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 ${\gamma }_0(x)\equiv x$, ψ 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 supported on a small neighborhood of $0\in {\mathbb{R}}^k$. Christ, Nagel, Stein and Wainger gave conditions on γ 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.

本文的目的是研究形式算子的稀疏界 $F\mapsto \psi （X）\int F（{\gamma }_{Ť}（X））ķ（Ť）dŤ$，在哪里 ${\gamma }_{Ť}（X）$ 是一个 $C^{\infty }$ 在以下位置的原点附近定义的函数 $（X，Ť）\in {\mathbb{[R}}^{ñ}\times {\mathbb{[R}}^{ķ}$，令人满意 ${\gamma }_0（X）\equiv X$，ψ是一个$C^{\infty }$ 在以下地区的小社区上支持截止功能 $0\in {\mathbb{[R}}^{ñ}$和ķ是支持的一小附近卡尔德龙，上Zygmund内核$0\in {\mathbb{[R}}^{ķ}$。Christ，Nagel，Stein和Wainger给出了关于γ的条件$Ť：{\mathrm{大号}}^p\mapsto {\mathrm{大号}}^p（\mathrm{1个}<p<\infty ）$有界。在这些相同条件下，我们证明了T的稀疏边界，这加强了它们的结果。作为推论，我们得出了此类算子的加权范数估计。