论文标题
Marcinkiewicz定理的尖锐变体，在Lorentz类型的Sobolev空间中具有乘数
A sharp variant of the Marcinkiewicz theorem with multipliers in Sobolev spaces of Lorentz type
论文作者
论文摘要
给定$ \ mathbb {r}^n $上有界的可测量函数$σ$，我们让$t_σ$是通过乘以傅立叶变换$σ$获得的运算符。令$ 0 <s_1 \ le s_2 \ le \ cdots \ le s_n <1 $和$ψ$是schwartz在真实行上的函数，其傅立叶变换$ \widehatψ$在$ [ - 2，-1/2] \ cup [1/2,2] $中得到支持，并且满足$ \ sum_ sum_ sum_ sum_ { \wideHatψ\ left（2^{ - j}ξ\ right）= 1 $ for All $ξ\ neq 0 $。在这项工作中，我们通过找到属性的几乎最佳函数空间来锐化Marcinkiewicz乘数定理的已知形式，如果函数\ begin \ begin {equation*}（ξ_1，\ dots，dots，ξ_n），ξ_n）\ mapsto \ mapsto \ prod_ = 1} \ prod_ {i = 1}^n \wideHatψ（ξ_i）σ（2^{j_1}ξ_1，\ dots，2^{j_n}ξ_n）\ big] \ big] \ big end {equation {qore {qore*}在$ j_1，$ j_1，$ j_ n $ tot_ $ j_ n $ tot_ $ n o; {l}^p（\ mathbb r^n）$时$ | \ frac {1} {p} {p} - \ frac {1} {2} {2} | <s_1 $和$ 1 <p <\ infty $。 In the case where $s_i\neq s_{i+1}$ for all $i$, it was proved in [Grafakos, Israel J. Math., to appear] that the Lorentz space $L ^{\frac{1}{s_1},1} (\mathbb{R}^n) $ is the function space sought.在这项工作中，当对于某些索引$ i $时，我们可能有$ s_i = s_ {i+1} $时，我们解决了更加困难的一般情况。我们获得了marcinkiewicz乘数定理的一个版本，其中$ l ^{\ frac {\ frac {1} {s_1}，1} $被与$ s_2，s_n $ s_n $ s_n $ s_1 $ s_1 $相关的一定凹形函数相关的适当lorentz空间代替了1} $。我们的结果是在Lorentz空间的定义凹面功能中的对数的任意小功率的最佳选择。
Given a bounded measurable function $σ$ on $\mathbb{R}^n$, we let $T_σ$ be the operator obtained by multiplication on the Fourier transform by $σ$. Let $0<s_1\le s_2\le \cdots \le s_n<1$ and $ψ$ be a Schwartz function on the real line whose Fourier transform $\widehatψ$ is supported in $[-2,-1/2]\cup[1/2,2]$ and which satisfies $\sum_{j \in \mathbb{Z}} \widehatψ\left(2^{-j} ξ\right)=1$ for all $ξ\neq 0$. In this work we sharpen the known forms of the Marcinkiewicz multiplier theorem by finding an almost optimal function space with the property that, if the function \begin{equation*} (ξ_1,\dots, ξ_n)\mapsto \prod_{i=1}^n (I-\partial_i^2)^{\frac {s_i}2} \Big[ \prod_{i=1}^n \widehatψ(ξ_i) σ(2^{j_1}ξ_1,\dots , 2^{j_n}ξ_n)\Big] \end{equation*} belongs to it uniformly in $j_1,\dots , j_n \in \mathbb Z$, then $T_σ$ is bounded on $ {L}^p(\mathbb R^n)$ when $ |\frac{1}{p}-\frac{1}{2} | < s_1$ and $1<p<\infty$. In the case where $s_i\neq s_{i+1}$ for all $i$, it was proved in [Grafakos, Israel J. Math., to appear] that the Lorentz space $L ^{\frac{1}{s_1},1} (\mathbb{R}^n) $ is the function space sought. In this work we address the significantly more difficult general case when for certain indices $i$ we might have $s_i=s_{i+1}$. We obtain a version of the Marcinkiewicz multiplier theorem in which the space $L ^{\frac{1}{s_1},1}$ is replaced by an appropriate Lorentz space associated with a certain concave function related to the number of terms among $s_2,\dots , s_n$ that equal $s_1$. Our result is optimal up to an arbitrarily small power of the logarithm in the defining concave function of the Lorentz space.
