The authors treat operators of the form \[ Tf(x)=\psi(x)\int f\bigl(\gamma_t(x)\bigr)K(t)\,dt, \] where \(\psi\in C_0^\infty(\mathbb R^n)\) is supported near \(0\), \(\gamma_t(x):\mathbb R_0^N\times\mathbb R_0^n \rightarrow \mathbb R^n\) is a germ of a real analytic function (defined on a neighborhood of \((0,0)\)) satisfying \(\gamma_0(x)\equiv x\), \(K\) is a multi-parameter distribution kernel, supported near \(0\in\mathbb R^N\). As a main example, they take a product kernel, i.e. a kernel satisfying \(|\partial_{t_1}^{\alpha_1}\cdots\partial_{t_\nu}^{\alpha_\nu}K(t)|\lesssim |t_1|^{-N_1-|\alpha_1|}\cdots|t_1|^{-N_1-|\alpha_1|}\), along with certain cancellation conditions, where \(\mathbb R^N=\mathbb R^{N_1}\times\cdots\times\mathbb R^{N_\nu}\). If \(\nu=1\), this is a standard Calderón-Zygmund kernel supported near \(0\), and \(L^p\) boundedness of \(T\) can be shown rather easily. But in the multi-parameter case the study of \(T\) is difficult. A prototype was studied in the work of M. Christ, A. Nagel, E. M. Stein and S. Wainger [Ann. Math. (2) 150, No. 2, 489–577 (1999; Zbl 0960.44001)]. The authors develop it in a series of three papers. This paper is the third one. The first two dealt with the general situation when \(\gamma\) is \(C^\infty\), instead of real analytic. They study also maximal operators of the form \[ \mathcal M f(x)=\sup_{0<\delta_1,\dots,\delta_\nu\leq a}\psi(x)\int|f\bigl( \gamma_{(\delta_1t,\dots,\delta_\nu t)}(x)\bigr)|\,dt_1\cdots dt_\nu, \] where \(\psi\) is as above with \(\psi\geq0\), and \(a>0\) is small.