#### Abstract

We establish a characterization for the homogeneous Weinstein-Besov spaces via the Weinstein heat semigroup. Next, we obtain the generalized Sobolev embedding theorems.

*Dedicated to Khalifa Trimèche*

#### 1. Introduction

We consider the Weinstein operator defined on by where is the Laplacian for the -first variables and is the Bessel operator for the last variable given by For , the operator is the Laplace-Beltrami operator on the Riemannian space equipped with the following metric: (cf. [1, 2]). The Weinstein operator has several applications in pure and applied mathematics especially in the fluid mechanics (cf. [3]).

The harmonic analysis associated with the Weinstein operator is studied by Ben Nahia and Ben Salem (cf. [1, 2]). In particular, the authors have introduced and studied the generalized Fourier transform associated with the Weinstein operator. This transform is called the Weinstein transform. We note that, for this transform, we have studied the uncertainty principle (cf. [4]) and the Gabor transform (cf. [5]).

In the present paper, we intend to continue our study of generalized spaces of type Sobolev associated with the Weinstein operator that started in [6].

In this paper we consider the Weinstein heat equation We study (4) to focus on the following problems.(1)Characterize the homogeneous Weinstein-Besov spaces via the Weinstein heat semigroup.(2)Prove the imbedding Sobolev theorems.

I have studied the generalized Sobolev spaces in the context of differential-differences operators (cf. [7–10]).

The remaining part of the paper is organized as follows. Section 2 is a summary of the main results in the harmonic analysis associated with the Weinstein operators. In Section 3, we introduce and study the homogeneous Weinstein-Besov spaces, the homogeneous Weinstein-Triebel-Lizorkin spaces and the homogeneous Weinstein-Riesz potential spaces. In Section 4, we characterize the homogeneous Weinstein Besov spaces via the Weinstein heat semigroup. Next, we prove the Sobolev embedding theorems.

#### 2. Preliminaries

In order to confirm the basic and standard notations, we briefly overview the Weinstein operator and the related harmonic analysis. Main references are [1, 2].

##### 2.1. Harmonic Analysis Associated with the Weinstein Operator

In this subsection, we collect some notations and results on the Weinstein kernel, the Weinstein intertwining operator and its dual, the Weinstein transform, and the Weinstein convolution.

In the following, We denote by the space of continuous functions on , even with respect to the last variable; the space of functions of class on , even with respect to the last variable; the space of -functions on , even with respect to the last variable; the Schwartz space of rapidly decreasing functions on , even with respect to the last variable; the space of -functions on which are of compact support, even with respect to the last variable; the space of temperate distributions on , even with respect to the last variable. It is the topological dual of .

We consider the Weinstein operator defined by where is the Laplace operator on and is the Bessel operator on given by

The Weinstein kernel is given by where is the normalized Bessel function. The Weinstein kernel satisfies the following properties.(i)For each , we have (ii)For all , we have (iii)For all , , and , we have where and . In particular,

The Weinstein intertwining operator is the operator defined on by

is a topological isomorphism from onto itself satisfying the following transmutation relation: where is the Laplacian on .

We denote by the space of measurable functions on such that where is the measure on given by

The Weinstein transform is given for in by Some basic properties of this transform are as follows.(i)For in , (ii)For in , we have (iii)For all in , if belongs to , then where (iv)For , if we define then

Proposition 1 (see [2]). *
(i) The Weinstein transform is a topological isomorphism from onto itself, and for all f in ,
**
(ii) In particular, the renormalized Weinstein transform can be uniquely extended to an isometric isomorphism from onto itself.*

In the Fourier analysis, the translation operator is given by .

In harmonic analysis associated for the operator , the generalized translation operator is defined by where .

By using the Weinstein kernel, we can also define a generalized translation. For functions and the generalized translation is defined by the following relation: By using the generalized translation, we define the generalized convolution product of functions as follows: This convolution is commutative and associative, and it satisfies the following.(i)For all , belongs to and (ii)Let such that . If and , then and

We define the tempered distribution associated with by for and denote by the integral in the righthand side.

*Definition 2. *The Weinstein transform of a distribution is defined by
for .

In particular, for , it follows that for ,

Proposition 3. *The Weinstein transform is a topological isomorphism from onto itself.*

*Definition 4. *The generalized convolution product of a distribution in and a function in is the function defined by

Proposition 5. *Let be in , , and in . Then, the distribution is given by the function and belongs to . Moreover, for all ,
**
where and
*

For each , we define the distribution by , and this distribution satisfies the following property: In the following, we denote given by (30) by for simplicity.

#### 3. , , Spaces and Basic Properties

##### 3.1. Homogeneous Weinstein-Littlewood-Paley Decomposition

One of the main tools in this paper is the homogeneous Littlewood-Paley decomposition of distribution associated with the Weinstein operators into dyadic blocs of frequencies.

Lemma 6. *One defines by the ring of center , of small radius and great radius . There exists two radial functions and the values of which are in the interval belonging to such that
*

*Notations*. We denote by
The distribution is called the th dyadic block of the homogeneous Littlewood-Paley decomposition of associated with the Weinstein operators.

Throughout this paper, we define and by and .

When dealing with the Littlewood-Paley decomposition, it is convenient to introduce the functions and belonging to such that on and on .

*Remark 7. *We remark that

We put

*Definition 8. *One denotes by the space of tempered distribution such that

Proposition 9 (Bernstein inequalities). *For all and , for all , for all , , and for all , one has the following
*

*Proof. *Using Remark 7, we deduce from Proposition 5 that
Thus, from the relation (29), we prove (i), (ii), and (iii).

