# A Class of Integral Operators and Bessel Plancherel Transform onLα2

Authors: M. Dziri
Journal: Journal of Function Spaces and Applications
 (2012)

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


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## Body

## 1. Introduction

Titchmarsh [1] studied the relation between the cosine transform and the modified Hardy operator H(f)(x)=(1/x)∫0xf(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 (1.1)G(x)=1x∫0∞φ(yx)g(y)dy
is the cosine transform of(1.2)F(y)=∫0∞1xφ(yx)f(x)dx.
The same result applies to sine transforms.The Riemann-Liouville transform and Weyl transform associated with the eigenfunctions of the differential operator(1.3)lα=d2dx2+2α+1xddx,α>-12
are, respectively, defined for all measurable functions by(1.4)Rα(f)(x)={Cαx-2α∫0x(x2-y2)α-1/2f(y)dy,ifx>0f(0)ifx=0,
respectively,(1.5)Wα(f)(x)=Cα∫x∞(y2-x2)α-1/2f(y)ydy,
where(1.6)Cα=2Γ(α+1)πΓ(α+1/2).
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 lα defined by(1.7)Sβ,α(f)(x)={Cβ,αx-2β∫0x(x2-y2)β-α-1f(y)y2α+1dy,ifx>0f(0)ifx=0,
where(1.8)α,β>-12,β-α>0,Cβ,α=2Γ(β+1)Γ(β-α)Γ(α+1)
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 functionφ which we denoted Tφ. We study the boundedness of these operators on Lα2 the space of all real-valued measurable functions f defined on [0,∞[ with norm ∥f∥2,α=(∫0∞|f(x)|2x2α+1dx)1/2is finite. Building on the continuity of the operator Tφ on Lα2, we establish a relation between Bessel-Plancherel transform Φα, 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 Φα. 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 Φα. Since the eigenfunction of the Dunkl operator defined on real axis ℝ by(1.9)Dαu(x)=dudx(x)+2α+1xu(x)-u(-x)2,α≥-12
is in connection with the special functions associated with the second-order differential operator lα 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 ℱα defined on the Shwartz space S(ℝ) by(1.10)∀λ∈R,Fα(f)(λ)=∫Rf(x)ψλα(x)dμα(x),
where the kernel ψλα is given by(1.11)ψλα(x)=jα(λx)+iλx2(α+1)jα+1(λx),
with jα the normalized Bessel function of index α defined by(1.12)jα(z)=Γ(α+1)∑n=0∞(-1)n(z/2)2nn!Γ(n+α+1),z∈C.
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 lα defined above. In Section 3 we study the boundedness of a class of operators generated by a measurable function φ≥0 on the space Lα2. Relation of the Bessel Plancherel transform, and this class of bounded operators on Lα2 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 lα.


## 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 μα on [0,∞[ depending on α by(2.1)dμα(x)=x2α+12αΓ(α+1)dx,x≥0.
We denote by Lαp, 1≤p<∞ the space of all real-valued measurable functions f defined on [0,∞[ and the norm(2.2)‖f‖p,α=(∫0∞|f(x)|pdμα(x))1/p
is finite. Whereas Lα∞=L∞ which does not depend on α denotes the space of those measurable functions defined on [0,∞[ for which(2.3)‖f‖∞=esssup⁡x>0|f(x)|
is finite.IfT is a linear transformation on Lα2 into itself then ∥T∥ is defined as(2.4)‖T‖=sup⁡g∈Lα2‖T(g)‖2,α‖g‖2,α.
We will make use of the Schwarz type inequality if f,g∈Lα2 then fg∈Lα1 and(2.5)‖fg‖1,α≤‖f‖2,α‖g‖2,α
and its converse if for each h∈Lα2, ∥Gh∥1,α≤A∥h∥2,α then G∈Lα2 and(2.6)‖G‖2,α≤A.The Bessel transform of orderα>-1/2 of a function f∈Lα1 is defined by(2.7)Bα(f)(λ)=∫0∞jα(λx)f(x)dμα(x)forλ≥0,
where jα is the normalized Bessel function defined by(2.8)jα(λx)=∑k=0∞(-1)kΓ(α+1)k!Γ(α+k+1)(λx2)2k.
The following properties are fundamental and are used to prove the main results of this paper.Theorem 2.1.
Letα>-1/2. Then(i)
iff∈Lα1, then ℬα(f) is a continuous function vanishing at infinity.(ii)
Letf be such that f and ℬα(f)∈Lα1. Then
(2.9)f(x)=∫0∞Bα(f)(t)jα(xt)dμα(t)a.e.(iii)
The Bessel transformℬα is an isomorphism from 𝒮*(ℝ) onto itself and its inverse denoted ℬα-1=ℬα, 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).
Letα>-1/2. Then there exists a unique isomorphism
(2.10)Φα:Lα2⟶Lα2,
satisfying(i)
for allf∈𝒮*(ℝ),Φα(f)=ℬα(f),(ii)
for allf∈Lα2,(2.11)‖Φα(f)‖2,α=‖f‖2,α.
The inverse of Φα denoted Φα-1=Φα almost everywhere. The transform Φα is called Bessel Plancherel transform.Proposition 2.3.
(i) For all f∈Lα1∩Lα2 one has ℬα=Φα almost everywhere.
(ii) Let f∈Lα2 then
(2.12)Φα(f)(x)=lim⁡R→∞∫0Rf(y)jα(xy)dμα(y),
where lim stands for lim in the Lα2 mean. That is,
(2.13)lim⁡R→+∞∫0∞|Φα(f)(x)-∫0Rf(y)jα(xy)dμα(y)|2dμα(x)=0.For more details of the previous results see [3, 16, 17].


