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

Abstract

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 𝑙_𝛼𝑢=𝑢^+((2𝛼+1)/𝑥)𝑢^, 𝛼>−1/2. We also construct a class of integral operators which commute with Bessel Plancherel transform.

1. Introduction

Titchmarsh [1] studied the relation between the cosine transform and the modified Hardy operator 𝐻(𝑓)(𝑥)=(1/𝑥)∫^𝑥0𝑓(𝑦)𝑑𝑦, 𝑓∈𝐿²([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∈𝐿²([0,∞[) then 𝐺(𝑥)=1𝑥^∞0𝜑𝑦𝑥𝑔(𝑦)𝑑𝑦(1.1) is the cosine transform of𝐹(𝑦)=^∞01𝑥𝜑𝑥𝑦𝑓(𝑥)𝑑𝑥.(1.2) The same result applies to sine transforms.

The Riemann-Liouville transform and Weyl transform associated with the eigenfunctions of the differential operator𝑙_𝛼=𝑑²𝑑𝑥²+2𝛼+1𝑥𝑑𝑑𝑥,𝛼>−21(1.3) are, respectively, defined for all measurable functions byℛ_𝛼(𝑓)(𝑥)=⎧⎪⎨⎪⎩𝐶^𝛼𝑥^−2𝛼^𝑥0𝑥²𝑓(0)−𝑦²^𝛼−1/2𝑓(𝑦)𝑑𝑦,if𝑥>0if𝑥=0,(1.4) respectively,𝒲_𝛼(𝑓)(𝑥)=𝐶_𝛼^∞𝑥𝑦²−𝑥²^𝛼−1/2𝑓(𝑦)𝑦𝑑𝑦,(1.5) where𝐶_𝛼=2Γ(𝛼+1)√𝜋Γ(𝛼+1/2).(1.6) 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 by𝒮_𝛽,𝛼(𝑓)(𝑥)=⎧⎪⎨⎪⎩𝐶^𝛽,𝛼𝑥^−2𝛽^𝑥0𝑥²𝑓(0)−𝑦²^{𝛽−𝛼−1}𝑓(𝑦)𝑦^2𝛼+1𝑑𝑦,if𝑥>0if𝑥=0,(1.7) where𝛼,𝛽>−12,𝛽−𝛼>0,𝐶^𝛽,𝛼=2Γ(𝛽+1)Γ(𝛽−𝛼)Γ(𝛼+1)(1.8) 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^2𝛼 the space of all real-valued measurable functions 𝑓 defined on [0,∞[ with norm ‖𝑓‖_2,𝛼=(∫^∞0|𝑓(𝑥)|²𝑥^2𝛼+1𝑑𝑥)^1/2is finite. Building on the continuity of the operator 𝑇_𝜑 on 𝐿^2𝛼, we 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 by𝐷_𝛼𝑢(𝑥)=𝑑𝑢𝑑𝑥(𝑥)+2𝛼𝑥+1𝑢(𝑥)−2𝑢(−𝑥),𝛼≥−12(1.9) 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∀𝜆∈ℝ,ℱ_𝛼(𝑓)(𝜆)=_ℝ𝑓(𝑥)𝜓^𝛼𝜆(𝑥)𝑑𝜇_𝛼(𝑥),(1.10) where the kernel 𝜓^𝛼𝜆 is given by𝜓^𝛼𝜆(𝑥)=𝑗_𝛼(𝜆𝑥)+𝑖𝜆𝑥2(𝛼+1)𝑗^𝛼+1(𝜆𝑥),(1.11) with 𝑗_𝛼 the normalized Bessel function of index 𝛼 defined by𝑗_𝛼(𝑧)=Γ(𝛼+1)^∞_𝑛=0(−1)^𝑛(𝑧/2)^2𝑛𝑛!Γ(𝑛+𝛼+1),𝑧∈ℂ.(1.12) 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^2𝛼. Relation of the Bessel Plancherel transform, and this class of bounded operators on 𝐿^2𝛼 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.

