A two-weight Sobolev inequality for Carnot-Carathéodory spaces

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