## 3. A Class of Bounded Operators onLα2

In this section, we will address the boundedness and some properties onLα2 of certain class of integral operators.Theorem 3.1.
Letφ be nonnegative and measurable function such that
(3.1)∫0∞φ(y)yα+1dμα(y)=Aα<∞.
Then the linear operator Tφ defined on Lα2 by
(3.2)Tφ(f)(x)=1x2α+2∫0∞φ(yx)f(y)dμα(y)
is a bounded operator on Lα2 and one has
(3.3)‖Tφ‖≤Aα.Proof.
For any functionh∈Lα2 we have
(3.4)∫0∞|Tφ(f)(x)h(x)|dμα(x)≤∫0∞|h(x)|x2α+2(∫0∞φ(yx)|f(y)|dμα(y))dμα(x).
By change of variable y=tx we obtain
(3.5)∫0∞|Tφ(f)(x)h(x)|dμα(x)≤∫0∞|h(x)|(∫0∞φ(t)|f(xt)|dμα(t))dμα(x).
So, Fubini-Tonelli theorem yields
(3.6)∫0∞|Tφ(f)(x)h(x)|dμα(x)≤∫0∞φ(t)(∫0∞|h(x)‖f(xt)|dμα(x))dμα(t).
But, by Schwarz inequality we have
(3.7)∫0∞|h(x)f(xt)|dμα(x)≤(∫0∞|h(x)|2dμα(x))1/2(∫0∞|f(xt)|2dμα(x))1/2.
Furthermore
(3.8)(∫0∞|f(xt)|2dμα(x))1/2=1tα+1(∫0∞|f(x)|2dμα(x))1/2.
Thus, (3.6) and (3.7) yield
(3.9)∫0∞|Tφ(f)(x)h(x)|dμα(x)≤Aα‖h‖2,α‖f‖2,α.
This proves that the last integral converges absolutely and we obtain
(3.10)‖Tφ(f)h‖1,α≤Aα‖h‖2,α‖f‖2,α.
Moreover the converse of Schwarz inequality allows us to get Tφ(f)∈Lα2 and we have
(3.11)‖Tφ(f)‖2,α≤Aα‖f‖2,α,∀f∈Lα2.
So, Tφ is a bounded linear transformation on Lα2 into itself and
(3.12)‖Tφ‖≤Aα.
The theorem is thus established.Now, choose anyf,h∈Lα2. Then with 〈f,h〉 defined as(3.13)〈f,h〉=∫0∞f(x)h(x)dμα(x)
the inner product in Lα2. Then for all nonnegative measurable function φ such that(3.14)∫0∞φ(t)tα+1dμα(t)
is finite, we have(3.15)〈Tφ(f),h〉=∫0∞Tφ(f)(x)h(x)dμα(x)=∫0∞1x2α+2(∫0∞φ(yx)f(y)dμα(y))h(x)dμα(x).
Using Fubini theorem we get(3.16)〈Tφ(f),h〉=∫0∞(∫0∞1x2α+2φ(yx)h(x)dmα(x))f(y)dmα(y).
That is by definition of adjoint shows that(3.17)〈Tφ(f),h〉=〈f,Tφ*(h)〉,
where Tφ* is the adjoint operator of Tφ defined on Lα2 by(3.18)Tφ*(f)(y)=∫0∞φ(yx)h(x)x2α+2dμα(x).
So, by Theorem 3.1 we get the following.Theorem 3.2.
Letφ be nonnegative and measurable function such that
(3.19)∫0∞φ(y)yα+1dμ(y)=Aα<∞.
Then the linear adjoint operator Tφ* of the operator Tφ defined on Lα2 by
(3.20)Tφ*(f)(y)=∫0∞φ(yx)h(x)x2α+2dμα(x)
is a bounded operator on Lα2 and one has
(3.21)‖Tφ*‖≤Aα.Remark 3.3.
The integrals definingTφ and Tφ* in Theorems 3.1 and 3.2 exist almost everywhere.


