标题： 径向傅里叶乘子在$\mathbb{R}^3$和$\mathbb{R}^4$ 显示英文标题

标题： Radial Fourier Multipliers in $\mathbb{R}^3$ and $\mathbb{R}^4$

摘要： 我们证明对于在原点处紧支撑的径向傅里叶乘子$m: \mathbb{R}^3\to\mathbb{C}$，如果$K=\hat{m}$属于$L^p(\mathbb{R}^3)$，则$T_m$在范围$1<p<\frac{13}{12}$内为限制强类型 (p,p)。 我们还证明了四维径向傅里叶乘子的$L^p$特征；即，对于在原点外紧支撑的径向傅里叶乘子$m: \mathbb{R}^4\to\mathbb{C}$，$T_m$在$L^p(\mathbb{R}^4)$上有界当且仅当$K=\hat{m}$属于$L^p(\mathbb{R}^4)$，在范围$1<p<\frac{36}{29}$内。我们的证明方法依赖于一种几何论证，该论证利用了三维环形区域的多重交集的大小的界限，以控制三维和四维中环形区域对之间的切线数量。 显示英文摘要

摘要： We prove that for radial Fourier multipliers $m: \mathbb{R}^3\to\mathbb{C}$ supported compactly away from the origin, $T_m$ is restricted strong type (p,p) if $K=\hat{m}$ is in $L^p(\mathbb{R}^3)$, in the range $1<p<\frac{13}{12}$. We also prove an $L^p$ characterization for radial Fourier multipliers in four dimensions; namely, for radial Fourier multipliers $m: \mathbb{R}^4\to\mathbb{C}$ supported compactly away from the origin, $T_m$ is bounded on $L^p(\mathbb{R}^4)$ if and only if $K=\hat{m}$ is in $L^p(\mathbb{R}^4)$, in the range $1<p<\frac{36}{29}$. Our method of proof relies on a geometric argument that exploits bounds on sizes of multiple intersections of three-dimensional annuli to control numbers of tangencies between pairs of annuli in three and four dimensions.