Real analytic multi-parameter singular Radon transforms: Necessity of the Stein-Street condition

We study operators of the form \begin{equation*} Tf(x)= \psi (x) \int f(\gamma _t(x))K(t)\,dt, \end{equation*} 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 Calderón-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.