- Research Article
- Open Access
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.
- Vector Field
- Unit Sphere
- Fundamental Solution
- Convergence Theorem
- Dimensional Vector
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.
where and .
Let and set . We will explicitly calculate the constant to show when . The method of calculation is similar to that used in .
where is the volume of , that is, the unit sphere in .
Firstly, we prove the following representation formula on , which is of its independent interest.
The proof is therefore completed.
We now prove the following Hardy inequalities on .
Set . Then, , and we get (3.11).
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
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