- Research Article
- Open Access
Weighted Rellich Inequality on H-Type Groups and Nonisotropic Heisenberg Groups
© Yongyang Jin and Yazhou Han. 2010
- Received: 19 March 2010
- Accepted: 15 June 2010
- Published: 30 June 2010
We prove a sharp weighted Rellich inequality associated with a class of Greiner-type vector fields on H-type groups. We also obtain some weighted Hardy- and Rellich-type inequalities on nonisotropic Heisenberg groups. As an application, we get a Rellich-Sobolev-type inequality on Heisenberg groups.
- Heisenberg Group
- Homogeneous Dimension
- Heisenberg Type
- Rellich Inequality
- Weighted Hardy Inequality
and is a positive integer. The space has a natural structure of a Heisenberg group, but the vector fields are not left or right invariant. In  Zhang and Niu studied the Greiner vector fields on for general parameter and got the corresponding Hardy-type inequality. Note that for nonintegral these vector fields do not satisfy the Hörmander condition and are not smooth.
H-type groups were introduced by Kaplan  as direct generalizations of Heisenberg groups. In  we define a class of vector fields (see (2.5)) on H-type groups generalizing the vector fields (1.2) considered in [5, 6] and find the fundamental solution of the corresponding -Laplacian with singularity at the identity element. Also we prove a Hardy-type inequality associated to .
In a recent paper  Yang obtains an version of Rellich inequality associated with the left-invariant vector fields in the setting of Heisenberg group, there is a similar Rellich inequality on the general carnot group in  with different approach. A natural question is to find an version of Rellich inequality in this general setting. The main purpose of the present paper is to prove some weighted -Rellich inequalities associated with Greiner-type vector fields on H-type groups. Our approach depends on the fundamental solution of the corresponding square operator and the weighted Hardy inequality proved in our earlier paper . We prove also some weighted Hardy and Rellich inequalities on nonisotropic Heisenberg groups by a different method caused by the absence of the explicit representation formula for fundamental solution. As an application, we get a Rellich-Sobolev-type inequality on Heisenberg groups.
The plan of the paper is as follows. In Section 2 we introduce a class of Greiner-type vector field and prove the corresponding weighted -Rellich inequality on H-type groups; Section 3 is devoted to the proof of weighted Hardy- and Rellich-type inequalities on nonisotropic Heisenberg groups and a Rellich-Sobolev-type inequality on Heisenberg groups.
for all where
For we write
Note that when and becomes the sub-Laplacian on the H-type group If , and , is a Greiner operator (see [5, 20]). Also we note that vector fields are neither left nor right invariant and they do not satisfy Hörmander's condition for
The main results in  are the following.
where is the gradient defined by the vector fields (2.5). Moreover, the constant is sharp.
Based on the above two theorems, we will prove the following version of weighted Rellich inequality on H-type groups.
Moreover, the constant is sharp.
In the abelian case the above result recovers the classical Rellich inequality (1.4) with under the condition
Now we prove Theorem 2.4.
we have thus proved (2.15).
Dividing both sides by and letting prove our claim.
The following is an immediate consequence of Theorem 2.4, which is an extension of the uncertainty principle inequality.
We mention that when and our inequality (2.34) goes back to inequality (5.7) in  in the setting of H-type group.
We end this section with the following Rellich-type inequality on the polarizable group which can be proved by the same method if we noted Theorem and Proposition in  and the weighted Hardy inequality (Theorem ) in .
Here is the homogeneous norm associated with the fundamental solution for the Kohn sub-Laplacian. Moreover, the constant is sharp.
In this section we prove firstly the Hardy-type inequality associated with the Greiner-type vector fields (3.2) on the nonisotropic Heisenberg group
For the proof of the above inequality, we need the following lemma which can be proved by a similar method in .
where denote the closure of in the norm
We now prove Theorem 3.1.
Hence Theorem 3.1 follows from Lemma 3.2 with .
Now it is time to prove the following Rellich inequality on nonisotropic Heisenberg group.
We also mention that to our knowledge, even in the special case our inequality (3.17) is new.
We now give the proof of Theorem 3.3.
Combining with (3.22), (3.23), and the weighted Hardy inequality associated with the Greiner-type vector fields (3.2) on the nonisotropic Heisenberg group (3.4), then we only need to do the same steps as in Theorem 2.4 and thus finish the proof of Theorem 3.3.
The following is also an uncertainty principle type inequality which is an immediate consequence of Theorem 3.3.
By a similar method, we can get the following inequality which does not contain the weight so we omit the proof.
With the help of Theorem 3.6, we can also obtain a Rellich-Sobolev-type inequality on the Heisenberg group.
Thus we obtain the desired result.
The authors would like to thank Professor Genkai Zhang for constant encouragement and the referee for his/her helpful comments. The research was supported by the Natural Science Foundation of Zhejiang Province (no. Y6090359, Y6090383) and Department of Education of Zhejiang Province (no. Z200803357).