2. 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 by𝑑𝜇_𝛼(𝑥)=𝑥^2𝛼+12^𝛼Γ(𝛼+1)𝑑𝑥,𝑥≥0.(2.1) We denote by 𝐿^𝑝𝛼, 1≤𝑝<∞ the space of all real-valued measurable functions 𝑓 defined on [0,∞[ and the norm‖𝑓‖_𝑝,𝛼=^∞0||𝑓(𝑥)||^𝑝𝑑𝜇_𝛼(𝑥)^1/𝑝(2.2) is finite. Whereas 𝐿^∞𝛼=𝐿^∞ which does not depend on 𝛼 denotes the space of those measurable functions defined on [0,∞[ for which‖𝑓‖_∞=esssup_𝑥>0||𝑓(𝑥)||(2.3) is finite.

If 𝑇 is a linear transformation on 𝐿^2𝛼 into itself then ‖𝑇‖ is defined as‖𝑇‖=sup_{𝑔∈𝐿2𝛼}‖𝑇(𝑔)‖_2,𝛼‖𝑔‖_2,𝛼.(2.4) We will make use of the Schwarz type inequality if 𝑓,𝑔∈𝐿^2𝛼 then 𝑓𝑔∈𝐿^1𝛼 and‖𝑓𝑔‖_1,𝛼≤‖𝑓‖_2,𝛼‖𝑔‖_2,𝛼(2.5) and its converse if for each ℎ∈𝐿^2𝛼, ‖𝐺ℎ‖_1,𝛼≤𝐴‖ℎ‖_2,𝛼 then 𝐺∈𝐿^2𝛼 and‖𝐺‖_2,𝛼≤𝐴.(2.6)

The Bessel transform of order 𝛼>−1/2 of a function 𝑓∈𝐿^1𝛼 is defined byℬ_𝛼(𝑓)(𝜆)=^∞0𝑗_𝛼(𝜆𝑥)𝑓(𝑥)𝑑𝜇_𝛼(𝑥)for𝜆≥0,(2.7) where 𝑗_𝛼 is the normalized Bessel function defined by𝑗_𝛼(𝜆𝑥)=^∞_𝑘=0(−1)^𝑘Γ(𝛼+1)𝑘!Γ(𝛼+𝑘+1)𝜆2𝑥^2𝑘.(2.8) The following properties are fundamental and are used to prove the main results of this paper.

Theorem 2.1. Let 𝛼>−1/2. Then(i)if 𝑓∈𝐿^1𝛼, then ℬ_𝛼(𝑓) is a continuous function vanishing at infinity.(ii)Let 𝑓 be such that 𝑓 and ℬ_𝛼(𝑓)∈𝐿^1𝛼. Then 𝑓(𝑥)=^∞0ℬ_𝛼(𝑓)(𝑡)𝑗_𝛼(𝑥𝑡)𝑑𝜇_𝛼(𝑡)𝑎.𝑒.(2.9)(iii)The Bessel transform ℬ_𝛼 is an isomorphism from 𝒮_∗(ℝ) onto itself and its inverse denoted ℬ^𝛼−1=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 Φ_𝛼∶𝐿^2𝛼⟶𝐿^2𝛼,(2.10) satisfying(i)for all 𝑓∈𝒮_∗(ℝ),Φ_𝛼(𝑓)=ℬ_𝛼(𝑓),(ii)for all 𝑓∈𝐿^2𝛼,‖‖Φ_𝛼(𝑓)‖‖_2,𝛼=‖𝑓‖_2,𝛼.(2.11) The inverse of Φ_𝛼 denoted Φ^𝛼−1=Φ_𝛼 almost everywhere. The transform Φ_𝛼 is called Bessel Plancherel transform.

Proposition 2.3. (i) For all 𝑓∈𝐿^1𝛼∩𝐿^2𝛼 one has ℬ_𝛼=Φ_𝛼 almost everywhere.
(ii) Let 𝑓∈𝐿^2𝛼 then Φ_𝛼(𝑓)(𝑥)=lim_𝑅→∞^𝑅0𝑓(𝑦)𝑗_𝛼(𝑥𝑦)𝑑𝜇_𝛼(𝑦),(2.12) where lim stands for lim in the 𝐿^2𝛼 mean. That is, lim_𝑅→+∞^∞0||||Φ_𝛼(𝑓)(𝑥)−^𝑅0𝑓(𝑦)𝑗_𝛼(𝑥𝑦)𝑑𝜇_𝛼(𝑦)||||²𝑑𝜇_𝛼(𝑥)=0.(2.13)

For more details of the previous results see [3, 16, 17].

