On the failure of multilinear multiplier theorem with endpoint smoothness conditions

Abstract

We study a multilinear version of Hörmander multiplier theorem, namely \begin{equation*} \Vert T_σ(f_1,\dots,f_n)\Vert_{L^p}\lesssim \sup_{k\in\mathbb{Z}}{\Vert σ(2^k\cdot,\dots,2^k\cdot)\widehat{ϕ^{(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 $T_{\sigma }$.

To appear in Potential Anal