Real Analytic Multi Parameter Singular Radon Transforms

We study operators of the formTf(x)= \psi(x) \int f(\gamma_t(x))K(t)\,dt, where \gamma_t(x) is a real analytic function of (t,x) mapping from a neighborhood of (0,0) in \mathbb{R}^N \times \mathbb{R}^n into \mathbb{R}^n satisfying \gamma_0(x)\equiv x, \psi(x) \in C_c^\infty(\mathbb{R}^n), and K(t) is a ``multi-parameter singular kernel'' with compact support in \mathbb{R}^N; for example when K(t) is a product singular kernel. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth \gamma_t(x), in the single-parameter case when K(t) is a Calder\'on-Zygmund kernel. Street and Stein generalized their work to the multi-parameter case, and gave sufficient conditions for the L^p-boundedness of such operators. This paper shows that when \gamma_t(x) is real analytic, the sufficient conditions of Street and Stein are also necessary for the L^p-boundedness of T, for all such kernels K.