In this paper sharp $L^2$-Caffarelli–Kohn–Nirenberg (CKN) inequalities on Euclidean space equipped with Grushin vector fields are established. By use of a Kelvin-like transform, and Grushin cylindrical coordinates, it is shown here that these inequalities are sharp for all possible parameter values for these inequalities.

中文翻译：

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尖锐 ${\mathrm{大号}}^2$Grushin向量场的-Caffarelli–Kohn–Nirenberg不等式

在本文中 ${\mathrm{大号}}^2$建立了配备有Grushin向量场的欧几里得空间上的-Caffarelli-Kohn-Nirenberg（CKN）不等式。通过使用像开尔文一样的变换和Grushin柱面坐标，这里显示出这些不等式对于这些不等式的所有可能参数值都是尖锐的。