A Two-Weight Sobolev Inequality for Carnot-Carathéodory Spaces
RICERCHE DI MATEMATICA(2022)
摘要
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.
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关键词
Carnot-Carathéodory spaces,Weighted Sobolev inequalities,Muckenhoupt and Gehring weights,35R03,39B62
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