数学物理学报 ›› 2011, Vol. 31 ›› Issue (5): 1181-1189.

• 论文 • 上一篇    下一篇

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

韩亚洲, 赵琼   

  1. 中国计量学院数学系 杭州 310018
  • 收稿日期:2009-10-09 修回日期:2010-09-08 出版日期:2011-10-25 发布日期:2011-10-25
  • 基金资助:

    浙江省自然科学基金(Y6110118)资助

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

 HAN Ya-Zhou, ZHAO Qiong   

  1. Department of Mathematics, College of Science, China Jiliang University, Hangzhou 310018
  • Received:2009-10-09 Revised:2010-09-08 Online:2011-10-25 Published:2011-10-25
  • Supported by:

    浙江省自然科学基金(Y6110118)资助

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、H\"{o}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.

关键词: Caffarelli-Kohn-Nirenberg型不等式, Baouendi-Grushin向量场

Abstract:

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 H\"{o}lder inequality, etc., it is proved  that the condition of necessity is also that of  sufficiency for the case p=2.

Key words: Caffarelli-Kohn-Nirenberg type inequalities, Baouendi-Grushin vector fields

中图分类号: 

  • 35H20