Multi-parameter singular Radon transforms

Abstract

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 \[ f\mapsto ψ(x) \int f(γ_t(x)) K(t)\: dt, \] where ${\gamma }_t(x)=\gamma (t,x)$ is a $C^{\infty }$ function defined on a neighborhood of the origin in $(t,x)\in {\mathbb{R}}^N\times {\mathbb{R}}^n$ satisfying ${\gamma }_0(x)\equiv x$, $K(t)$ 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.

18 pages; an announcement for a three part series of papers; final version to appear in Math. Res. Lett