#### Abstract

We define and study Sobolev-type spaces associated with singular second-order differential operator on . Some properties are given; in particular we establish a compactness-type imbedding result which allows a Reillich-type theorem. Next, we introduce a generalized Weierstrass transform and, using the theory of reproducing kernels, some applications are given.

#### 1. Introduction

The Sobolev spaces have served as a very useful tool in the theory of partial differential equations, mostly those related to continuum mechanics or physics. Their uses and the study of their properties were facilitated by the theory of distributions and Fourier analysis. The Sobolev space is defined by the use of the classical Fourier transform as the set of all tempered distribution such that its classical Fourier transform satisfying Generalization of the Sobolev space has been studied by replacing the classical Fourier transform by a generalized one.

In this paper we consider the differential operator on , where is the Chebli-Trimeche function (cf. [1, Section 3.5]) defined on and satisfies the following conditions. (i)There exists a positive even infinitely differentiable function on , with , , such that , .(ii) is increasing on and .(iii) is decreasing on , and .(iv)There exists a constant , such that for all , we have where is on , bounded together with its derivatives.

For , , and , we regain the Bessel operator

For , , , and , we regain the Jacobi operator The purpose of this paper is to introduce and study new Sobolev-type spaces , associated with the singular operator that generalizes the corresponding classical spaces. The Bessel case was treated by Assal and Nessibi [2], while Ben Salem and Dachraoui [3] studied the generalized Soblev spaces in the Jacobi setting theory.

The paper is organized as follows. In Section 2 we recall the main results about the harmonic analysis associated with the operator . In Section 3 Sobolev-type spaces on the dual of the Chébli-Trimèche hypergroup are studied. Some properties including completeness and Sobolev embedding theorems are established. Next, we prove a Reillich-type theorem. Finally, in Section 4, as applications, we give practical real inversion formulas using the theory of reproducing kernels for the generalized Weierstrass transform.

#### 2. Preliminaries

In this section, we collect some harmonic analysis results related to the operator . For details, we refer the reader to [1, 4–8].

##### 2.1. Eigenfunctions of the Operator

In the following, we denote by the space of even -functions on , the subspace of , consisting of functions rapidly decreasing together with their derivatives, , where is the eigenfunction of the operator associated with the value , the dual topological space of , the dual topological space of , the dual topological space of , the space of even entire functions on which are of exponential type and slowly increasing, the subspace of satisfying We have .

For every , let us denote by the unique solution of the eigenvalue problem:

*Remark 1. *This function satisfies the following properties. (i), the function is analytic on .(ii) Product formula:
where is a measurable positive function on , with support in .(iii)(iv) For , we have
where is a positive constant.(v) For , we have
(vi) We have the following integral representation of Mehler type,
where is an even positive function on with support in .

##### 2.2. Generalized Fourier Transform

For a Borel positive measure on , and , we write for the Lebesgue space equipped with the norm defined by and . When , with a nonnegative function on , we replace the in the norms by .

For , the generalized Fourier transform is defined by

The inverse generalized Fourier transform of a suitable function on is given by where is the spectral measure given by

*Remark 2. *The function satisfies the following properties. (i)For , we have .(ii)The function is continuous on .(iii)There exist positive constants , , and , such that

If , , ;

If , , ;

If , ;

Proposition 3 (see [7, 9]). *(i) The generalized transform and its inverse are topological isomorphisms between the generalized Schwartz space and the Schwartz space .**(ii) The transform is a topological isomorphism from onto . Moreover, for all , we have if and only if .*

Next, we give some properties of this transform.(i)For in we have (ii)For in we have

Proposition 4 (see [7, 9]). *Plancherel formula for . For all in , we have
**
(ii) Plancherel Theorem. The transform extends uniquely to an isomorphism from onto .*

*Remark 5. *We have for all , but for all .

Proposition 6. *Let . The Fourier transform (resp. ) can be extended as a continuous mapping from onto (resp. from onto ) and we have
**
with .*

##### 2.3. Generalized Convolution

*Definition 7 (see [10]). *The translation operator associated with the operator is defined on , by
where is the function defined in the relation (9).

Proposition 8 (see [10]). *For a suitable function on , we have*(i),(ii),(iii),(iv),(v),(vi).

