Riesz Potential on the Heisenberg Group

The classical Riesz potential is defined on by

(1.1)

where is the Laplacian operator. By virtue of the equations

(1.2)

where , one can get the explicit expression of Riesz potential

(1.3)

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 ,

(1.4)

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

(2.1)

where . The dilation of is defined by with . For , the homogeneous norm of is given by

(2.2)

Note that . In addition, satisfies the quasi-triangle inequality

(2.3)

The ball of radius centered at is given by

(2.4)

For , let be the space of measurable functions on , such that

(2.5)

Let be the Schrödinger representations which acts on by

(2.6)

where . Suppose that is a Schwartz function on , that is, . The Fourier transform of is defined by

(2.7)

This means that, for each ,

(2.8)

where denotes the inner product.

Let us write with and define

(2.9)

Then (2.7) can be written as

(2.10)

If we set

(2.11)

then . Let ; one has the inversion of Fourier transform

(2.12)

where denotes the adjoint of .

The convolution of and is defined by

(2.13)

It is clear that . In addition, we have the generalized Yong inequality

(2.14)

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

(2.15)

where does not depend on . Similarly, is of weak type if

(2.16)

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])

(2.17)

We set , then

(2.18)

Hence

(2.19)

A direct calculation shows that the area of is

(2.20)

In addition, we have the volume of unit ball in

(2.21)

and thus the volume of

(2.22)

For a radial function , we have

(2.23)

The Hardy-Littlewood maximal operator is defined on by

(2.24)

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

As it is known, the following vector fields

(3.1)

form a basis for the Lie algebra of left-invariant vector fields on . The sublaplacian is defined by

(3.2)

which also has another explicit form

(3.3)

where is the standard Laplacian on and

(3.4)

For the Schrödinger representations one easily calculates that

(3.5)

So that .

Let ( ) be the normalized Hermite functions given by

(3.6)

where . For and , we define

(3.7)

Then forms an orthonormal basis for .

We set and denote

(3.8)

Therefore,

(3.9)

Moreover, one has

(3.10)

Now let , then has the form

(3.11)

From (3.9) we know that the functions

(3.12)

are eigenfunctions of the operator :

(3.13)

Let be the Laguerre functions defined on by

(3.14)

and set for . Then from [19, (2.3.26)] we have

(3.15)

In view of this equation we have the following.

Proposition 3.1.

One has

(3.16)

where stands for the projection of onto the th eigenspace of , that is,

(3.17)

Now we consider the heat equation associated to the sublaplacian

(3.18)

with the initial condition . In fact, the function given by

(3.19)

is just the solution of the heat equation and satisfies

(3.20)

Moreover, we have the Fourier transform of (see [18, page 86])

(3.21)

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

(4.1)

From above definition and (3.10) it is easy to see that

(4.2)

If , , then we have

(4.3)

which suggests that . Especially, for , one has

(4.4)

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

(4.5)

Proof.

By (3.17), (3.21), and Proposition 3.1 we have

(4.6)

Then we get the desired result.

Lemma 4.3.

The heat kernel satisfies the estimate

(4.7)

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

(4.8)

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

(4.9)

Let be the ball of radius centered at (0,0), and let be its characteristic function. We set

(4.10)

with . Obviously, is radial and decreasing, then we can write

(4.11)

where and is the ball centered at origin. By (2.23) and (2.24) we have

(4.12)

Let be the conjugate exponent of . Since , we have

(4.13)

Then by Hölder's inequality we get

(4.14)

Therefore,

(4.15)

We choose such that

(4.16)

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

(4.17)

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 .

1. (b)

   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

(4.18)

Then by (4.9) we get

(4.19)

Thus for , we have

(4.20)

Case 1 ( ).

It follows from the hypothesis that

(4.21)

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

Case 2 ( ).

Similarly, we have

(4.22)

which implies that .

Then we complete the proof of this theorem.