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↦ψ(x)∫f(γt(x))K(t)dt, where γt(x) is a C∞ function defined on a neighborhood of the origin in (x,t)∈Rn×Rk, satisfying γ0(x)≡x, ψ is a C∞ cut-off function supported on a small neighborhood of 0∈Rn and K is a Calderón-Zygmund kernel supported on a small neighborhood of 0∈Rk. Christ, Nagel, Stein and Wainger gave conditions on γ under which T:Lp↦Lp(1<p<∞) 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.