# Some Weighted Hardy-Type Inequalities on Anisotropic Heisenberg Groups

- Bao-Sheng Lian
^{1}Email author, - Qiao-Hua Yang
^{2}and - Fen Yang
^{1}

**2011**:924840

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

© Bao-Sheng Lian et al. 2011

**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

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].

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

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.

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

Proof.

## 3. Hardy-Type Inequality

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

Lemma 3.1.

Proof.

The proof is therefore completed.

We now prove the following Hardy inequalities on .

Theorem 3.2.

Proof.

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

Remark 3.3.

From inequality (3.17), we have the following corollary which generalizes the result of [21] when and .

Corollary 3.4.

Proof.

## 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

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