turkmath.org

We discuss operators of the form

$Tf(x)=\psi (x)\int f(\gamma_{t}(x))K (t)dt$,

where $\psi (x) \in C_{c}^{\infty}(\mathbb{R}^{n})$, $\gamma_{t}(x)$ 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 $\gamma_{0}(x)\equiv x$, and $K (t)$ is a product kernel with small support in $\mathbb{R}^{N}$. The celebrated work of Christ, Nagel, Stein, and Wainger studied such operators with smooth $\gamma_{t}(x)$, in the special case when $K (t)$ is a Calderón-Zygmund kernel. Street and Stein generalized their work to (for instance) the product kernel case, and gave sufficient conditions for the $L^{p}$ boundedness of such operators for all such kernels $K$. In this talk, we will state that when $\gamma_{t}(x)$ is real analytic, the Stein-Street condition is also necessary, and will also use several simple examples and graphs to illustrate this necessary and sufficient condition and explain the main ideas of the proof methods.