Acta mathematica scientia,Series A ›› 2011, Vol. 31 ›› Issue (5): 1181-1189.

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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

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

CLC Number: 

  • 35H20
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