Sparse domination of singular Radon transform

Abstract

The purpose of this paper is to study the sparse bound of the operator of the form $f\mapsto \psi (x)\int f({\gamma }_t(x))K(t)dt$, where ${\gamma }_t(x)$ is a $C^{\infty }$ function defined on a neighborhood of the origin in $(x,t)\in {\mathbb{R}}^n\times {\mathbb{R}}^k$, satisfying ${\gamma }_0(x)\equiv x$, $\psi$ is a $C^{\infty }$ cut-off function supported on a small neighborhood of $0\in {\mathbb{R}}^n$ and $K$ is a Calderón-Zygmund kernel suppported on a small neighborhood of $0\in {\mathbb{R}}^k$. Christ, Nagel, Stein and Wainger gave conditions on $\gamma$ under which $T:L^p\mapsto L^p(1<p<\infty )$ is bounded. Under the these same conditions, we prove sparse bounds for $T$, which strengthens their result. As a corollary, we derive weighted norm estimates for such operators.

80 pages, 4 figures. Based on the framework developed by Stein and Street in arXiv:1005.4400 and arXiv:1105.4590

References in corpus (1)

• The Sparse T1 Theorem