{k ext{ in }J} (s_k/d - 1/p_k) = -1/2.

Provides necessary and sufficient conditions for boundedness of T_\sigma.

Abstract

We study a multilinear version of H\"ormander multiplier theorem, namely \begin{equation*} \Vert T_{\sigma}(f_1,\dots,f_n)\Vert_{L^p}\lesssim \sup_{k\in\mathbb{Z}}{\Vert \sigma(2^k\cdot,\dots,2^k\cdot)\widehat{\phi^{(n)}}\Vert_{L^{2}_{(s_1,\dots,s_n)}}}\Vert f_1\Vert_{H^{p_1}}\cdots\Vert f_n\Vert_{H^{p_n}}. \end{equation*} We show that the estimate does not hold in the limiting case $\min{(s_1,\dots,s_n)}=d/2$ or $\sum_{k\in J}{({s_k}/{d}-{1}/{p_k})}=-{1}/{2}$ for some $J \subset \{1,\dots,n\}$. This provides the necessary and sufficient condition on $(s_1,\dots,s_n)$ for the boundedness of $T_{\sigma}$.