Multi-parameter singular Radon transforms

The purpose of this announcement is to describe a development given in a series of forthcoming papers by the authors that concern operators of the form \begin{equation*} f\mapsto \psi\q(x\w) \int f\q(\gamma_t\q(x\w)\w) K\q(t\w)\: dt, \end{equation*} where $\gamma_t\q(x\w)=\gamma\q(t,x\w)$ is a $C^\infty$ function defined on a neighborhood of the origin in $\q(t,x\w)\in \R^N\times \R^n$ satisfying $\gamma_0\q(x\w)\equiv x$, $K\q(t\w)$ is a “multi-parameter singular kernel” supported near $t=0$, and $\psi$ is a cutoff function supported near $x=0$. This note concerns the case when $K$ is a “product kernel.” The goal is to give conditions on $\gamma$ such that the above operator is bounded on $L^p$ for $1<p<\infty$. Associated maximal functions are also discussed. The “single-parameter” case when $K$ is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger. The theory here extends these results to the multi-parameter context and also deals effectively with the case when $\gamma$ is real-analytic.

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Published 20 May 2011