arXiv 1105.45902011-05math.CA

多参数奇异Radon变换II：L^p理论

Multi-parameter singular Radon transforms II: the L^p theory

Elias M. Stein, Brian Street

备注：: 41 pages; part 2 in a three part series

摘要：

研究了\[ f\mapsto ψ(x) \int f(γ_t(x))K(t)\: dt, \]型算子的$L^p$有界性，其中$γ_t(x)$是定义在$(t,x)\in \R^N\times \R^n$中原点邻域上的$C^\infty$函数，满足$γ_0(x)\equiv x$，$ψ$是支撑在$0\in \R^n$小邻域上的$C^\infty$截断函数，$K$是支撑在$0\in \R^N$小邻域上的“多参数奇异核”.我们还研究了相关的极大算子。我们的目标是，给定一个适当的类的内核$K$，给出$γ$的条件，使上述形式的每个运营商是有界的$L^p$（$1<p<\infty$）。当$K$是一个Calderón-Zygmund内核的情况下，研究了基督，Nagel，Stein和Wainger;我们推广他们的工作的情况下，$K$是（例如）由“产品内核。“即使$K$是Calderón-Zygmund内核，我们的方法也会产生一些新的结果。这是三部分系列中的第二篇论文。第一篇论文涉及的情况下$p=2$，而第三篇论文涉及的特殊情况下，$γ$是真正的分析。

英文摘要：

The purpose of this paper is to study the $L^p$ boundedness of operators of the form \[ f\mapsto ψ(x) \int f(γ_t(x))K(t)\: dt, \] where $γ_t(x)$ is a $C^\infty$ function defined on a neighborhood of the origin in $(t,x)\in \R^N\times \R^n$, satisfying $γ_0(x)\equiv x$, $ψ$ is a $C^\infty$ cutoff function supported on a small neighborhood of $0\in \R^n$, and $K$ is a "multi-parameter singular kernel" supported on a small neighborhood of $0\in \R^N$. We also study associated maximal operators. The goal is, given an appropriate class of kernels $K$, to give conditions on $γ$ such that every operator of the above form is bounded on $L^p$ ($1<p<\infty$). The case when $K$ is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their work to the case when $K$ is (for instance) given by a "product kernel." Even when $K$ is a Calderón-Zygmund kernel, our methods yield some new results. This is the second paper in a three part series. The first paper deals with the case $p=2$, while the third paper deals with the special case when $γ$ is real analytic.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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