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

TL;DR

This paper proves that for real analytic multi-parameter singular Radon transforms, the previously known sufficient conditions for $L^p$ boundedness are also necessary, establishing a complete characterization.

Contribution

It demonstrates the necessity of Street and Stein's conditions for $L^p$ boundedness in the real analytic setting, extending prior sufficiency results to necessity.

Findings

Necessity of Street and Stein's conditions established for real analytic $oldsymbol{ ext{γ}_t(x)}$.

Complete characterization of $L^p$ boundedness conditions for these operators.

Extension of prior work from smooth to real analytic functions.

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 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 $K$.