  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.
