On the failure of multilinear multiplier theorem with endpoint smoothness conditions

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 $\min (s_1,\dots ,s_n)=d\mathrm{/}2$ or ${\sum }_{k\in J}(s_k\mathrm{/}d-1\mathrm{/}{p}_k)=-1\mathrm{/}2$ for some $J \subset {1,\dots,n}$\mathrm{.}Thisprovidesthenecessaryandsufficientconditionon$(s_1,\dots,s_n)$fortheboundednessof$T_{\sigma}$.