## 4. Bessel Plancherel Transform and OperatorsTφ, 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 onLα2.Theorem 4.1.
Letα>-1/2 and φ a measurable nonnegative function such that φ∈Lα1 and
(4.1)∫0∞φ(y)yα+1dμα(y)<∞.
Then
(4.2)TφΦα=ΦαTφ*,
where Tφ and Tφ* are as in Theorems 3.1 and 3.2 and Φα is Bessel Plancherel transform as in Theorem 2.2 and Proposition 2.3.Proof.
Using Proposition2.3 and since Lα1∩Lα2 is dense in Lα2 and Tφ and Tφ* are continuous on Lα2 then to prove the theorem it is sufficient to prove that
(4.3)TφBα=BαTφ*,forf∈Lα1∩Lα2,
where ℬα is the Bessel transform defined in Section 2. Accordingly, choose any f∈Lα1∩Lα2(4.4)Tφ(Bα(f))(x)=1x2α+2∫0∞φ(yx)Bα(f)(y)dμα(y).
Making the change of variable y=sx yields
(4.5)Tφ(Bα(f))(x)=∫0∞φ(s)Bα(f)(sx)dμα(s)=∫0∞φ(s)(∫0∞jα(sxt)f(t)dμα(t))dμα(s).
Interchanging the order of integration and making the change of variable u=st yields
(4.6)Tφ(Bα(f))(x)=∫0∞f(t)t2α+2(∫0∞φ(ut)jα(xu)dμα(u))dμα(t).
Interchanging again the order of integration, so
(4.7)Tφ(Bα(f))(x)=∫0∞jα(xu)(∫0∞f(t)t2α+2φ(ut)dμα(t))dμα(u)=∫0∞jα(xu)Tφ*(f)(u)dμα(u).
The integral (4.5) converges absolutely since φ,f∈Lα1 and |jα(u)|≤1. This justifies the changes in order of integration and also shows that Tφ*(f) belongs to Lα1. Thus from (4.7) we get
(4.8)TφBα=BαTφ*,
which is what we wanted to show.A more general result, we may drop the hypothesis thatφ∈Lα1 in Theorem 4.1.Theorem 4.2.
Ifφ is measurable nonnegative function such that
(4.9)∫0∞φ(y)yα+1dμ(y)<∞.
Then
(4.10)TφΦα=ΦαTφ*.Proof.
Choose any nonnegative function such that∫0∞φ(y)yαdy=Aα<∞ (but not necessarily such that φ∈Lα1). For n=1,2,… define
(4.11)φn(y)=φ(y),ify∈[1n,n],φn(y)=0,ify∈[0,1n[∪]n,∞[.
Then
(4.12)∫0∞φn(y)yαdy=∫1/nnφn(y)yαdy=An,α<∞,
and by the Lebesgue convergence theorem
(4.13)lim⁡n→∞An,α=Aα.
Moreover if Tφ, Tφn are defined by φ in Theorem 3.2 then
(4.14)Tφ-Tφn=Tφ-φn,
and thus by Theorem 3.2(4.15)‖Tφ-Tφn‖≤Aα-An,α.
But Tφn obeys the hypothesis of Theorem 4.1. Hence
(4.16)TφnBα=BαTφn*.
Letting n→∞ and using (4.15) we get
(4.17)TφBα=BαTφ*.
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.
Letφ be nonnegative on (0,1) with
(4.18)∫01φ(t)tα+1dμα(t)
is finite. Define
(4.19)φ(t)=1t2α+2φ(1t),fort>1.
Then the operator Tφ is as in Theorem 3.1 commutes with the Bessel Plancherel transform Φα, that is
(4.20)TφΦα=ΦαTφ.


