Multi-parameter singular Radon transforms III: real analytic surfaces

The goal of this paper is to study operators of the form, $$Tf(x)=\psi (x)\int f({\gamma }_t(x))K(t)dt,$$ 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, $$\mathcal{M}f(x)=\psi (x)\underset{0<{\delta }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\delta }_N<<1}{\sup }{\int }_{\mathrm{\mid }t\mathrm{\mid }<1}\mathrm{\mid }f({\gamma }_{{\delta }_1{t}_1,\mathrm{.}\mathrm{.}\mathrm{.},{\delta }_N{t}_N}(x))\mathrm{\mid }dt\mathrm{.}$$ We show that $\mathcal{M}$ is bounded on $L^p$ ($1<p\le \mathrm{\infty }$). We give conditions on $\gamma$ under which $T$ is bounded on $L^p$ ($1<p<\mathrm{\infty }$); these conditions hold automatically when $K$ is a Calder'on-Zygmund kernel. This is the final paper in a three part series. The first two papers consider the more general case when $\gamma$ is $C^{\mathrm{\infty }}$.