##### 3.2. Definitions

In the following, we define analogues of the homogeneous Besov, Triebel-Lizorkin, and Riesz potential spaces associated with the Weinstein operators on and obtain their basic properties.

From now, we make the convention that for all nonnegative sequence , the notation stands for in the case .

*Definition 10. *Let and . The homogeneous Weinstein-Besov spaces are the spaces of distribution in such that

Proposition 11 (see [6]). *Let and and two elements of ; the space is the set of verifying
**
where , for all and .*

*Definition 12. *For and , one writes

The nonhomogeneous Besov space associated with the Weinstein operators is defined by

We give now another definition equivalent to the nonhomogeneous Besov space .

Proposition 13. *Let and and two elements of ; the space is the set of verifying
*

*Definition 14. *Let and , , the homogeneous Weinstein-Triebel-Lizorkin space is the space of distribution in such that

*Definition 15. *For , the operator from to is defined by
The operator is called Weinstein-Riesz potential space.

*Definition 16. *For and , the homogeneous Weinstein-Riesz potential space is defined as the space , equipped with the norm .

Proposition 17. *Let and .**The operator is a linear continuous operator from into and from into .*

*Proof. *
We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [6]).

Proposition 18. *Let and . The operator is a linear continuous injective operator from onto and from onto .*

*Proof. *
We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [6]).

##### 3.3. Embeddings

As in the Euclidean case (cf. [11]) the monotone character of -spaces and the Minkowskis inequality yield the following.

Proposition 19. *If one has
**
Moreover,
**
If one also has
**
where .*

Proposition 20. *One assumes that . Then the following inclusion holds
*

*Proof. *In order to prove the inclusion, we use the estimate
Proposition 9(i) gives that
By definition of the homogeneous Weinstein-Besov spaces, we therefore infer
since . This gives the inclusion.

Proposition 21. *
(1) If belongs to , then belongs to for all and
**
(2) If belongs to and , then belongs to for all and there exists a positive constant such that
**
(3) If belongs to and , then belongs to and there exists a positive constant such that
*

*Proof. *(1) is obvious from the Hölder’s inequality. As for , we write as
where is chosen here after. By the definition of the homogeneous Weinstein-Besov norms, we see that
and thus is dominated by
Hence, in order to complete the proof of , it suffices to choose such that
As for , it is easy to see that is dominated as
Hence, letting
we can obtain the desired estimate.

Proposition 22. *Let and let such that , then one has
*

*Proof. *We obtain these results by the similar ideas used in the nonhomogeneous case (cf. [6]).

Theorem 23. *Let and let . Let, and let . Then, there exists a constant such that for every ; then one has
**
In particular, one gets
*

*Proof. *Let be a Schwartz class; we have
We define as the largest index such that
and we write
Thus, (69) is proved. In order to obtain (70), it is enough to apply the Hölder inequality in the expression above since we have and let .

Corollary 24. *Let and let such that , then one has
**
where .*

*Proof. *By choosing , , , and , we deduce (74) from the relations (70) and (67). In the same way, we deduce (75) from the relations (70) and (68).

#### 4. Generalized Heat Equation

##### 4.1. Characterization for the Weinstein-Besov Spaces

The Weinstein heat equation reads We introduce the Weinstein heat semigroup for the Weinstein-Laplace operator where is the Weinstein heat kernel defined by where Thus In practice, we use the integral formulation of (76)

*Remark 25. *The function is the Gauss kernel associated with Weinstein operators. This function satisfies

Proposition 26. *Let and let . Then the operator maps continuously to and
**
Moreover,
**
for all .*

*Proof. *It follows from the relations (80) and (29) combined with scaling property of the kernel .

In this section, we prove estimates for the Weinstein heat semigroup. These estimates are based on the following result.

Lemma 27. *Let be an annulus. Positive constants and exist such that for any in and any couple of positive real numbers, one has
*

*Proof. *We again consider a function in , the value of which is identically in neighborhood of annulus . We can also assume without loss of generality that . We then have
where
The lemma is proved provided that we can find positive real numbers and such that
To begin, we perform integrations by parts in (87). We get
Using Leibniz's formula, we obtain
and (88) follows.

For any interval of (bounded or unbounded), we define the mixed space-time Banach space of (classes of) measurable functions such that , with

Corollary 28. *Let be an annulus and a positive real number. Let (resp., ) satisfy (resp., for all in ). Consider a solution of
**
and a solution of
**
There exist positive constants and , depending only on , such that for any and , we have
*

*Proof. *It suffices to use the fact that
Combining Lemma 27 and Young's inequality (29) with scaling property of the kernel now yields the result.

Theorem 29. *Let be a positive real number and . A constant exists which satisfies the following property. For , one has
*

To prove this result we need the following lemma.

Lemma 30. *There exist two positive constants and depending only on such that for all , and , one has
*

*Proof. *The result follows immediately by applying Lemma 27 and because .

*Proof of Theorem 29. *Using Lemma 30 and considering the fact that the operator commutes with the operator and the definition of the homogeneous Weinstein-Besov (semi) norm we get
where denotes, as in all this proof, a generic element of the unit sphere of . In the case when , the required inequality comes immediately from the following easy result. For any positive , we have
In the case , using the Hölder inequality with the weight , (99), and the Fubini theorem, we obtain

In order to prove the other inequality, let us observe that for any greater than , we have
Then, Lemma 30, Proposition 9, and the fact that the operator commutes with the operator lead to the following:
In the case , we simply write
In the case , Hölder’s inequality with the weight gives
Thanks to (99) and Fubini’s theorem, we infer from (102) that
The theorem is proved.

*Second Proof of Theorem 29*. We only consider the case