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arXiv · 1912.03873

On the failure of multilinear multiplier theorem with endpoint smoothness conditions

Bae Jun Park
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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 $T_σ$.

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Bae Jun Park. 2021-03-15. On the failure of multilinear multiplier theorem with endpoint smoothness conditions. https://arxiv.org/abs/1912.03873

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