- Hörmander L: Hypoelliptic second order differential equations. Acta Mathematica 1967, 119: 147–171. 10.1007/BF02392081MathSciNetView ArticleMATHGoogle 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
- Nagel A, Stein EM, Wainger S: Balls and metrics defined by vector fields. I. Basic properties. Acta Mathematica 1985, 155(1–2):103–147.MathSciNetView ArticleMATHGoogle Scholar
- Kohn JJ: Hypoellipticity and loss of derivatives. Annals of Mathematics. Second Series 2005, 162(2):943–986. With an appendix by Makhlouf Derridj and David S. Tartakoff 10.4007/annals.2005.162.943MathSciNetView ArticleMATHGoogle Scholar
- Beals R, Gaveau B, Greiner P: Uniforms hypoelliptic Green's functions. Journal de Mathématiques Pures et Appliqués 1998, 77(3):209–248.MathSciNetView ArticleMATHGoogle Scholar
- Zhang H, Niu P: Hardy-type inequalities and Pohozaev-type identities for a class of -degenerate subelliptic operators and applications. Nonlinear Analysis: Theory, Methods & Applications 2003, 54(1):165–186. 10.1016/S0362-546X(03)00062-2MathSciNetView ArticleMATHGoogle Scholar
- Kaplan A: Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Transactions of the American Mathematical Society 1980, 258(1):147–153. 10.1090/S0002-9947-1980-0554324-XMathSciNetView ArticleMATHGoogle Scholar
- Jin Y, Zhang G: Fundamental solutions for a class of degenerate p-Laplacian operators and applications to Hardy type inequalities. to appear in Canadian Journal of MathematicsGoogle Scholar
- Davies EB, Hinz AM: Explicit constants for Rellich inequalities in . Mathematische Zeitschrift 1998, 227(3):511–523. 10.1007/PL00004389MathSciNetView ArticleMATHGoogle Scholar
- Adimurthi , Grossi M, Santra S: Optimal Hardy-Rellich inequalities, maximum principle and related eigenvalue problem. Journal of Functional Analysis 2006, 240(1):36–83. 10.1016/j.jfa.2006.07.011MathSciNetView ArticleMATHGoogle Scholar
- Barbatis G: Best constants for higher-order Rellich inequalities in . Mathematische Zeitschrift 2007, 255(4):877–896. 10.1007/s00209-006-0056-5MathSciNetView ArticleMATHGoogle Scholar
- Barbatis G, Tertikas A: On a class of Rellich inequalities. Journal of Computational and Applied Mathematics 2006, 194(1):156–172. 10.1016/j.cam.2005.06.020MathSciNetView ArticleMATHGoogle Scholar
- Tertikas A, Zographopoulos NB: Best constants in the Hardy-Rellich inequalities and related improvements. Advances in Mathematics 2007, 209(2):407–459. 10.1016/j.aim.2006.05.011MathSciNetView ArticleMATHGoogle 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.2007.12.014MathSciNetView ArticleMATHGoogle Scholar
- Kombe I: Hardy, Rellich and uncertainty principle inequalities on Carnot groups. http://arxiv.org/abs/math/0611850
- Cowling M, Dooley A, Korányi A, Ricci F: An approach to symmetric spaces of rank one via groups of Heisenberg type. The Journal of Geometric Analysis 1998, 8(2):199–237.MathSciNetView ArticleMATHGoogle Scholar
- Cowling M, Dooley AH, Korányi A, Ricci F: -type groups and Iwasawa decompositions. Advances in Mathematics 1991, 87(1):1–41. 10.1016/0001-8708(91)90060-KMathSciNetView ArticleMATHGoogle Scholar
- Kaplan A: Lie groups of Heisenberg type. Rendiconti del Seminario Matematico. Università e Politecnico di Torino 1983, 117–130. Conference on Differential Geometry on Homogeneous Spaces (Turin, 1983)Google Scholar
- Korányi A: Geometric properties of Heisenberg-type groups. Advances in Mathematics 1985, 56(1):28–38. 10.1016/0001-8708(85)90083-0MathSciNetView ArticleMATHGoogle Scholar
- Greiner PC: A fundamental solution for a nonelliptic partial differential operator. Canadian Journal of Mathematics 1979, 31(5):1107–1120. 10.4153/CJM-1979-101-3MathSciNetView ArticleMATHGoogle Scholar
- Balogh ZM, Tyson JT: Polar coordinates in Carnot groups. Mathematische Zeitschrift 2002, 241(4):697–730. 10.1007/s00209-002-0441-7MathSciNetView ArticleMATHGoogle Scholar
- Kombe I: Sharp Hardy type inequalities on the Carnot group. http://arxiv.org/abs/math/0501522
- Chang D-C, Greiner P: Harmonic analysis and sub-Riemannian geometry on Heisenberg groups. Bulletin of the Institute of Mathematics. Academia Sinica 2002, 30(3):153–190.MathSciNetMATHGoogle Scholar
- Chang DC, Tie JZ: A note on Hermite and subelliptic operators. Acta Mathematica Sinica 2005, 21(4):803–818. 10.1007/s10114-004-0336-0MathSciNetView ArticleMATHGoogle 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/S0002-9939-01-06011-7MathSciNetView ArticleMATHGoogle Scholar
- Varopoulos NTh, Saloff-Coste L, Coulhon T: Analysis and Geometry on Groups, Cambridge Tracts in Mathematics. Volume 100. Cambridge University Press, Cambridge, Mass, USA; 1992:xii+156.MATHGoogle Scholar
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