LAUSR

Let $$X = \{X_1,X_2, \ldots ,X_m\}$$ be a system of smooth vector fields in $${{\mathbb R}^n}$$ satisfying the Hormander’s finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Caratheodory space $$\mathbb G$$ associated to system X $$\begin{aligned} \left( \frac{1}{\int _{B_R} K(x)\; dx} \int _{B_R} |u|^{t} K(x) \; dx \right) ^{1/t} \le C\, R \left( \frac{1}{\int _{B_R}\frac{1}{K(x)} \; dx} \int _{B_R} \frac{|X u|^2}{K(x)} \; dx \right) ^{1/2}, \end{aligned}$$ where Xu denotes the horizontal gradient of u with respect to X. We assume that the weight K belongs to Muckenhoupt’s class $$A_2$$ and Gehring’s class $$G_{\tau }$$ , where $$\tau $$ is a suitable exponent related to the homogeneous dimension.