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 Calder\'on-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.