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Some Weighted Hardy-Type Inequalities on Anisotropic Heisenberg Groups
Journal of Inequalities and Applications volume 2011, Article number: 924840 (2011)
Abstract
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.
1. Introduction
The Hardy inequality in states that, for all and ,
In the case of the Heisenberg group , Garofalo and Lanconelli (cf. [1]) firstly proved the following Hardy inequality:
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 [2]). Inequality (1.2) was generalized by Niu et al. [3] (see also [4]) using the Picone-type identify. For more Hardy-Sobolev inequalities on nilpotent groups, we refer the reader to [5–19].
More recently, Jin and Han (cf. [20, 21]), using the method by Niu et al. [3], have proved the following Hardy inequalities on anisotropic Heisenberg groups :
where are the nonisotropic Greiner-type vector fields, is a positive integer,
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. [22]). 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
Recall that the anisotropic Heisenberg groups are the Carnot group of step two whose group structure is given by (cf. [23])
where , (), and are positive constants, numbered so that
We consider the following nonisotropic Greiner-type vector fields which are introduced by Jin and Han [21]:
(). These vector fields are not left or right invariant when . The horizontal gradient is the dimensional vector given by
A natural family of anisotropic dilations related to is
For simplicity, we denote by . The Jacobian determinant of is , where is the homogenous dimension. The anisotropic norm on is
For simplicity, we use the notation and . Then,
and . With this norm, we can define the metric ball centered at neutral element and with radius by
and the unit sphere . Furthermore, we have the following polar coordinates for all (cf. [24]):
where and .
Let and set . We will explicitly calculate the constant to show when . The method of calculation is similar to that used in [22].
Lemma 2.1.
For ,
where is the volume of , that is, the unit sphere in .
Proof.
To compute , let , then,
Next, if ,
Therefore,
Thus, if ,
3. Hardy-Type Inequality
Firstly, we prove the following representation formula on , which is of its independent interest.
Lemma 3.1.
Let and . Then,
where is a diagonal matrix given by
Proof.
We argue as in the proof of Theorem 1.2 in [22]. Since ,
Therefore,
Notice that
we have, by (3.4),
To finish the proof, it is enough to show that
vanishes. Notice that the operator annihilates functions of , and, for , the integrand above is absolutely integrable. We have, for any , though integration by parts,
Let . By dominated convergence theorem,
The proof is therefore completed.
We now prove the following Hardy inequalities on .
Theorem 3.2.
Let and . There holds, for all ,
Proof.
Set with . Replacing by in Lemma 3.1, we obtain, for any ,
It is easy to check that the following equations hold
Therefore, by (3.11),
By dominated convergence, letting , we have
By Hölder's inequality,
Canceling and raising both sides to the power , we obtain
Set . Then, , and we get (3.11).
Remark 3.3.
Notice that , we have, by Theorem 3.2, for all ,
From inequality (3.17), we have the following corollary which generalizes the result of [21] when and .
Corollary 3.4.
Let , and . There holds, for all ,
Proof.
Since ,
We have, by inequality (3.17),
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Acknowledgments
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).
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Lian, BS., Yang, QH. & Yang, F. Some Weighted Hardy-Type Inequalities on Anisotropic Heisenberg Groups. J Inequal Appl 2011, 924840 (2011). https://doi.org/10.1155/2011/924840
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DOI: https://doi.org/10.1155/2011/924840