- Research Article
- Open Access

# Riesz Potential on the Heisenberg Group

- Jinsen Xiao
^{1}and - Jianxun He
^{1}Email author

**2011**:498638

https://doi.org/10.1155/2011/498638

© J. Xiao and J. He 2011

**Received:**24 November 2010**Accepted:**17 February 2011**Published:**9 March 2011

## Abstract

The relation between Riesz potential and heat kernel on the Heisenberg group is studied. Moreover, the Hardy-Littlewood-Sobolev inequality is established.

## Keywords

- Heat Kernel
- Maximal Operator
- Heisenberg Group
- Theta Function
- Weak Type

## 1. Introduction

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.

## 2. Preliminaries

where denotes the inner product.

where denotes the adjoint of .

where . More details about the harmonic analysis on Heisenberg group can be found in [14–16].

where does not depend on or ( ).

which is of type for and is of weak type (1,1) (see [17, 18]).

## 3. The Sublaplacian and the Heat Kernel on the Heisenberg Group

So that .

Then forms an orthonormal basis for .

In view of this equation we have the following.

Proposition 3.1.

## 4. Riesz Potential on the Heisenberg Group

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.

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.

Proof.

Then we get the desired result.

Lemma 4.3.

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.

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.

That is, . Substituting this in the above then gives (4.9).

This proves our main theorem.

Theorem 4.6.

Let and . For , one has the following.

- (b)
If , then the condition is necessary and sufficient for the type of .

Proof.

Case 1 ( ).

If , then as . If , then as . Thus we have .

Case 2 ( ).

which implies that .

Then we complete the proof of this theorem.

## Declarations

### Acknowledgments

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.

## Authors’ Affiliations

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