Let \(X = \{X_1,X_2, \ldots ,X_m\}\) be a system of smooth vector fields in \({{\mathbb R}^n}\) satisfying the Hörmander’s finite rank condition. We prove the following Sobolev inequality with reciprocal weights in Carnot-Carathéodory 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.

中文翻译：

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Carnot-Carathéodory空间的两个权重Sobolev不等式

令\（X = \ {X_1，X_2，\ ldots，X_m \} \）是\（{{\ mathbb R} ^ n} \）中满足Hörmander有限秩条件的光滑矢量场系统。我们证明了与系统X相关的卡诺·卡托西多利空间\（\ mathbb G \）中具有倒数的下列Sobolev不等式

$$ \ begin {aligned} \ left（\ frac {1} {\ int _ {B_R} K（x）\; dx} \ int _ {B_R} | u | ^ {t} K（x）\; dx \ right）^ {1 / t} \ le C \，R \ left（\ frac {1} {\ int _ {B_R} \ frac {1} {K（x）} \; dx} \ int _ {B_R } \ frac {| X u | ^ 2} {K（x）} \; dx \ right）^ {1/2}，\ end {aligned} $$

其中Xu表示u相对于X的水平梯度。我们假设权重K属于Muckenhoupt的\（A_2 \）类和Gehring的\（G _ {\ tau} \）类，其中\（\ tau \）是与齐次维相关的合适指数。