Real Analytic Multi-parameter Singular Radon Transforms: necessity of the Stein-Street condition

Abstract

We study operators of the form $$Tf(x)=\psi (x)\int f({\gamma }_t(x))K(t)dt,$$ 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 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 $K$.

60 pages