中文摘要：

受Caffarelli等建立欧氏空间上Caffarelli—Kohn—Nirenberg（CKN）不等式的思想方法启发，该文结合广义Baouendi-Grushin（B—G）向量场上极坐标变换，通过选取不同的辅助函数，给出广义B—G向量场上CKN型不等式成立的必要条件；从广义B-G向量场上Hardy—Sobolev型不等式出发，结合插值、HSlder不等式等工具，通过对参数的精细讨论，证明广义B-G向量场上P=2时CKN型不等式成立的必要条件也是充分条件．

英文摘要：

Inspired by the method of Caffarelli et al., which establishes the CKN inequality on Rn, this paper is devoted to study the CKN type inequality for the generalized B-G vector fields. By using the polar coordinates transform of the generalized B-G vector fields, the necessity of the CKN type inequality is given by constructing suitable auxiliary functions. Combining the Hardy-Sobolev type inequality of the generalized B-G vector fields and some tools, such as interpolation method and Hoelder inequality, etc., it is proved that the condition of necessity is also that of sufficiency for the case p - 2.