## 5. Applications

Consider the differential operatorlα defined on]0,∞[by(5.1)lαu=u′′+2α+1xu′=1x2α+1ddx(x2α+1ddx)(u).
Then for α>-1/2 the system(5.2)lαu(x)=-λ2u(x),u(0)=1,u′(0)=0,
admits the unique solution given by the function(5.3)x⟶jα(λx),
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 transformℛα, α>-1/2 associated with the differential operator lα is the integral transform defined on ξ*(ℝ) by
(5.4)Rα(f)(x)={Cαx-2α∫0x(x2-y2)α-1/2f(y)dy,ifx>0,f(0)ifx=0.
(2) The Weyl transform𝒲α associated with lα is defined on 𝒟*(ℝ) by
(5.5)Wα(f)(x)=Cα∫x∞(y2-x2)α-1/2f(y)ydy,
where
(5.6)Cα=2Γ(α+1)πΓ(α+1/2).We note that the relation between the normalized Bessel function and the Riemann-Liouville transform is the following(5.7)jα(λx)=Rα(cos⁡(λ.)(x)).
Trimèche [3, 16] has studied properties of ℛα on ξ*(ℝ) and 𝒲α 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 onLα2.Theorem 5.2.
Let-1/2<α<0. Then(i)
there exists a positive constantAα such that
(5.8)‖Rα(f)‖2,α≤Aα‖f‖2,α,‖1x2α+1Wα(f)‖2,α≤Aα‖f‖2,α.(ii)
And(5.9)RαΦα=Φα(1x2α+1Wα).Proof.
An easy calculation gives(5.10)Rα(f)(x)=C1,αTφ(f)(x),1x2α+1Wα(f)(x)=C2,αTφ*(f)(x),
where
(5.11)φ(t)=t-2α-1(1-t2)α-1/2⋅χ]0,1[(t),
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
(5.12)∫0∞φ(t)tα+1dμα(t)=kα∫01(1-t2)α-1/2tα+1dt.
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 operatorlα, namely, Sonine transform.Definition 5.3.
Letα,β≥-1/2, β-α>0. The Sonine transform associated with lα is defined on C*(ℝ) by
(5.13)Sβ,α(f)(x)={Cβ,αx-2β∫0x(x2-y2)β-α-1f(y)dμα(y),ifx>0,f(0)ifx=0,
where
(5.14)Cβ,α=2Γ(β+1)Γ(β-α)Γ(α+1).We note that(5.15)Sβ,α(f)(x)=Cβ,αTφ(f)(x),
where(5.16)φ(t)=(1-t2)β-α-1χ]0,1[(t),
with Cβ,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(5.17)∫0∞φ(t)tα+1dμα(t)=k(α,β)∫01(1-t2)β-α-1tαdt<∞,
where k(α,β) is a nonnegative constant. This allows us to obtain the following result.Theorem 5.4.
Forα,β>-1/2, β-α>0 one has the following.(i)
There exists a nonnegative constantC1,α,β such that
(5.18)‖Sβ,α(f)‖2,α≤C1,α,β‖f‖2,α,‖Sβ,α*(f)‖2,α≤C1,α,β‖f‖2,α.(ii)
And(5.19)Sβ,αΦα=ΦαSβ,α*,
where
(5.20)Sβ,α*(f)(x)=Cβ,α∫x∞(y2-x2)β-α-1f(y)y2β+1dμα(y).Finally we give a particular integral transform which commutes with Bessel Plancherel transform. Letα>-1/2 and(5.21)φ(t)=1(1+t2)α+1,fort>0.
It is clear that φ satisfies the hypothesis of Theorem 4.3. Thus the integral operator Iα defined by(5.22)Iα(f)(x)=∫0∞f(y)(x2+y2)α+1dμα(y)
is a self adjoint operator and we have the following.Proposition 5.5.
Letα>-1/2. Then(i)
there exists a nonnegative constantC1,α such that
(5.23)‖Iα(f)‖2,α≤C1,α‖f‖2,α,(ii)
And(5.24)IαΦα=ΦαIα.



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