Multi-parameter singular Radon transforms

TL;DR

This paper extends the theory of singular Radon transforms to multi-parameter settings with product kernels, providing conditions for boundedness on L^p spaces and discussing associated maximal functions.

Contribution

It introduces new conditions on the phase function b3 for boundedness of multi-parameter singular Radon transforms, extending previous single-parameter results to more complex multi-parameter cases.

Findings

Established L^p boundedness criteria for multi-parameter Radon transforms.

Extended single-parameter Calderf3n-Zygmund theory to multi-parameter context.

Analyzed the case when b3 is real-analytic.

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 \[ f\mapsto \psi(x) \int f(\gamma_t(x)) K(t)\: dt, \] 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 Calder\'on-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.