1. Introduction
The classical Hardy inequality
holds for
when
and for
when
. Hardy’s inequality plays an important role in analysis and has extensive applications in partial differential equations and physics. Since Hardy in [
1] firstly proved this inequality in the case of one dimension, many researchers devoted themselves to it and made great progress, not only in Euclidean spaces, there are many counterparts in Carnot groups and Riemannian manifolds, see [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12] and the references therein.
Davis and Hinz obtained in [
13] the following Rellich inequality
where
, and the constant
is sharp. It is a generalization to the second-order derivative of Hardy inequality. In [
14], Tertikas and Zographopoulos obtained a Hardy–Rellich type inequality
where
,
, and
The constant
was proven to be sharp. Particularly, when
,
, whereas when
,
. Furthermore, a new proof of inequality (
3) for
was given in [
15]. Particularly, the author proved that, for
, the sharp constants
are
and 3, respectively. Moreover, the authors in [
16] obtained an improved Hardy–Rellich inequality associated with operators
where
is a non-negative, self-adjoint operator on
.
In recent years, there has been considerable interest in studying the Hardy-type inequality for Dunkl operators. It is well known that there exists a constant such that the following Hardy inequality
holds for all
(see [
17,
18,
19]). When
,
is defined in
Section 2, the best constant
was obtained by different method in [
17,
18]. When
, Velicu proved in [
19] for any
and
, the following sharp inequality
holds. This results is based on a
norm comparison of
and
for any
which is obtained in [
20] by investigating the carré-du-champ operator. For any
, the authors in [
17,
19] get explicit constants
and
, respectively. However, the best constant of
Dunkl–Hardy inequality for any
is still an open question.
The author in [
19] also obtained a Rellich inequality for Dunkl–Laplacian
and the constant
is sharp.
In this paper, we proved the following weighted Hardy–Rellich inequality for Dunkl operators with an explicit constant
It is an extension of inequality (
3) in the case of Dunkl operators. In [
18], the authors proved that for
and
the best constants of inequality (
7) are, respectively,
(
) and
(
). When
, we have
for any
, where
is defined as the largest real zero points of cubic function
Note that
, so our result improved the inequality in [
18]. When
and
,
and
degenerate, respectively, to
and ∇, and the inequality (
7) return to the inequality (
3).
The plan of this paper is as follows. In
Section 2, we introduce some definitions and basic conceptions of Dunkl operators. In
Section 3, we obtained weighted Hardy–Rellich type inequalities for Dunkl operators by using the spherical h-harmonic decomposition.
2. Dunkl Operators
In this section, we will introduce some fundamental concepts and notations of Dunkl operators, see also [
21,
22] for more details.
We call
R a root system, if
is a finite set such that
and
for any
, then denote
as a reflection on the hyperplane which is orthogonal to the root
, written as
We write
G as the group generated by all the reflections
for
, it is a finite group. Let
be a G-invariant function, i.e.,
for all
and all
, simply written
.
R can be decomposed as
, when
, then
, and
is called a positive subsystem. Fix a positive subsystem
in a root system
R. Without loss of generality, we assume that for all
,
. For
the Dunkl operators on
is defined as
By this definition, we can see that even if the decomposition of
R is not unique, the different choices of positive subsystems make no difference in the definitions due to the G-invariance of
k. Denote by
the Dunkl gradient,
the Dunkl–Laplacian. Especially, for
we have
and
. The Dunkl–Laplacian can be written in terms of the usual gradient and Laplacian as follows,
The weight function naturally associated with Dunkl operators is
This is a homogeneous function of degree
, where
We will work in spaces
, where
is the weighted measure. For this weighted measure, we have a formula of integration by parts
If at least one of the functions
u,
v is G-invariant, the following Leibniz rule holds.
Spherical h-harmonics. We introduce some concepts and basic facts for spherical
h-harmonic theory, see [
21] for more details. then we called homogeneous polynomial
p of degree
n an
h-harmonic polynomial of degree
n if it satisfies
Spherical
h-harmonics (or shortly
h-harmonics) of degree
n are the restrictions of
h-harmonic polynomials of degree
n to the unit sphere
. We denote the space of
h-harmonics of degree
n as
. Denote the dimension of
as
, which is finite and given by following equation:
Furthermore, one can decompose the space as the orthogonal direct sum of the spaces , for .
Let
be a set of orthogonal basis of
, In the spherical polar coordinates
, for
and
, we can write the Dunkl–Laplacian as
where
is a generalization of the classical Laplace = Beltrami operator on the sphere, which only acts on the
variable. The spherical h-harmonics
are all eigenfunctions of
, and the corresponding eigenvalues are given by
The
h-harmonic expansion of a function
can be expressed as
where
and
is the surface measure on the sphere
.
3. Hardy–Rellich Type Inequalities for Dunkl Operators
In this section, we prove the weighted Hardy–Rellich inequalities for Dunkl operators.
Theorem 1. Assume , and . If one of the following conditions is satisfied:
(1) , and ;
(2) , and ;
(3) , , andthen, for any , the following inequality holds Remark 1. Note that when , the functionit follows that Thus, Theorem 1 improves the results given in [18]. Remark 2. , , so . This means that the condition is reasonable when in inequality (8). Remark 3. Especially, we have . Therefore, inequality (8) holds for any . Proposition 1. Assume , then .
Proof. We only need to prove
. By direct computation we have
Since , it is clear that . □
We prove firstly an estimate of the right-hand side of inequality (
8) which is different from the result for Euclidean gradient. In fact, in the case of the Euclidean gradient, the following inequality (
9) is exactly an equality for any
.
Lemma 1. For any , we have the inequalities:
Proof. By integration by parts,
where
Inserting (
13) into (
12),
When
, since
and
are G-invariant, by Hölder’s inequality we have
It follows from (
14) and (
15),
By using the spherical decomposition for Dunkl operators,
When
, inequality (
10) can be obtained similarly. On the other hand, by spherical decomposition we have
where
where
is the spherical measure. Note that
is G-invariant, by a change of variables
, we obtain
From Parseval’s identity, we have
By (
16) and spherical decomposition of Dunkl operators, we have
□
Now it’s time to prove Theorem 1.
Proof of Theorem 1. Since
when
, by Lemma 1 we have
Using the following weighted Hardy inequality
we have
Let
, then
. Taking
, we have
If
,
, then
. When
,
, then
if
, thus
By integration by parts, we obtain
since
, then
.
Denote
then from inequality (
17) we have
For
,
so we get
.
For
, take
, from inequality (
18) we get
where
If
, we can rewrite
as
Thus,
if
. Moreover,
can be seen as a quadratic function of
. Since
for
and
, we have
So
for any
, i.e.,
for any
. Thus, the inequality (
8) holds.
If
, we can also have
for
Moreover,
if
or
and
. Computing directly we have
Combining all the arguments above, we obtain the inequality (
8).
Next we prove the optimality of the constant
. For any
, take
Recall that
, calculating directly we have
□
Theorem 2. Assume , , and . Then, for any , we have the inequalitywhere Proof. When
, choosing
, then we have from the inequalities (
18) and (
19)
where
Firstly we can rewrite
as
From
we have
if and only if
. Taking
, then,
for any
.
On the other hand, for any
,
we have
. Thus,
, which implies that inequality (
20) holds. □
Remark 4. If , , thenwhich recovers the results in [15].