Some Weighted Hardy-Type Inequalities on Anisotropic Heisenberg Groups
© Bao-Sheng Lian et al. 2011
Received: 9 December 2010
Accepted: 4 March 2011
Published: 15 March 2011
We prove some weighted Hardy type inequalities associated with a class of nonisotropic Greiner-type vector fields on anisotropic Heisenberg groups. As an application, we get some new Hardy type inequalities on anisotropic Heisenberg groups which generalize a result of Yongyang Jin and Yazhou Han.
where is the neutral element of , is the Korányi-Folland nonisotropic gauge induced by the fundamental solution, and is the homogenous dimension of (see also ). Inequality (1.2) was generalized by Niu et al.  (see also ) using the Picone-type identify. For more Hardy-Sobolev inequalities on nilpotent groups, we refer the reader to [5–19].
and . However, the inequalities above do not cover the case of and . So, it is an interesting problem to study a Hardy-type inequality related to for on and . In this note, we will consider some Hardy inequalities on for . In fact, we prove a representation formula associated with , which is analogous to the Korányi-Folland nonisotropic gauge on Heisenberg group (cf. ). Using this representation formula, we prove some new Hardy inequalities on , which include the case of and .
This paper is organized as follows. We start in Section 2 with the necessary background on anisotropic Heisenberg groups . In Section 3, we prove a representation formula and use it to obtain some Hardy-type inequalities.
2. Notations and Preliminaries
Let and set . We will explicitly calculate the constant to show when . The method of calculation is similar to that used in .
3. Hardy-Type Inequality
The proof is therefore completed.
From inequality (3.17), we have the following corollary which generalizes the result of  when and .
This paper was supported by the Fundamental Research Funds for the Central Universities under Grant no. 1082001 and the National Natural Science Foundation of China (Grant no. 10901126).
- Garofalo N, Lanconelli E: Frequency functions on the Heisenberg group, the uncertainty principle and unique continuation. Annales de l'Institut Fourier (Grenoble) 1990,40(2):313–356. 10.5802/aif.1215MATHMathSciNetView ArticleGoogle Scholar
- Goldstein JA, Zhang QS: On a degenerate heat equation with a singular potential. Journal of Functional Analysis 2001,186(2):342–359. 10.1006/jfan.2001.3792MATHMathSciNetView ArticleGoogle Scholar
- Niu P, Zhang H, Wang Y: Hardy type and Rellich type inequalities on the Heisenberg group. Proceedings of the American Mathematical Society 2001,129(12):3623–3630. 10.1090/S0002-9939-01-06011-7MATHMathSciNetView ArticleGoogle Scholar
- D'Ambrozio L: Some Hardy inequalities on the Heisenberg group. Differential Equations 2004,40(4):552–564.MathSciNetView ArticleGoogle Scholar
- Dou J: Picone inequalities for -sub-Laplacian on the Heisenberg group and its applications. Communications in Contemporary Mathematics 2010,12(2):295–307. 10.1142/S0219199710003804MATHMathSciNetView ArticleGoogle Scholar
- Goldstein JA, Kombe I: The Hardy inequality and nonlinear parabolic equations on Carnot groups. Nonlinear Analysis: Theory, Methods & Applications 2008,69(12):4643–4653. 10.1016/j.na.2007.11.020MATHMathSciNetView ArticleGoogle Scholar
- Han J: A class of improved Sobolev-Hardy inequality on Heisenberg groups. Southeast Asian Bulletin of Mathematics 2008,32(3):437–444.MATHMathSciNetGoogle Scholar
- Han JQ, Niu PC, Han YZ: Some Hardy-type inequalities on groups of Heisenberg type. Journal of Systems Science and Mathematical Sciences 2005,25(5):588–598.MATHMathSciNetGoogle Scholar
- Han Y, Niu P, Luo X: A Hardy type inequality and indefinite eigenvalue problems on the homogeneous group. Journal of Partial Differential Equations 2002,15(4):28–38.MATHMathSciNetGoogle Scholar
- Han J, Niu P, Qin W: Hardy inequalities in half spaces of the Heisenberg group. Bulletin of the Korean Mathematical Society 2008,45(3):405–417. 10.4134/BKMS.2008.45.3.405MATHMathSciNetView ArticleGoogle Scholar
- Jin Y, Zhang G: Degenerate p-Laplacian operators on H-type groups and applications to Hardy type inequalities. to appear in Canadian Journal of MathematicsGoogle Scholar
- Luan J-W, Yang Q-H: A Hardy type inequality in the half-space on and Heisenberg group. Journal of Mathematical Analysis and Applications 2008,347(2):645–651. 10.1016/j.jmaa.2008.06.048MATHMathSciNetView ArticleGoogle Scholar
- Kombe I: Sharp weighted Rellich and uncertainty principle inequalities on Carnot groups. Communications in Applied Analysis 2010,14(2):251–271.MATHMathSciNetGoogle Scholar
- Wang W-C, Yang Q-H: Improved Hardy-Sobolev inequalities for radial derivative. Mathematical Inequalities and Applications 2011,14(1):203–210.MathSciNetView ArticleGoogle Scholar
- Xiao Y-X, Yang Q-H: An improved Hardy-Rellich inequality with optimal constant. Journal of Inequalities and Applications 2009, 2009:-10.Google Scholar
- Xiao Y-X, Yang Q-H: Some Hardy and Rellich type inequalities on anisotropic Heisenberg groups. preprintGoogle Scholar
- Yang Q: Best constants in the Hardy-Rellich type inequalities on the Heisenberg group. Journal of Mathematical Analysis and Applications 2008,342(1):423–431. 10.1016/j.jmaa.2007.12.014MATHMathSciNetView ArticleGoogle Scholar
- Yang Q: Improved Sobolev inequalities on groups of Iwasawa type in presence of symmetry. Journal of Mathematical Analysis and Applications 2008,341(2):998–1006. 10.1016/j.jmaa.2007.11.009MATHMathSciNetView ArticleGoogle Scholar
- Yang QH, Lian BS: On the best constant of weighted Poincaré inequalities. Journal of Mathematical Analysis and Applications 2011,377(1):207–215. 10.1016/j.jmaa.2010.10.027MATHMathSciNetView ArticleGoogle Scholar
- Jin Y: Hardy-type inequalities on H-type groups and anisotropic Heisenberg groups. Chinese Annals of Mathematics Series B 2008,29(5):567–574. 10.1007/s11401-006-0291-4MATHMathSciNetView ArticleGoogle Scholar
- Jin Y, Han Y: Weighted Rellich inequality on H-type groups and nonisotropic Heisenberg groups. Journal of Inequalities and Applications 2010, 2010:-17.Google Scholar
- Cohn WS, Lu G: Best constants for Moser-Trudinger inequalities on the Heisenberg group. Indiana University Mathematics Journal 2001,50(4):1567–1591. 10.1512/iumj.2001.50.2138MATHMathSciNetView ArticleGoogle Scholar
- Beals R, Gaveau B, Greiner PC: Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. Journal de Mathématiques Pures et Appliquées. Neuvième Série 2000,79(7):633–689.MATHMathSciNetView ArticleGoogle Scholar
- Folland GB, Stein EM: Hardy spaces on Homogeneous Groups, Mathematical Notes. Volume 28. Princeton University Press, Princeton, NJ, USA; 1982:xii+285.Google Scholar
This article is published under license to BioMed Central Ltd. 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.