3. A Class of Bounded Operators on 𝐿^2𝛼

In this section, we will address the boundedness and some properties on 𝐿^2𝛼 of certain class of integral operators.

Theorem 3.1. Let 𝜑 be nonnegative and measurable function such that ^∞0𝜑(𝑦)𝑦^𝛼+1𝑑𝜇_𝛼(𝑦)=𝐴_𝛼<∞.(3.1) Then the linear operator 𝑇_𝜑 defined on 𝐿^2𝛼 by 𝑇_𝜑(𝑓)(𝑥)=1𝑥^2𝛼+2^∞0𝜑𝑦𝑥𝑓(𝑦)𝑑𝜇^𝛼(𝑦)(3.2) is a bounded operator on 𝐿^2𝛼 and one has ‖‖𝑇_𝜑‖‖≤𝐴_𝛼.(3.3)

Proof. For any function ℎ∈𝐿^2𝛼 we have ^∞0||𝑇_𝜑(𝑓)(𝑥)ℎ(𝑥)||𝑑𝜇_𝛼(𝑥)≤^∞0||ℎ(𝑥)||𝑥^2𝛼+2^∞0𝜑𝑦𝑥||𝑓(𝑦)||𝑑𝜇^𝛼(𝑦)𝑑𝜇_𝛼(𝑥).(3.4) By change of variable 𝑦=𝑡𝑥 we obtain ^∞0||𝑇_𝜑(𝑓)(𝑥)ℎ(𝑥)||𝑑𝜇_𝛼(𝑥)≤^∞0||ℎ(𝑥)||^∞0𝜑(𝑡)||𝑓(𝑥𝑡)||𝑑𝜇_𝛼(𝑡)𝑑𝜇_𝛼(𝑥).(3.5) So, Fubini-Tonelli theorem yields ^∞0||𝑇_𝜑(𝑓)(𝑥)ℎ(𝑥)||𝑑𝜇_𝛼(𝑥)≤^∞0𝜑(𝑡)^∞0||ℎ(𝑥)‖𝑓(𝑥𝑡)||𝑑𝜇_𝛼(𝑥)𝑑𝜇_𝛼(𝑡).(3.6) But, by Schwarz inequality we have ^∞0||ℎ(𝑥)𝑓(𝑥𝑡)||𝑑𝜇_𝛼(𝑥)≤^∞0||ℎ(𝑥)||²𝑑𝜇_𝛼(𝑥)^1/2^∞0||𝑓(𝑥𝑡)||²𝑑𝜇_𝛼(𝑥)^1/2.(3.7) Furthermore ^∞0||𝑓(𝑥𝑡)||²𝑑𝜇_𝛼(𝑥)^1/2=1𝑡^𝛼+1^∞0||𝑓(𝑥)||²𝑑𝜇_𝛼(𝑥)^1/2.(3.8) Thus, (3.6) and (3.7) yield ^∞0||𝑇_𝜑(𝑓)(𝑥)ℎ(𝑥)||𝑑𝜇_𝛼(𝑥)≤𝐴_𝛼‖ℎ‖_2,𝛼‖𝑓‖_2,𝛼.(3.9) This proves that the last integral converges absolutely and we obtain ‖‖𝑇_𝜑(𝑓)ℎ‖‖_1,𝛼≤𝐴_𝛼‖ℎ‖_2,𝛼‖𝑓‖_2,𝛼.(3.10) Moreover the converse of Schwarz inequality allows us to get 𝑇_𝜑(𝑓)∈𝐿^2𝛼 and we have ‖‖𝑇_𝜑(𝑓)‖‖_2,𝛼≤𝐴_𝛼‖𝑓‖_2,𝛼,∀𝑓∈𝐿^2𝛼.(3.11) So, 𝑇_𝜑 is a bounded linear transformation on 𝐿^2𝛼 into itself and ‖‖𝑇_𝜑‖‖≤𝐴_𝛼.(3.12) The theorem is thus established.

