Dear all,
Tomorrow's speaker in the Analysis and Probability Seminar is Lingxiao Zhang from UConn. The title of her talk is "Real Analytic Singular Radon Transforms With Product Kernels: Necessity of the Stein-Street condition."
The talk will begin at 2:30pm tomorrow in Mont 313.
Abstract: 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 Calder’on-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.
Best,
Sean
