Multi-parameter singular Radon transforms III: real analytic surfaces

Abstract

The goal of this paper is to study operators of the form, \[ Tf(x)= ψ(x)\int f(γ_t(x))K(t)\: dt, \] where $\gamma$ is a real analytic function defined on a neighborhood of the origin in $(t,x)\in {\mathbb{R}}^N\times {\mathbb{R}}^n$, satisfying ${\gamma }_0(x)\equiv x$, $\psi$ is a cutoff function supported near $0\in {\mathbb{R}}^n$, and $K$ is a "multi-parameter singular kernel" supported near $0\in {\mathbb{R}}^N$. A main example is when $K$ is a "product kernel." We also study maximal operators of the form, \[ \mathcal{M} f(x) = ψ(x)\sup_{0<δ_1,..., δ_N<<1} \int_{|t|<1} |f(γ_{δ_1 t_1,...,δ_N t_N}(x))|\: dt. \] We show that $\mathcal{M}$ is bounded on $L^p$ ($1<p\le \infty$). We give conditions on $\gamma$ under which $T$ is bounded on $L^p$ ($1<p<\infty$); these conditions hold automatically when $K$ is a Calderón-Zygmund kernel. This is the final paper in a three part series. The first two papers consider the more general case when $\gamma$ is $C^{\infty }$.

22 pages, part 3 in a three part series