Riesz Potential on the Heisenberg Group
© J. Xiao and J. He 2011
Received: 24 November 2010
Accepted: 17 February 2011
Published: 9 March 2011
The relation between Riesz potential and heat kernel on the Heisenberg group is studied. Moreover, the Hardy-Littlewood-Sobolev inequality is established.
In addition, one has the following Hardy-Littlewood-Sobolev theorem (see ).
In recent years many interesting works about the Riesz potential have been done by many authors. Thangavelu and Xu  discussed the Riesz potential for the Dunkl transform. Garofalo and Tyson  proved superposition principle Riesz potentials of nonnegative continuous function on Lie groups of Heisenberg type. Huang and Liu  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.
3. The Sublaplacian and the Heat Kernel on the Heisenberg Group
In view of this equation we have the following.
4. Riesz Potential on the Heisenberg Group
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.
Then we get the desired result.
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.
Using Theorems 4.2 and 4.4, we get the Hardy-Littlewood-Sobolev theorem on the Heisenberg group.
This proves our main theorem.
Then we complete the proof of this theorem.
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.
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