On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions

Abstract

We study a multilinear version of the Hörmander multiplier theorem, namely

$$ \Vert T_{\sigma}(f_{1},\dots,f_{n})\Vert_{L^{p}}\lesssim \sup_{k\in\mathbb{Z}}{\Vert \sigma(2^{k}\cdot,\dots,2^{k}\cdot)\widehat{\phi^{(n)}}\Vert_{L^{2}_{(s_{1},\dots,s_{n})}}}\Vert f_{1}\Vert_{H^{p_{1}}}\cdots\Vert f_{n}\Vert_{H^{p_{n}}}. $$

We show that the estimate does not hold in the limiting case $\min \limits {(s_{1},\dots ,s_{n})}=d/2$ 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_σ.