Now, choose any 𝑓,ℎ∈𝐿^2𝛼. Then with ⟨𝑓,ℎ⟩ defined as⟨𝑓,ℎ⟩=^∞0𝑓(𝑥)ℎ(𝑥)𝑑𝜇_𝛼(𝑥)(3.13) the inner product in 𝐿^2𝛼. Then for all nonnegative measurable function 𝜑 such that^∞0𝜑(𝑡)𝑡^𝛼+1𝑑𝜇_𝛼(𝑡)(3.14) is finite, we have𝑇_𝜑(𝑓),ℎ==^∞0𝑇_𝜑(𝑓)(𝑥)ℎ(𝑥)𝑑𝜇_𝛼(𝑥)_∞01𝑥^2𝛼+2^∞0𝜑𝑦𝑥𝑓(𝑦)𝑑𝜇^𝛼(𝑦)ℎ(𝑥)𝑑𝜇_𝛼(𝑥).(3.15) Using Fubini theorem we get𝑇_𝜑(𝑓),ℎ=^∞0^∞01𝑥^2𝛼+2𝜑𝑦𝑥ℎ(𝑥)𝑑𝑚^𝛼(𝑥)𝑓(𝑦)𝑑𝑚_𝛼(𝑦).(3.16) That is by definition of adjoint shows that𝑇_𝜑(𝑓),ℎ=𝑓,𝑇^∗𝜑(ℎ),(3.17) where 𝑇^∗𝜑 is the adjoint operator of 𝑇_𝜑 defined on 𝐿^2𝛼 by𝑇^∗𝜑(𝑓)(𝑦)=^∞0𝜑𝑦𝑥𝑥ℎ(𝑥)^2𝛼+2𝑑𝜇_𝛼(𝑥).(3.18) So, by Theorem 3.1 we get the following.

Theorem 3.2. Let 𝜑 be nonnegative and measurable function such that ^∞0𝜑(𝑦)𝑦^𝛼+1𝑑𝜇(𝑦)=𝐴_𝛼<∞.(3.19) Then the linear adjoint operator 𝑇^∗𝜑 of the operator 𝑇_𝜑 defined on 𝐿^2𝛼 by 𝑇^∗𝜑(𝑓)(𝑦)=^∞0𝜑𝑦𝑥𝑥ℎ(𝑥)^2𝛼+2𝑑𝜇_𝛼(𝑥)(3.20) is a bounded operator on 𝐿^2𝛼 and one has ‖‖𝑇^∗𝜑‖‖≤𝐴_𝛼.(3.21)

Remark 3.3. The integrals defining 𝑇_𝜑 and 𝑇^∗𝜑 in Theorems 3.1 and 3.2 exist almost everywhere.

4. Bessel Plancherel Transform and Operators 𝑇_𝜑, and Its Adjoint

This section deals with the relation between Bessel Plancherel transform and a class of integral operators. In particular we construct a class of integral operators which commute with the Bessel Plancherel transform on 𝐿^2𝛼.

Theorem 4.1. Let 𝛼>−1/2 and 𝜑 a measurable nonnegative function such that 𝜑∈𝐿^1𝛼 and ^∞0𝜑(𝑦)𝑦^𝛼+1𝑑𝜇_𝛼(𝑦)<∞.(4.1) Then 𝑇_𝜑Φ_𝛼=Φ_𝛼𝑇^∗𝜑,(4.2) where 𝑇_𝜑 and 𝑇^∗𝜑 are as in Theorems 3.1 and 3.2 and Φ_𝛼 is Bessel Plancherel transform as in Theorem 2.2 and Proposition 2.3.

