广义Baouendi—Grushin向量场上的一类Caffarelli-Kohn-Nirenberg型不等式

A Class of Caffarelli-Kohn-Nirenberg Type Inequalities for Generalized Baouendi-Grushin Vector Fields

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、Hlder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.

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.