Abstract
The relation between Riesz potential and heat kernel on the Heisenberg group is studied. Moreover, the Hardy-Littlewood-Sobolev inequality is established.
Journal of Inequalities and Applications volume 2011, Article number: 498638 (2011)
The relation between Riesz potential and heat kernel on the Heisenberg group is studied. Moreover, the Hardy-Littlewood-Sobolev inequality is established.
The classical Riesz potential is defined on by
where is the Laplacian operator. By virtue of the equations
where , one can get the explicit expression of Riesz potential
In addition, one has the following Hardy-Littlewood-Sobolev theorem (see [1]).
Theorem 1.1.
Let , , . One has the following.
(a)If , then ;
(b)if , then for all ,
In recent years many interesting works about the Riesz potential have been done by many authors. Thangavelu and Xu [2] discussed the Riesz potential for the Dunkl transform. Garofalo and Tyson [3] proved superposition principle Riesz potentials of nonnegative continuous function on Lie groups of Heisenberg type. Huang and Liu [4] studied the Hardy-Littlewood-Sobolev inequality of this operator on the Laguerre hypergroup. For more results about the Riesz potential, we refer the readers to see [5–9].
It is a remarkable fact that the Heisenberg group, denoted by , arises in two aspects. On the one hand, it can be realized as the boundary of the unit ball in several complex variables. On the other hand, an important aspect of the study of the Heisenberg group is the background of physics, namely, the mathematical ideas connected with the fundamental notions of quantum mechanics. In other words, there is its genesis in the context of quantum mechanics which emphasizes its symplectic role in the theory of theta functions and related parts of analysis. Due to this reason, many interesting works were devoted to the theory of harmonic analysis on in [10–15] and the references therein.
In present paper, we consider the Riesz potential associated with the Heisenberg group. We will show a connection between the Riesz potential and the heat kernel, and then get the Hardy-Littlewood-Sobolev inequality.
The Heisenberg group is a Lie group with the underlying manifold , the multiplication law is
where . The dilation of is defined by with . For , the homogeneous norm of is given by
Note that . In addition, satisfies the quasi-triangle inequality
The ball of radius centered at is given by
For , let be the space of measurable functions on , such that
Let be the Schrödinger representations which acts on by
where . Suppose that is a Schwartz function on , that is, . The Fourier transform of is defined by
This means that, for each ,
where denotes the inner product.
Let us write with and define
Then (2.7) can be written as
If we set
then . Let ; one has the inversion of Fourier transform
where denotes the adjoint of .
The convolution of and is defined by
It is clear that . In addition, we have the generalized Yong inequality
where . More details about the harmonic analysis on Heisenberg group can be found in [14–16].
Let be a mapping from to , , . Then is of type if
where does not depend on . Similarly, is of weak type if
where does not depend on or ().
Let be the unit sphere in and the unit Euclidean sphere in . Suppose that is a measurable function on , and we have (see [10])
We set , then
Hence
A direct calculation shows that the area of is
In addition, we have the volume of unit ball in
and thus the volume of
For a radial function , we have
The Hardy-Littlewood maximal operator is defined on by
which is of type for and is of weak type (1,1) (see [17, 18]).
As it is known, the following vector fields
form a basis for the Lie algebra of left-invariant vector fields on . The sublaplacian is defined by
which also has another explicit form
where is the standard Laplacian on and
For the Schrödinger representations one easily calculates that
So that .
Let () be the normalized Hermite functions given by
where . For and , we define
Then forms an orthonormal basis for .
We set and denote
Therefore,
Moreover, one has
Now let , then has the form
From (3.9) we know that the functions
are eigenfunctions of the operator :
Let be the Laguerre functions defined on by
and set for . Then from [19, (2.3.26)] we have
In view of this equation we have the following.
Proposition 3.1.
One has
where stands for the projection of onto the th eigenspace of , that is,
Now we consider the heat equation associated to the sublaplacian
with the initial condition . In fact, the function given by
is just the solution of the heat equation and satisfies
Moreover, we have the Fourier transform of (see [18, page 86])
In Section 1 we have recalled some properties about the Riesz potential on ; now we are going to discuss the Riesz potential on the Heisenberg group.
Definition 4.1.
For , the Riesz potential is defined on by
From above definition and (3.10) it is easy to see that
If , , then we have
which suggests that . Especially, for , one has
At present we do not prepare to gain the expression of analogues to (1.3) because it is hard to calculate the Fourier transform of . But the following theorem will give us another expression of , which provides a bridge to discuss the boundedness of the Riesz potential.
Theorem 4.2.
Let be the heat kernel on . For , one has for
Proof.
By (3.17), (3.21), and Proposition 3.1 we have
Then we get the desired result.
Lemma 4.3.
The heat kernel satisfies the estimate
with some positive constants and .
Proof.
Since , then by [19, Proposition 2.8.2] we obtain this lemma.
The following theorem is an immediate consequence of Theorem 4.2 and Lemma 4.3.
Theorem 4.4.
The Riesz potential satisfies the estimate
where is a positive constant.
Using Theorems 4.2 and 4.4, we get the Hardy-Littlewood-Sobolev theorem on the Heisenberg group.
Theorem 4.5.
Let , , and . For , one has the following.
(a)If , then is of type .
(b)If , then is of weak type .
Proof.
Let be the maximal operator defined by (2.24). We claim that
Let be the ball of radius centered at (0,0), and let be its characteristic function. We set
with . Obviously, is radial and decreasing, then we can write
where and is the ball centered at origin. By (2.23) and (2.24) we have
Let be the conjugate exponent of . Since , we have
Then by Hölder's inequality we get
Therefore,
We choose such that
That is, . Substituting this in the above then gives (4.9).
Now if , we obtain (a) by the virtue of the type of the maximal operator . If and , we have
This proves our main theorem.
Theorem 4.6.
Let and . For , one has the following.
(a)If , then the condition is necessary and sufficient for the weak type of .
If , then the condition is necessary and sufficient for the type of .
Proof.
The sufficiency follows from Theorem 4.5. We now begin to prove the necessity. Suppose that , and let . Note that
Then by (4.9) we get
Thus for , we have
Case 1 ().
It follows from the hypothesis that
If , then as . If , then as . Thus we have .
Case 2 ().
Similarly, we have
which implies that .
Then we complete the proof of this theorem.
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The work for this paper is supported by the National Natural Science Foundation of China (no. 10971039) and the Doctoral Program Foundation of the Ministry of China (no. 200810780002). The authors are also grateful for the referee for the valuable suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Xiao, J., He, J. Riesz Potential on the Heisenberg Group. J Inequal Appl 2011, 498638 (2011). https://doi.org/10.1155/2011/498638
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DOI: https://doi.org/10.1155/2011/498638