Proof. Using Proposition 2.3 and since 𝐿^1𝛼∩𝐿^2𝛼 is dense in 𝐿^2𝛼 and 𝑇_𝜑 and 𝑇^∗𝜑 are continuous on 𝐿^2𝛼 then to prove the theorem it is sufficient to prove that 𝑇_𝜑ℬ_𝛼=ℬ_𝛼𝑇^∗𝜑,for𝑓∈𝐿^1𝛼∩𝐿^2𝛼,(4.3) where ℬ_𝛼 is the Bessel transform defined in Section 2. Accordingly, choose any 𝑓∈𝐿^1𝛼∩𝐿^2𝛼𝑇_𝜑ℬ_𝛼(𝑓)(𝑥)=1𝑥^2𝛼+2^∞0𝜑𝑦𝑥ℬ^𝛼(𝑓)(𝑦)𝑑𝜇_𝛼(𝑦).(4.4) Making the change of variable 𝑦=𝑠𝑥 yields 𝑇_𝜑ℬ_𝛼(𝑓)(𝑥)==^∞0𝜑(𝑠)ℬ_𝛼(𝑓)(𝑠𝑥)𝑑𝜇_𝛼(𝑠)_∞0𝜑(𝑠)^∞0𝑗_𝛼(𝑠𝑥𝑡)𝑓(𝑡)𝑑𝜇_𝛼(𝑡)𝑑𝜇_𝛼(𝑠).(4.5) Interchanging the order of integration and making the change of variable 𝑢=𝑠𝑡 yields 𝑇_𝜑ℬ_𝛼(𝑓)(𝑥)=^∞0𝑓(𝑡)𝑡^2𝛼+2^∞0𝜑𝑢𝑡𝑗^𝛼(𝑥𝑢)𝑑𝜇_𝛼(𝑢)𝑑𝜇_𝛼(𝑡).(4.6) Interchanging again the order of integration, so 𝑇_𝜑ℬ_𝛼(𝑓)(𝑥)=^∞0𝑗_𝛼(𝑥𝑢)^∞0𝑓(𝑡)𝑡^2𝛼+2𝜑𝑢𝑡𝑑𝜇^𝛼=(𝑡)𝑑𝜇_𝛼(𝑢)_∞0𝑗_𝛼(𝑥𝑢)𝑇^∗𝜑(𝑓)(𝑢)𝑑𝜇_𝛼(𝑢).(4.7) The integral (4.5) converges absolutely since 𝜑,𝑓∈𝐿^1𝛼 and |𝑗_𝛼(𝑢)|≤1. This justifies the changes in order of integration and also shows that 𝑇^∗𝜑(𝑓) belongs to 𝐿^1𝛼. Thus from (4.7) we get 𝑇_𝜑ℬ_𝛼=ℬ_𝛼𝑇^∗𝜑,(4.8) which is what we wanted to show.

A more general result, we may drop the hypothesis that 𝜑∈𝐿^1𝛼 in Theorem 4.1.

Theorem 4.2. If 𝜑 is measurable nonnegative function such that ^∞0𝜑(𝑦)𝑦^𝛼+1𝑑𝜇(𝑦)<∞.(4.9) Then 𝑇_𝜑Φ_𝛼=Φ_𝛼𝑇^∗𝜑.(4.10)

Proof. Choose any nonnegative function such that ∫^∞0𝜑(𝑦)𝑦^𝛼𝑑𝑦=𝐴_𝛼<∞ (but not necessarily such that 𝜑∈𝐿^1𝛼). For 𝑛=1,2,… define 𝜑_𝑛(𝑦)=𝜑(𝑦),if𝑦∈1𝑛,𝑛,𝜑^𝑛(𝑦)=0,if𝑦∈0,1𝑛∪]𝑛,∞[.(4.11) Then ^∞0𝜑_𝑛(𝑦)𝑦^𝛼𝑑𝑦=^𝑛1/𝑛𝜑_𝑛(𝑦)𝑦^𝛼𝑑𝑦=𝐴_𝑛,𝛼<∞,(4.12) and by the Lebesgue convergence theorem lim_𝑛→∞𝐴_𝑛,𝛼=𝐴_𝛼.(4.13) Moreover if 𝑇_𝜑, 𝑇_𝜑𝑛 are defined by 𝜑 in Theorem 3.2 then 𝑇_𝜑−𝑇_𝜑𝑛=𝑇_{𝜑−𝜑𝑛},(4.14) and thus by Theorem 3.2‖‖𝑇_𝜑−𝑇_𝜑𝑛‖‖≤𝐴_𝛼−𝐴_𝑛,𝛼.(4.15) But 𝑇_𝜑𝑛 obeys the hypothesis of Theorem 4.1. Hence 𝑇_𝜑𝑛ℬ_𝛼=ℬ_𝛼𝑇^∗𝜑_𝑛.(4.16) Letting 𝑛→∞ and using (4.15) we get 𝑇_𝜑ℬ_𝛼=ℬ_𝛼𝑇^∗𝜑.(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.

