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Weighted Rellich Inequality on H-Type Groups and Nonisotropic Heisenberg Groups
Journal of Inequalities and Applications volume 2010, Article number: 158281 (2010)
Abstract
We prove a sharp weighted Rellich inequality associated with a class of Greiner-type vector fields on H-type groups. We also obtain some weighted Hardy- and Rellich-type inequalities on nonisotropic Heisenberg groups. As an application, we get a Rellich-Sobolev-type inequality on Heisenberg groups.
1. Introduction
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

where ,

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
,

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

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.
2. Rellich Inequality on H-Type 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

defined by the condition

satisfies

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

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

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

where

We denote

There is a one-parameter group of dilations on
:

The volume element is transformed by via

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

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
,

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

Theorem 2.3.
Let be an H-type group,
, and
Suppose that
Then the following Hardy-type inequality holds for
:

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
:

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

for ; we have

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

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

this implies that

Applying Hölder's inequality, we get

Noticing that

we have thus proved (2.15).
It remains to show the sharpness of the constant . Let
be any constant satisfying the inequality

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

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

Here we used the fact that

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

The first integration is

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

where is a positive constant. Similarly,

The first integration is precisely the same as above and is

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

The inequality (2.23) now becomes

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:

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
:

Here is the homogeneous norm associated with the fundamental solution
for the Kohn sub-Laplacian. Moreover, the constant
is sharp.
3. Rellich Inequality on Nonisotropic Heisenberg Groups
Let Let
be the corresponding nonisotropic Heisenberg group with the product

We consider the following nonisotropic Greiner-type vector fields:

where Denote

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:

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

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

where denote the closure of
in the norm
We now prove Theorem 3.1.
Proof.
Take and
. Noting that

we get

where

By direct computations we have

Then

where

Using Cauchy inequality

and that (since, by assumption, )

we find

so for ,

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:

where
Remark 3.4.
If , then we get the Rellich inequality on the Heisenberg group
with homogeneous dimension
:

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

By a direct computation we have

Noting that then

we take and thus deduce from (3.19) that

On the other hand,

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:

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:

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:

Proof.
By Hölder inequality we have

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

Thus we obtain the desired result.
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Acknowledgments
The authors would like to thank Professor Genkai Zhang for constant encouragement and the referee for his/her helpful comments. The research was supported by the Natural Science Foundation of Zhejiang Province (no. Y6090359, Y6090383) and Department of Education of Zhejiang Province (no. Z200803357).
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Jin, Y., Han, Y. Weighted Rellich Inequality on H-Type Groups and Nonisotropic Heisenberg Groups. J Inequal Appl 2010, 158281 (2010). https://doi.org/10.1155/2010/158281
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DOI: https://doi.org/10.1155/2010/158281
Keywords
- Heisenberg Group
- Homogeneous Dimension
- Heisenberg Type
- Rellich Inequality
- Weighted Hardy Inequality