Singular integrals on $C^{1,\alpha}$ regular curves in Carnot groups

TL;DR

This paper proves that certain singular integrals, initially bounded on horizontal lines in Carnot groups, are also bounded on smooth curves, extending classical results to a broader geometric setting.

Contribution

It establishes $L^p$ boundedness of convolution singular integrals on $C^{1,eta}$ curves in Carnot groups based on their boundedness on horizontal lines.

Findings

Boundedness of singular integrals on horizontal lines implies boundedness on smooth curves.

Results extend classical Euclidean singular integral theory to Carnot groups.

Provides a new criterion for $L^p$ boundedness in sub-Riemannian geometry.

Abstract

Let $\mathbb{G}$ be any Carnot group. We prove that if a convolution type singular integral associated with a $1$-dimensional Calder\'on-Zygmund kernel is $L^2$-bounded on horizontal lines, with uniform bounds, then it is bounded in $L^p, p \in (1,\infty),$ on any compact $C^{1,\alpha}, \alpha \in (0,1],$ regular curve in $\mathbb{G}$.