A two-weight Sobolev inequality for Carnot-Carathéodory spaces

Let X = {X_1,X_2, … ,X_m} be a system of smooth vector fields in ℝ^n satisfying the Hörmander’s finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Carathéodory space 𝔾 associated to system X ( 1/∫ _B_R K(x) dx∫ _B_R |u|^t K(x) dx ) ^1/t≤ C R ( 1/∫ _B_R1/K(x) dx∫ _B_R|X u|^2/K(x) dx ) ^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 and Gehring’s class G_τ , where τ is a suitable exponent related to the homogeneous dimension.