In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [10] where similar estimates were established for the "constant coefficient" Baouendi-Grushin operator. Consequently, we obtain: (ⅰ) a Bourgain-Kenig type quantitative uniqueness result in the variable coefficient setting; (ⅱ) and a strong unique continuation property for a class of degenerate sublinear equations. We also derive a subelliptic version of a scaling critical Carleman estimate proven by Regbaoui in the Euclidean setting using which we deduce a new unique continuation result in the case of scaling critical Hardy type potentials.

中文翻译：

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一类可变系数退化椭圆算子的 Carleman 估计及其在唯一延拓中的应用

在本文中，我们获得了一类变系数退化椭圆算子的新 Carleman 估计，其某一点的常系数模型是所谓的 Baouendi-Grushin 算子。这概括了我们两个与 Garofalo 在 [10] 其中为“常数系数”Baouendi-Grushin 算子建立了类似的估计。因此，我们得到： (ⅰ) 可变系数设置中的 Bourgain-Kenig 型定量唯一性结果；(ⅱ) 和一类退化次线性方程的强唯一连续性质。我们还推导出由 Regbaoui 在欧几里德设置中证明的缩放临界卡尔曼估计的亚椭圆版本，使用它我们在缩放临界哈代型势的情况下推导出新的唯一连续结果。