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.

中文翻译：

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带有端点光滑度条件的多线性乘法定理的失败

我们研究了Hörmander乘子定理的多线性形式，即

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

我们证明，在极限情况下，估计不成立\（\ min \ limits {{s_ {1}，\ dots，s_ {n}）} = d / 2 \）或\（{\ sum} _ {k \ in J} {（{s_ {k}} / {d}-{1} / {p_ {k}}）} =-{1} / {2} \）对于某些\（J \ subset \ {1 ，\ dots，n \} \）。这提供了充分必要条件\（（S_ {1}，\点，S_ {N}）\）用于有界Ť _σ。