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

Abstract

Let $\mathbb{G}$ be any Carnot group. We prove that if a convolution type singular integral associated with a $1$-dimensional Calderón-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 $\mathbb{G}$.