A Class of Integral Operators and Bessel Plancherel Transform on L 2 α

For the relation between Bessel Plancherel transform and a wide class of integral operators we establish some results generalizing the corresponding results for the cosine transform, given by Goldberg 1972 and Titchmarsh 1937 . Building on these results we obtain a new properties of certain well-known integral transforms associated with the eigenfunction of the Bessel differential operator defined on 0, ∞ by lαu u′′ 2α 1 /x u′, α > −1/2. We also construct a class of integral operators which commute with Bessel Plancherel transform.


Introduction
Titchmarsh 1 studied the relation between the cosine transform and the modified Hardy operator H f x 1/x x 0 f y dy, f ∈ L 2 0, ∞ .Then, Goldberg 2 considered a wide class of integral transform and established some results generalizing the corresponding theorems obtained by Goldberg in 2 .More precisely Goldberg proved that if g is the cosine transform of f ∈ L 2 0, ∞ then is the cosine transform of The same result applies to sine transforms.

Journal of Function Spaces and Applications
The Riemann-Liouville transform and Weyl transform associated with the eigenfunctions of the differential operator are, respectively, defined for all measurable functions by respectively, where These operators have been studied on regular spaces of functions.In particular in 3 Trimèche has proved that the Riemann-Liouville transform is an isomorphism from ξ * R the space of even infinitely differentiable functions on R onto itself and that the Weyl transform is an isomorphism from D * R the space of even infinitely differentiable functions on R, with compact support onto itself.The Weyl transform has also been studied on Schwartz spaces S * R .As for the Sonine transform associated with l α defined by where is linked with the Riemann-Liouville transform and Weyl transform.Such integral transforms and many types integral operators have been studied by many authors 4-9 .They have many applications to science and engineering 5, 10 .
In this work we consider a class of integral operators generated by a measurable nonnegative function ϕ which we denoted T ϕ.We study the boundedness of these operators on L 2 α the space of all real-valued measurable functions f defined on 0, ∞ with norm f 2,α establish a relation between Bessel-Plancherel transform Φ α , α > −1/2 which we will define in Section 2 and this class of operators.We also construct a class of self-adjoint operator which commutes with Φ α .Then, we derive new results concerning the relation between the Bessel-Plancherel transform, and the Riemann-Liouville transform, Weyl transform, and Sonine transform.Finally we give a self adjoint operator which commutes with the transform Φ α .Since the eigenfunction of the Dunkl operator defined on real axis R by is in connection with the special functions associated with the second-order differential operator l α defined above, the present paper paves the way for the coming paper which deals with the relation between a class of integral operators and the Dunkl transform F α defined on the Shwartz space S R by where the kernel ψ α λ is given by with j α the normalized Bessel function of index α defined by For more details see 11 .The content of this paper is as follows.In Section 2 we recall some properties of the Bessel transform also called Hankel transform associated with the singular differential operator l α defined above.In Section 3 we study the boundedness of a class of operators generated by a measurable function ϕ ≥ 0 on the space L 2 α .Relation of the Bessel Plancherel transform, and this class of bounded operators on L 2 α is presented in Section 4. Section 5 deals with the connection between Plancherel transform, Riemann-Liouville transform, Weyl transform and Sonine transform.In this section we give an operator which commutes with Plancherel transform associated with l α .

Bessel Transform and Bessel Plancherel Transform
In the following we give some definitions and some results concerning Bessel transform and Bessel Plancherel transform.For more details see 3, 12-16 .
For fixed, α > −1/2 we define a measure μ α on 0, ∞ depending on α by We denote by L p α , 1 ≤ p < ∞ the space of all real-valued measurable functions f defined on 0, ∞ and the norm which does not depend on α denotes the space of those measurable functions defined on 0, ∞ for which We will make use of the The Bessel transform of order α > −1/2 of a function f ∈ L 1 α is defined by where j α is the normalized Bessel function defined by

2.8
The following properties are fundamental and are used to prove the main results of this paper. 2.9 iii The Bessel transform B α is an isomorphism from S * R onto itself and its inverse denoted B −1 α B α , where S * R is the space of even infinitely differentiable functions on R, rapidly decreasing together with all their derivatives equipped with its usual topology.Theorem 2.2 Plancherel .Let α > −1/2.Then there exists a unique isomorphism The inverse of where lim stands for lim in the L 2 α mean.That is, lim For more details of the previous results see 3, 16, 17 .

