Journal of Inequalities and Applications
Volume 2007 (2007), Article ID 32585, 24 pages
Research Article

A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem

Jingbo Dou, Pengcheng Niu, and Zixia Yuan

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi 710072, China

Received 22 March 2007; Revised 26 May 2007; Accepted 20 October 2007

Academic Editor: László Losonczi

Copyright © 2007 Jingbo Dou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the p-sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.