*Definition 9 (see [10]). *For suitable functions and , we define the convolution product by

*Remark 10. *It is clear that this convolution product is both commutative and associative:(i). (ii).

Proposition 11 (see [10]). *(i) Assume that , satisfy . Then, for every and , we have , and
**(ii) If and , then
**(iii) If and such that , then
**
where is the conjugate exponent of .**(iv) If and such that , then
*

Proposition 12. *For and , with we have
*

Proposition 13. *Let . Then if and only if belongs to , and in this case we have
*

*Definition 14. *The generalized Fourier transform of a distribution in is defined by

Proposition 15. *The generalized Fourier transform is a topological isomorphism from onto .*

Let be in . We define the distribution , by This distribution satisfy the following property:

#### 3. Sobolev-Type Spaces on the Dual of the Chébli-Trimèche Hypergroup

*Definition 16. *Let and . We define the Sobolev-type spaces as the set of tempered distributions such that

We provide the space with the norm:

In the sequel, we will give some properties of the space .

Proposition 17. *Let . The space is dense in , for
*

*Proof. *Firstly, we want to prove that the space is a subset of . Indeed, let . By Proposition 3(i) the function . Thus, using Remark 5, we deduce the claim. Now, we prove the density. Let . Then, from the density of in , we deduce the existence of a sequence in such that
On the other hand, according to Proposition 3(i), for all , the function is in and we have
Therefore, the result follows by combining (38) and (39).

Proposition 18. *(i) Let and let and in such that then
**(ii) Let , and let , and be three real numbers: . Then, for all , there exists a nonnegative constant such that for all in *

*Proof. *(i) is clear.

(ii) We consider , with . Moreover it is easy to see
Thus,
Hence, the proof is completed for .

Proposition 19. *(i) Let if and in , if . The space provided with the norm is a Banach space.**(ii) Let and . Then, the operator defined, on , by
**
is an isometric isomorphism from onto . Moreover, for all , , and for all , the function
**
belongs to the space , with .*

*Proof. *(i) Let be a Cauchy sequence in . Then is a Cauchy sequence in . But is complete, so, there exists a function such that and
But since when and when , then and consequently . This implies that and from relation (46), we get
This achieves the proof of (i).

(ii) Let . By remarking that is an isomorphism from onto and using the fact that
we deduce the first part of (ii). Now, let , . Then, for all , the function
belongs to the space . Therefore, we obtain the second part of (ii) by using inequality (23).

In the following, we prove a Hardy-Littlewood-Paley type inequality for the transform .

Proposition 20. *(1) Let . Then, for and , there exists a positive constant such that for all **
(2) (i) For , , , and , we have for all **(ii) For , , , and , we have (51) for all .*

We start with the following lemma deduced from the hypothesis of the function .

Lemma 21. *(i) For any real , there exist positive constants and such that for all ,
**(ii) For ,
**(iii) For ,
*

*Proof of Proposition 20. *(1) Let and . Clearly, the operator defined on , , by
is of strong type between the spaces and . Therefore, according to the Marcinkiewicz theorem (cf. [11]), to obtain the result, it suffices to show that is of weak type between the spaces under consideration. Indeed, using assertions (i) and (iii) of Lemma 21 and inequality (23), we obtain for all and
and the desired result follows.

(2) Let , , and . According to Proposition 3(ii), for all , it follows that .(i) If and , one can easily see by using Holder inequality, Lemma 21(i), and inequality (23) that for (ii) If , then by virtue of Plancherel Theorem for the transform , we deduce that for all ,
This completes the proof of the proposition.

Proposition 22. *(1) Let . Then for , , and ,
**
and we have
**(2) (i) Let . Then for , , and , we have for all **(ii) For . Then for , , and , we have for all ,
*

*Proof. *(1) The result follows from Proposition 20(1) and the fact that, for all ,

(2) (i) If and using the fact that, for all ,
it follows, from Holder inequality, that for all and
Thus, we deduce the result using Proposition 20(2) and inequality (23). (ii)By virtue of (64), we obtain the result, for , from Plancherel Theorem and Proposition 20(2).

Proposition 23. *Let be a non negative real number. Then, we have *(i) *For **, **, **; *. (ii) *If **, **, **; *,
*where ** is the space of even functions with class ** on **.*

*Proof. *(i) Let be in with . It is clear that belongs to .