Theorem 4.3. Let 𝜑 be nonnegative on (0,1) with ¹⁰𝜑(𝑡)𝑡^𝛼+1𝑑𝜇_𝛼(𝑡)(4.18) is finite. Define 𝜑(𝑡)=1𝑡^2𝛼+2𝜑1𝑡,for𝑡>1.(4.19) Then the operator 𝑇_𝜑 is as in Theorem 3.1 commutes with the Bessel Plancherel transform Φ_𝛼, that is 𝑇_𝜑Φ_𝛼=Φ_𝛼𝑇_𝜑.(4.20)

5. Applications

Consider the differential operator 𝑙_𝛼 defined on]0,∞[by𝑙_𝛼𝑢=𝑢^′+2𝛼+1𝑥𝑢^=1𝑥^2𝛼+1𝑑𝑑𝑥𝑥^2𝛼+1𝑑𝑑𝑥(𝑢).(5.1) Then for 𝛼>−1/2 the system𝑢(𝑙0_𝛼)𝑢=(𝑥1),=−𝜆𝑢²_𝑢(𝑥),(0)=0,(5.2) admits the unique solution given by the function𝑥⟶𝑗_𝛼(𝜆𝑥),(5.3) defined in Section 1 [12, 17–19].

In 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 ℛ_𝛼(𝑓)(𝑥)=⎧⎪⎨⎪⎩𝐶^𝛼𝑥^−2𝛼^𝑥0𝑥²𝑓(0)−𝑦²^𝛼−1/2𝑓(𝑦)𝑑𝑦,if𝑥>0,if𝑥=0.(5.4)
(2) The Weyl transform 𝒲_𝛼 associated with 𝑙_𝛼 is defined on 𝒟_∗(ℝ) by 𝒲_𝛼(𝑓)(𝑥)=𝐶_𝛼^∞𝑥𝑦²−𝑥²^𝛼−1/2𝑓(𝑦)𝑦𝑑𝑦,(5.5) where 𝐶_𝛼=2Γ(𝛼+1)√𝜋Γ(𝛼+1/2).(5.6)

We note that the relation between the normalized Bessel function and the Riemann-Liouville transform is the following𝑗_𝛼(𝜆𝑥)=ℛ_𝛼(cos(𝜆.)(𝑥)).(5.7) Trimèche [3, 16] has studied properties of ℛ_𝛼 on 𝜉_∗(ℝ) and 𝒲_𝛼 on 𝒟_∗(ℝ) and gave the inversion formula of these operators.

The objective here is to give a relation between these operators and Bessel Plancherel transform on 𝐿^2𝛼.

Theorem 5.2. Let −1/2<𝛼<0. Then(i)there exists a positive constant 𝐴_𝛼 such that ‖‖‖‖‖1ℛ_𝛼(𝑓)‖‖_2,𝛼≤𝐴_𝛼‖𝑓‖_2,𝛼,𝑥^2𝛼+1𝒲_𝛼(𝑓)‖‖‖_2,𝛼≤𝐴_𝛼‖𝑓‖_2,𝛼.(5.8)(ii)And ℛ_𝛼Φ_𝛼=Φ_𝛼1𝑥^2𝛼+1𝒲_𝛼.(5.9)

Proof. An easy calculation gives ℛ_𝛼(𝑓)(𝑥)=𝐶_1,𝛼𝑇_𝜑(𝑓)(𝑥),1𝑥^2𝛼+1𝒲_𝛼(𝑓)(𝑥)=𝐶_2,𝛼𝑇^∗𝜑(𝑓)(𝑥),(5.10) where 𝜑(𝑡)=𝑡^−2𝛼−11−𝑡²^𝛼−1/2⋅𝜒_]0,1[(𝑡),(5.11) 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 ^∞0𝜑(𝑡)𝑡^𝛼+1𝑑𝜇_𝛼(𝑡)=𝑘_𝛼¹⁰1−𝑡²^𝛼−1/2𝑡^𝛼+1𝑑𝑡.(5.12) 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.

