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- W4301882816 abstract "The purpose of this paper is to study the $L^2$ boundedness 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 $(t,x)in R^Ntimes R^n$, satisfying $gamma_0(x)equiv x$, $psi$ is a $C^infty$ cutoff function supported on a small neighborhood of $0in R^n$, and $K$ is a multi-parameter singular kernel supported on a small neighborhood of $0in R^N$. The goal is, given an appropriate class of kernels $K$, to give conditions on $gamma$ such that every operator of the above form is bounded on $L^2$. The case when $K$ is a Calder'on-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when $K$ has a multi-parameter structure. For example, when $K$ is given by a product kernel. Even when $K$ is a Calder'on-Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of $L^p$ boundedness, while the third paper deals with the special case when $gamma$ is real analytic." @default.
- W4301882816 created "2022-10-05" @default.
- W4301882816 creator A5077939778 @default.
- W4301882816 date "2010-05-24" @default.
- W4301882816 modified "2023-10-16" @default.
- W4301882816 title "Multi-parameter singular Radon transforms I: the $L^2$ theory" @default.
- W4301882816 doi "https://doi.org/10.48550/arxiv.1005.4400" @default.
- W4301882816 hasPublicationYear "2010" @default.
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