Weighted Rellich Inequality on H-Type Groups and Nonisotropic Heisenberg Groups

The study of partial differential operators constructed from noncommutative vector fields satisfying the Hörmander condition [1] has had much development. We refer to [2, 3] and the references therein for a systematic account of the study. Recently there have been considerable interests in studying the sub-Laplacians as square sums of vector fields that are not invariant or do not satisfy the Hörmander condition. Among the examples of such sub-Laplacians are the Grushin operators, Greiner-type operators, and the sub-Laplacian constructed by Kohn [4]. Those noninvariant sub-Laplacians also appear naturally in complex analysis. In [5] Beals et al. considered the CR operators on as boundary of the complex domain

(1.1)

where ,

(1.2)

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 [6] 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 [7] as direct generalizations of Heisenberg groups. In [8] 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 .

The goal of the present paper is to continue our study on analysis associated with Greiner-type vector fields introduced in [8]. We will throughout study the Rellich inequality which is a generalization of Hardy inequality to higher-order derivatives. They have various applications in the study of elliptic and parabolic PDEs. The classical Rellich inequality in states that for and ,

(1.3)

The constant is sharp and is never achieved. Davies and Hinz [9] generalized (1.3) to the case and showed that for any there holds

(1.4)

See also the works in [10–13].

In a recent paper [14] 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 [15] 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 [8]. 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.

We recall that a simply connected nilpotent group is of Heisenberg type, or of H-type, if its Lie algebra is of step two, and if there is an inner product on such that the linear map

(2.1)

defined by the condition

(2.2)

satisfies

(2.3)

for all where

Groups of H-type were introduced by Kaplan in [7] as direct generalizations of Heisenberg groups, and they have been studied quite extensively; see [16–19] and the references therein.

We identify with its Lie algebra via the exponential map, The Lie group product is given by

(2.4)

For we write

In [8] the authors constructed a family of Greiner-type vector fields on :

(2.5)

where , are the directional derivatives, is an orthonormal basis of , and is a fixed parameter. If are the left-invariant vector fields defined by the orthonormal basis on The corresponding degenerate -sub-Laplacian is

(2.6)

where

(2.7)

We denote

(2.8)

There is a one-parameter group of dilations on :

(2.9)

The volume element is transformed by via

(2.10)

where with and    will be called the degree of homogeneity and is the homogeneous dimension if . We define a corresponding homogeneous norm by

(2.11)

Remark 2.1.

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 [8] are the following.

Theorem 2.2.

Let be an H-type group, , and Then for ,

(2.12)

is a fundamental solution of with singularity at the identity element . Here is defined in (2.11),

(2.13)

Theorem 2.3.

Let be an H-type group, , and Suppose that Then the following Hardy-type inequality holds for :

(2.14)

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.

Theorem 2.4.

Let be an H-type group, , and Suppose that Then the following Rellich-type inequality holds for :

(2.15)

Moreover, the constant is sharp.

Remark 2.5.

In the abelian case the above result recovers the classical Rellich inequality (1.4) with under the condition

Remark 2.6.

When we take and our inequality (2.15) is just inequality (1.5) in [14] and inequality (5.2) in [15] for Heisenberg groups.

Now we prove Theorem 2.4.

Proof.

We denote , where is as in (2.11), then

(2.16)

for ; we have

(2.17)

since is the fundamental solution of at the origin by Theorem 2.2; however the left-hand side is

(2.18)

Thus, by (2.17), (2.18), and the corresponding weighted Hardy inequality (Theorem 2.3), we have

(2.19)

this implies that

(2.20)

Applying Hölder's inequality, we get

(2.21)

Noticing that

(2.22)

we have thus proved (2.15).

It remains to show the sharpness of the constant . Let be any constant satisfying the inequality

(2.23)

We will prove that . The idea is to find functions so that the difference between the left- and right-hand sides approximates to . Given any positive integer it is elementary that there exists in such that , on , and on , where is a constant independent of . Let

(2.24)

Clearly which is radial. Denoting by , it is easy to see that

(2.25)

Here we used the fact that

(2.26)

The left-hand side of the above inequality (2.23) is

(2.27)

The first integration is

(2.28)

which can be evaluated as the last computations in the proof of Theorem in [8], and is

(2.29)

where is a positive constant. Similarly,

(2.30)

The first integration is precisely the same as above and is

(2.31)

It is easy to estimate the error terms which they are all bounded:

(2.32)

The inequality (2.23) now becomes

(2.33)

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.

Corollary 2.7.

Let be a Heisenberg-type group, , and Suppose that Then for all the following inequality holds:

(2.34)

Remark 2.8.

We mention that when and our inequality (2.34) goes back to inequality (5.7) in [15] 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 [21] and the weighted Hardy inequality (Theorem ) in [22].

Theorem 2.9.

Let be a polarizable group with homogeneous dimension Suppose that Then the following inequality holds for all :

(2.35)

Here is the homogeneous norm associated with the fundamental solution for the Kohn sub-Laplacian. Moreover, the constant is sharp.

Let Let be the corresponding nonisotropic Heisenberg group with the product

(3.1)

We consider the following nonisotropic Greiner-type vector fields:

(3.2)

where Denote

(3.3)

For further information on the nonisotropic Heisenberg group, see, for example, [23, 24].

In this section we prove firstly the Hardy-type inequality associated with the Greiner-type vector fields (3.2) on the nonisotropic Heisenberg group

Theorem 3.1.

Let be the anisotropic Heisenberg group with Let and Then the following inequality is valid:

(3.4)

where

For the proof of the above inequality, we need the following lemma which can be proved by a similar method in [25].

Lemma 3.2.

Let be a weight function in and Suppose that for some there exists such that

(3.5)

for some , in the sense of distribution acting on nonnegative test functions. Then for any it holds that

(3.6)

where denote the closure of in the norm

We now prove Theorem 3.1.

Proof.

Take and . Noting that

(3.7)

we get

(3.8)

where

(3.9)

By direct computations we have

(3)

Then

(3.11)

where

(3.12)

Using Cauchy inequality

(3.13)

and that (since, by assumption, )

(3.14)

we find

(3.15)

so for ,

(3.16)

Hence Theorem 3.1 follows from Lemma 3.2 with .

Now it is time to prove the following Rellich inequality on nonisotropic Heisenberg group.

Theorem 3.3.

Let be a nonisotropic Heisenberg group with Suppose that and Then the following inequality is valid:

(3.17)

where

Remark 3.4.

If , then we get the Rellich inequality on the Heisenberg group with homogeneous dimension :

(3.18)

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.

Proof.

We have

(3.19)

By a direct computation we have

(3.20)

Noting that then

(3.21)

we take and thus deduce from (3.19) that

(3.22)

On the other hand,

(3.23)

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.

Corollary 3.5.

Let be an anisotropic Heisenberg group with Then for the following inequality holds:

(3.24)

where

By a similar method, we can get the following inequality which does not contain the weight so we omit the proof.

Theorem 3.6.

Let be a nonisotropic Heisenberg group with Let and Then the following inequality is valid:

(3.25)

With the help of Theorem 3.6, we can also obtain a Rellich-Sobolev-type inequality on the Heisenberg group.

Corollary 3.7.

Let be the Heisenberg group with homogeneous dimension Suppose that Then there exists a positive constant such that for any the following inequality holds:

(3.26)

Proof.

By Hölder inequality we have

(3.27)

Thanks to the Rellich inequality (3.25) with and the Sobolev inequality on the nilpotent Lie group (Chapter IV Theorem 3.3.1 in [26]) we get

(3.28)

Thus we obtain the desired result.