Definition 5.3. Let 𝛼,𝛽≥−1/2, 𝛽−𝛼>0. The Sonine transform associated with 𝑙_𝛼 is defined on 𝐶_∗(ℝ) by 𝒮_𝛽,𝛼(𝑓)(𝑥)=⎧⎪⎨⎪⎩𝐶^𝛽,𝛼𝑥^−2𝛽^𝑥0𝑥²𝑓(0)−𝑦²^{𝛽−𝛼−1}𝑓(𝑦)𝑑𝜇_𝛼(𝑦),if𝑥>0,if𝑥=0,(5.13) where 𝐶_𝛽,𝛼=2Γ(𝛽+1)Γ(𝛽−𝛼)Γ(𝛼+1).(5.14)

We note that𝒮_𝛽,𝛼(𝑓)(𝑥)=𝐶_𝛽,𝛼𝑇_𝜑(𝑓)(𝑥),(5.15) where𝜑(𝑡)=1−𝑡²^{𝛽−𝛼−1}𝜒_]0,1[(𝑡),(5.16) with 𝐶_𝛽,𝛼 is given by relation (5.14). Moreover the function 𝜑 is measurable and nonnegative function, also for 𝛽−𝛼>0 and 𝛼,𝛽>−1/2^∞0𝜑(𝑡)𝑡^𝛼+1𝑑𝜇_𝛼(𝑡)=𝑘(𝛼,𝛽)¹⁰1−𝑡²^{𝛽−𝛼−1}𝑡^𝛼𝑑𝑡<∞,(5.17) where 𝑘(𝛼,𝛽) is a nonnegative constant. This allows us to obtain the following result.

Theorem 5.4. For 𝛼,𝛽>−1/2, 𝛽−𝛼>0 one has the following.(i)There exists a nonnegative constant 𝐶_1,𝛼,𝛽 such that ‖‖‖‖𝒮𝒮_{𝛽,𝛼∗𝛽,𝛼}(𝑓)‖‖_2,𝛼≤𝐶_1,𝛼,𝛽‖𝑓‖_2,𝛼,(𝑓)‖‖_2,𝛼≤𝐶_1,𝛼,𝛽‖𝑓‖_2,𝛼.(5.18)(ii)And 𝒮_𝛽,𝛼Φ_𝛼=Φ_𝛼𝒮^∗𝛽,𝛼,(5.19) where 𝒮^∗𝛽,𝛼(𝑓)(𝑥)=𝐶_𝛽,𝛼^∞𝑥𝑦²−𝑥²^{𝛽−𝛼−1}𝑓(𝑦)𝑦^2𝛽+1𝑑𝜇_𝛼(𝑦).(5.20)

Finally we give a particular integral transform which commutes with Bessel Plancherel transform. Let 𝛼>−1/2 and𝜑(𝑡)=11+𝑡²^𝛼+1,for𝑡>0.(5.21) It is clear that 𝜑 satisfies the hypothesis of Theorem 4.3. Thus the integral operator 𝐼_𝛼 defined by𝐼_𝛼(𝑓)(𝑥)=^∞0𝑓(𝑦)𝑥²+𝑦²^𝛼+1𝑑𝜇_𝛼(𝑦)(5.22) is a self adjoint operator and we have the following.

Proposition 5.5. Let 𝛼>−1/2. Then(i)there exists a nonnegative constant 𝐶_1,𝛼 such that ‖‖𝐼_𝛼(𝑓)‖‖_2,𝛼≤𝐶_1,𝛼‖𝑓‖_2,𝛼,(5.23)(ii)And 𝐼_𝛼Φ_𝛼=Φ_𝛼𝐼_𝛼.(5.24)

Acknowledgment

The author would like to thank the referee for his suggestions and remarks.