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

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 $C^{1,α}, α\in (0,1],$ regular curve in $\mathbb{G}$.

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