A Class of Bounded Operators on L 2 α
In this section, we will address the boundedness and some properties on L 2 α of certain class of integral operators.A α < ∞.

3.1
Then the linear operator T ϕ defined on L 2 α by Journal of Function Spaces and Applications is a bounded operator on L 2 α and one has By change of variable y tx we obtain But, by Schwarz inequality we have

3.8
Thus, 3.6 and 3.7 yield This proves that the last integral converges absolutely and we obtain Moreover the converse of Schwarz inequality allows us to get T ϕ f ∈ L 2 α and we have

3.11
So, T ϕ is a bounded linear transformation on L 2 α into itself and The theorem is thus established.

3.15
Using Fubini theorem we get h x dm α x f y dm α y .

3.16
That is by definition of adjoint shows that where T * ϕ is the adjoint operator of T ϕ defined on L 2 α by α then to prove the theorem it is sufficient to prove that where B α is the Bessel transform defined in Section 2. Accordingly, choose any Making the change of variable y sx yields

4.5
Interchanging the order of integration and making the change of variable u st yields Interchanging again the order of integration, so

4.7
The integral 4.5 converges absolutely since ϕ, f ∈ L 1 α and |j α u | ≤ 1.This justifies the changes in order of integration and also shows that T * ϕ f belongs to L 1 α .Thus from 4.7 we get which is what we wanted to show.
A more general result, we may drop the hypothesis that ϕ ∈ L 1 α in Theorem 4.1. Then Proof.Choose any nonnegative function such that A n,α A α .

4.13
Moreover if T ϕ , T ϕ n are defined by ϕ in Theorem 3.2 then and thus by Theorem 3.2 But T ϕ n obeys the hypothesis of Theorem 4.1.Hence Letting n → ∞ and using 4.15 we get 4.17 We have thus shown the result.
In the following we will construct a class of self-adjoint bounded operators which commute with the Bessel Plancherel transform.
Using the same techniques as those used by Goldberg 2 , we get the following result.

4.19
Then the operator T ϕ is as in Theorem 3.1 commutes with the Bessel Plancherel transform Φ α , that is 4.20

Applications
Consider the differential operator l α defined on 0, ∞ by Then for α > −1/2 the system admits the unique solution given by the function x −→ j α λx , 5.3 defined in Section 1 12, 17-19 .the following we introduce some function spaces, namely, i C * R the space of even continuous function on R, ii ξ * R the space of even infinitely differentiable functions on R, iii D * R the space of even infinitely differentiable function with compact support.
Each of these space is equipped with usual topology.
Definition 5.1.1 The Riemann-Liouville transform R α , α > −1/2 associated with the differential operator l α is the integral transform defined on ξ * R by 5.4 2 The Weyl transform W α associated with l α is defined on D * R by where We note that the relation between the normalized Bessel function and the Riemann-Liouville transform is the following j α λx R α cos λ. x .5.7 5.8 ii And 5.9 Proof.An easy calculation gives with χ 0,1 is the characteristic function of 0, 1 .To obtain the theorem it suffices to verify the hypothesis of Theorems 3.1, 3.2, and 4.2.Indeed it is clear that ϕ is a measurable nonnegative function.Furthermore But this integral is finite if −1/2 < α < 0. Thus the theorem is proved.
In the following we define and study another important integral transform associated with the differential operator l α , namely, Sonine transform.where k α, β is a nonnegative constant.This allows us to obtain the following result.