论文标题
具有Lorentz-Sobolev条件的多线性荷尔曼德乘数定理
The multilinear Hormander multiplier theorem with a Lorentz-Sobolev condition
论文作者
论文摘要
在本文中，我们提供了具有Lorentz-Sobolev空间条件的Hörmander乘数定理的多线性版本。这项工作是由第一作者和斯拉维科瓦的最新结果激发的，在该结果中，获得了古典Hörmander乘数定理的类似版本。此版本在许多方面都很清晰，并减少了定理语句中出现的索引数量。作为线性案例的自然扩展，在这项工作中，我们证明，如果$ mn/2 <s <mn $，则$ \ big \ big \ big \ vertt_σ（f_1，\ dots，f_m）\ big big \ vert_ {l^p（（（（\ mathbb {r}）\ simemim \ sup_ {k \ in \ mathbb {z}}} \ big \ vert σ（2^k \; \ vec {\ cdot} \;）\ wideHat {ψ^{（m）}}} \ big \ big \ vert_ {l_ {l_ {s}^{mn/s，1}（\ Mathbb {\ Mathbb {r}^{mn}^{mn}}} \ vert f_1 \ vert_ {l^{p_1}（（（\ Mathbb {r}）^n）}}} \ cdots \ vert f_m \ vert_ {l^{p_m}（（\ Mathbb {r}） 对于某些$ p，p_1，\ dots，p_m $，$ 1/p = 1/p_1+\ dots+1/p_m $。我们还表明，从某种意义上说，上面的估计值是清晰的，从lorentz-sobolev space $ l_s^{mn/s，1} $无法用$ l_ {s}^{r，q} $代替，$ r <mn/s $，$ 0 <q \ q \ leq \ leq \ leq \ leq \ leq \ infty $ l_s $ l_s $ n/q} $ n/q} $ n/q}，
In this article, we provide a multilinear version of the Hörmander multiplier theorem with a Lorentz-Sobolev space condition. The work is motivated by the recent result of the first author and Slavíková where an analogous version of classical Hörmander multiplier theorem was obtained; this version is sharp in many ways and reduces the number of indices that appear in the statement of the theorem. As a natural extension of the linear case, in this work, we prove that if $mn/2<s<mn$, then $$ \big\Vert T_σ(f_1,\dots,f_m)\big\Vert_{L^p((\mathbb{R})^n)}\lesssim \sup_{k\in\mathbb{Z}}\big\Vert σ(2^k\;\vec{\cdot}\;)\widehat{Ψ^{(m)}}\big\Vert_{L_{s}^{mn/s,1}(\mathbb{R}^{mn})}\Vert f_1\Vert_{L^{p_1}((\mathbb{R})^n)}\cdots \Vert f_m\Vert_{L^{p_m}((\mathbb{R})^n)} $$ for certain $p,p_1,\dots,p_m$ with $1/p=1/p_1+\dots+1/p_m$. We also show that the above estimate is sharp, in the sense that the Lorentz-Sobolev space $L_s^{mn/s,1}$ cannot be replaced by $L_{s}^{r,q}$ for $r<mn/s$, $0<q\leq \infty$, or by $L_s^{mn/s,q}$ for $q>1$.
