Open Access

Some Weighted Hardy-Type Inequalities on Anisotropic Heisenberg Groups

Journal of Inequalities and Applications20112011:924840

https://doi.org/10.1155/2011/924840

Received: 9 December 2010

Accepted: 4 March 2011

Published: 15 March 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 ,
(1.1)
In the case of the Heisenberg group , Garofalo and Lanconelli (cf. [1]) firstly proved the following Hardy inequality:
(1.2)

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 [519].

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 :
(1.3)
where are the nonisotropic Greiner-type vector fields, is a positive integer,
(1.4)

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])
(2.1)
where , ( ), and are positive constants, numbered so that
(2.2)
We consider the following nonisotropic Greiner-type vector fields which are introduced by Jin and Han [21]:
(2.3)
( ). These vector fields are not left or right invariant when . The horizontal gradient is the dimensional vector given by
(2.4)
A natural family of anisotropic dilations related to is
(2.5)
For simplicity, we denote by . The Jacobian determinant of is , where is the homogenous dimension. The anisotropic norm on is
(2.6)
For simplicity, we use the notation and . Then,
(2.7)
and . With this norm, we can define the metric ball centered at neutral element and with radius by
(2.8)
and the unit sphere . Furthermore, we have the following polar coordinates for all (cf. [24]):
(2.9)

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 ,
(2.10)

where is the volume of , that is, the unit sphere in .

Proof.

To compute , let , then,
(2.11)
Next, if ,
(2.12)
Therefore,
(2.13)
Thus, if ,
(2.14)

3. Hardy-Type Inequality

Firstly, we prove the following representation formula on , which is of its independent interest.

Lemma 3.1.

Let and . Then,
(3.1)
where is a diagonal matrix given by
(3.2)

Proof.

We argue as in the proof of Theorem 1.2 in [22]. Since ,
(3.3)
Therefore,
(3.4)
Notice that
(3.5)
we have, by (3.4),
(3.6)
To finish the proof, it is enough to show that
(3.7)
vanishes. Notice that the operator annihilates functions of , and, for , the integrand above is absolutely integrable. We have, for any , though integration by parts,
(3.8)
Let . By dominated convergence theorem,
(3.9)

The proof is therefore completed.

We now prove the following Hardy inequalities on .

Theorem 3.2.

Let and . There holds, for all ,
(3.10)

Proof.

Set with . Replacing by in Lemma 3.1, we obtain, for any ,
(3.11)
It is easy to check that the following equations hold
(3.12)
Therefore, by (3.11),
(3.13)
By dominated convergence, letting , we have
(3.14)
By Hölder's inequality,
(3.15)
Canceling and raising both sides to the power , we obtain
(3.16)

Set . Then, , and we get (3.11).

Remark 3.3.

Notice that , we have, by Theorem 3.2, for all ,
(3.17)

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 ,
(3.18)

Proof.

Since ,
(3.19)
We have, by inequality (3.17),
(3.20)

Declarations

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).

Authors’ Affiliations

(1)
College of Science, Wuhan University of Science and Technology
(2)
School of Mathematics and Statistics, Wuhan University

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© Bao-Sheng Lian et al. 2011

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