On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions

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

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

We show that the estimate does not hold in the limiting case (min limits {(s_{1},dots ,s_{n})}=d/2) or ({sum}_{kin 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.