Thus, from (14) and Proposition 4(ii), we have
We identify with the second member, then we deduce that belongs to and the injection of into is continuous.

Now, let be in with such that with . From (12), for all , and such that , we have
Using the same method as for and the derivation theorem under the integral sign, we deduce that
Then, for all such that belongs to Thus, is in and the injection of into is continuous.

(ii) If and in ; then using assertions (i) and (iii) of Lemma 21 and Holder inequality, we deduce that for all , the function belongs to .

Therefore, (ii) follows from (i).

Proposition 24. *For and , the space is separable.*

*Proof. *Let . It is well known that is separable. More precisely, the set
is countable and dense in . Thus, for all , , there exists a sequence in such that
On the other hand, for all , , and so . Therefore, for all , there exists such that . Hence, from (70), we obtain
This implies that is countable and dense in and the proposition is proved.

##### 3.1. Reillich-Type Theorem

In this subsection, using Hahn Banach’s and Riesz’s theorems [12, 13], we describe the dual space of . We prove also that a compact imbedding theorem and a Reillich-type theorem are established. We need firstly the following lemmas.

Lemma 25. *Let and . For all positive continuous function , we have
*

*Proof. *The result follows by using the following classical Peetre’s inequality:
and the fact that the kernel is positive with support in .

Lemma 26. *For all and , we have
**
where, for in and in , the function is the generalized convolution product of the distribution and the function defined by
*

*Proof. *For all and , the function belongs to (cf. [4]) and we have
Therefore, the result follows by using the fact that is an isomorphism from (resp. ) onto (resp. ).

Theorem 27. *Let , , and .*(i)*If , then the mapping
* *is continuous.*(ii) *If . We have the same result as in (i) if .*(iii) *If and , then the mapping
* *is continuous.*(iv)*Let , , and . Then, the mapping
* *is continuous.*

*Proof. *(1) (i) According to Lemmas 25 and 26, we have, for and ,
On the other hand, from the hypothesis on and , there exists
Thus, using Proposition 11(i), we obtain the result.

(ii) As , it is easy to see that for . Moreover, we proceed as above and using Proposition 11(i), we obtain
for
Hence, using the fact that the imbedding is continuous, the desired result follows.(iii)Let . Then . Therefore, from (80) and using Proposition 11(iii), we deduce the result.(iv)Using (23) and Holder’s inequality, we obtain, for all and ,
where and are,respectively, the conjugates of and . Therefore, we deduce the result by remarking that the embedding is continuous. This achieves the proof of theorem.

Now, we shall characterize the dual space of .

Theorem 28. *Let when and when . The dual space of can be identified with , where is the conjugate of .*

*Proof. *Let when and when . Then, for , we have, for all ,
This proves, from the density of in , that admits a unique continuous extension to .

Conversely, suppose that . Then the mapping
is continuous, where is the isometric isomorphism from into , defined by
with inverse
Thus, and using the fact that , there exists such that
Hence, from the density of in and using Riez’s theorem, we deduce that belongs to . This completes the proof of Theorem 28.

Proposition 29. *Let when and when . Let be in and . Then for all such that if and if , the mapping
**
is compact.*

*Proof. *Let be a sequence in such that , for all . Then, from Theorem 28, we deduce that can be regarded as a sequence in with , and using Holder inequality, we obtain for all
Therefore, by virtue of Propositions 24 and 19(i), is a separable Banach space. Which implies, from Alaoglu theorem (cf. [14]), that there exists a subsequence weakly converging in . We denote by its weak limit. Using Lemma 26, for all , we have
But is weakly converging to zero in , then it follows that for all .

Now, according to Theorem 27, for all , and using (82) we get
Hence, by Lebesgue’s theorem, we deduce that for all ,
On the other hand, for with , , and , it follows from Theorem 27(i) that for and satisfying , we have
And for with , , and , we obtain from Theorem 27(iv)
Which implies that these last integrals, (95) and (96), tend to zero when tends to uniformly with respect to and so, by virtue of (94), we conclude that is strongly converging in ; that is, . This achieves the proof of Proposition 29.

*Notation*. Let be a compact contained in . We denote by