Sobolev spaces associated to singular and fractional Radon transforms
Sobolev spaces associated to singular and fractional Radon transforms
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与奇异和分数 Radon 变换相关的 Sobolev 空间
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发表时间：
2015
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通讯作者：
B. Street
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作者：
B. Street
本文的目的是研究$fmapsto psi(x) int f(gamma_t(x)) K(t)的光滑性质（在$L^p$ Sobolev空间中）：dt$，其中$gamma_t(x)$是定义在$(t,x)inmathbb{R}^ n乘以mathbb{R}^n$的原点的邻域$C^ inty $上的函数，满足$gamma_0(x)等价x$, $psi$是在$ mathbb{R}^n$的小邻域$0上支持的$C^ inty $截止函数，$K$是在$ mathbb{R}^n$的小邻域$0上支持的“多参数分数核”。当$K$是Calder'on-Zygmund核时，这些算子由Christ、Nagel、Stein和Wainger研究；当$K$是多参数奇异核时，这些算子由作者和Stein研究。在这两种情况下，给出了$gamma$上上述算子在$L^p$ （$1<p<infty$）上有界的条件。在这些相同的条件下，我们引入了与$gamma$相关的非各向同性$L^p$ Sobolev空间。更进一步，当K$是一个阶数趋近于0$平滑的分数核（即非常趋近于奇异核）时，我们证明了上述算子在这些非各向同性Sobolev空间上的映射性质。作为推论，在Christ、Nagel、Stein和Wainger在$gamma$上引入的条件下，我们证明了当$K$是一个分数阶核，且为极低阶平滑时，上述算子在各向同性$L^p$ Sobolev空间中的最优平滑性质。
The purpose of this paper is to study the smoothing properties (in $L^p$ Sobolev spaces) of operators of the form $fmapsto psi(x) int f(gamma_t(x)) K(t): dt$, where $gamma_t(x)$ is a $C^infty$ function defined on a neighborhood of the origin in $(t,x)inmathbb{R}^N imes mathbb{R}^n$, satisfying $gamma_0(x)equiv x$, $psi$ is a $C^infty$ cut-off function supported on a small neighborhood of $0in mathbb{R}^n$, and $K$ is a "multi-parameter fractional kernel" supported on a small neighborhood of $0in mathbb{R}^N$. When $K$ is a Calder'on-Zygmund kernel these operators were studied by Christ, Nagel, Stein, and Wainger, and when $K$ is a multi-parameter singular kernel they were studied by the author and Stein. In both of these situations, conditions on $gamma$ were given under which the above operator is bounded on $L^p$ ($1<p<infty$). Under these same conditions, we introduce non-isotropic $L^p$ Sobolev spaces associated to $gamma$. Furthermore, when $K$ is a fractional kernel which is smoothing of an order which is close to $0$ (i.e., very close to a singular kernel) we prove mapping properties of the above operators on these non-isotropic Sobolev spaces. As a corollary, under the conditions introduced on $gamma$ by Christ, Nagel, Stein, and Wainger, we prove optimal smoothing properties in isotropic $L^p$ Sobolev spaces for the above operator when $K$ is a fractional kernel which is smoothing of very low order.