Some Weighted Hardy-Type Inequalities on Anisotropic Heisenberg Groups

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 [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 :

(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.

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)

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)