**INVERSION FORMULAS FOR RIEMANN-LIOUVILLE** **TRANSFORM AND ITS DUAL ASSOCIATED WITH** **SINGULAR PARTIAL DIFFERENTIAL OPERATORS**

C. BACCAR, N. B. HAMADI, AND L. T. RACHDI

*Received 21 May 2005; Revised 27 September 2005; Accepted 20 October 2005*

We define Riemann-Liouville transform*α*and its dual* ^{t}*

*α*associated with two singu- lar partial diﬀerential operators. We establish some results of harmonic analysis for the Fourier transform connected with

*α*. Next, we prove inversion formulas for the opera- tors

*α*,

**

^{t}*α*and a Plancherel theorem for

**

^{t}*α*.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

**1. Introduction**

The mean operator is defined for a continuous function *f* onR^{2}, even with respect to the
first variable by

R0(*f*)(r,x)*=* 1
2π

_{2π}

0 *f*(rsinθ,x+*r*cosθ)dθ, (1.1)
which means thatR0(*f*)(r,x) is the mean value of *f* on the circle centered at (0,*x) and*
radius*r. The dual of the mean operator** ^{t}*R0is defined by

*t*R0(*f*)(r,x)*=*1
*π*

R*f*^{}*r*^{2}+ (x*−**y)*^{2},*y*^{}*d y.* (1.2)
The mean operatorR0and its dual* ^{t}*R0play an important role and have many applica-
tions, for example, in image processing of the so-called synthetic aperture radar (SAR)
data [11,12] or in the linearized inverse scattering problem in acoustics [6].

Our purpose in this work is to define and study integral transforms which general-
ize the operatorsR0and* ^{t}*R0. More precisely, we consider the following singular partial
diﬀerential operators:

Δ1*=* *∂*

*∂x*,
Δ2*=* *∂*^{2}

*∂r*^{2}+2α+ 1
*r*

*∂*

*∂r*^{−}

*∂*^{2}

*∂x*^{2}, (r,*x)**∈*]0, +*∞*[*×R*,*α*0.

(1.3)

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 86238, Pages1–26

DOI10.1155/IJMMS/2006/86238

We associate toΔ1andΔ2the Riemann-Liouville transformR*α*, defined onᏯ*∗*(R^{2}) (the
space of continuous functions onR^{2}, even with respect to the first variable) by

R*α*(*f*)(r,x)_{=}

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩
*α*
*π*

1

*−*1*f*^{}*rs*^{}1*−**t*^{2},*x*+*rt*^{}

*×*

1*−**t*^{2}^{}^{α}^{−}^{1/2}^{}1*−**s*^{2}^{}^{α}^{−}^{1}*dt ds,* if*α >*0,
1

*π*
1

*−*1*f*^{}*r*^{}1*−**t*^{2},*x*+*rt*^{}_{√}*dt*

1*−**t*^{2}, if*α**=*0.

(1.4)

The dual operator* ^{t}*R

*α*is defined on the space

*∗*(R

^{2}) (the space of infinitely diﬀeren- tiable functions onR

^{2}, rapidly decreasing together with all their derivatives, even with respect to the first variable) by

*t*R*α*(*f*)(r,x)*=*

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ 2α

*π*
+*∞*

*r*

^{√}_{u}^{2}_{−}_{r}^{2}

*−**√*

*u*^{2}*−**r*^{2}*f*(u,x+*v)*^{}*u*^{2}*−**v*^{2}*−**r*^{2}^{}^{α}^{−}^{1}*u du dv,* if*α >*0,
1

*π*

R*f*^{}*r*^{2}+ (x*−**y)*^{2},*y*^{}*d y,* if*α**=*0.

(1.5) For more general fractional integrals and fractional diﬀerential equations, we can see the works of Debnath [3,4] and Debnath with Bhatta [5].

We establish for the operatorsR*α*and* ^{t}*R

*α*the same results given by Helgason, Ludwig, and Solmon for the classical Radon transform onR

^{2}[10,14,17] and we find the results given in [15] for the spherical mean operator. Especially

(i) we give some harmonic analysis results related to the Fourier transform associ-
ated with the Riemann-Liouville transformR*α*;

(ii) we define and characterize some spaces of the functions on whichR*α*and* ^{t}*R

*α*

are isomorphisms;

(iii) we give the following inversion formulas forR*α*and* ^{t}*R

*α*:

*f*

*=*R

*α*

*K*

_{α}^{1}

*R*

^{t}*α*(

*f*),

*f*

*=*

*K*

_{α}^{1}

*R*

^{t}*α*R

*α*(

*f*),

*f* *=** ^{t}*R

*α*

*K*

_{α}^{2}R

*α*(

*f*),

*f*

*=*

*K*

_{α}^{2}R

*α*

*t*R

*α*(

*f*), (1.6) where

*K*

_{α}^{1}and

*K*

_{α}^{2}are integro-diﬀerential operators;

(iv) we establish a Plancherel theorem for* ^{t}*R

*α*;

(v) we show thatR*α*and* ^{t}*R

*α*are transmutation operators.

This paper is organized as follows. InSection 2, we show that for (μ,λ)*∈*C^{2}, the dif-
ferential system

Δ1*u(r,x)**= −**iλu(r,x),*
Δ2*u(r,x)**= −**μ*^{2}*u(r,x),*
*u(0, 0)** _{=}*1,

*∂u*

*∂r*(0,*x)** _{=}*0,

*∀*

*x*

_{∈}_{R},

(1.7)

admits a unique solution*ϕ**μ,λ*given by
*ϕ**μ,λ*(r,x)*=**j**α*

*r*^{}*μ*^{2}+*λ*^{2}^{}exp(*−**iλx),* (1.8)
where*j**α*is the modified Bessel function defined by

*j** _{α}*(s)

*=*2

*Γ(α+ 1)*

^{α}*J*

*(s)*

_{α}*s** ^{α}* , (1.9)

and*J** _{α}*is the Bessel function of first kind and index

*α. Next, we prove a Mehler integral*representation of

*ϕ*

*μ,λ*and give some properties ofR

*α*.

InSection 3, we define the Fourier transformF*α*connected withR*α*, and we establish
some harmonic analysis results (inversion formula, Plancherel theorem, Paley-Wiener
theorem) which lead to new properties of the operatorR*α*and its dual* ^{t}*R

*α*.

In Section 4, we characterize some subspaces of*∗*(R^{2}) on whichR*α*and * ^{t}*R

*α*are isomorphisms, and we prove the inversion formulas cited below where the operators

*K*

_{α}^{1}and

*K*

_{α}^{2}are given in terms of Fourier transforms. Next, we introduce fractional powers of the Bessel operator,

*α**=* *∂*^{2}

*∂r*^{2}+2α+ 1
*r*

*∂*

*∂r*, (1.10)

and the Laplacian operator,

Δ*=* *∂*^{2}

*∂r*^{2}+ *∂*^{2}

*∂x*^{2}, (1.11)

that we use to simplify*K*_{α}^{1}and*K*_{α}^{2}.

Finally, we prove the following Plancherel theorem for* ^{t}*R

*α*:

R

+*∞*
0

*f*(r,x)^{}^{2}*r*^{2α+1}*dr dx**=*

R

+*∞*
0

*K*_{α}^{3}^{}* ^{t}*R

*α*(

*f*)

^{}(r,

*x)*

^{}

^{2}

*dr dx,*(1.12) where

*K*

_{α}^{3}is an integro-diﬀerential operator.

InSection 5, we show thatR*α*and* ^{t}*R

*α*satisfy the following relations of permutation:

*t*R*α*

Δ2*f*^{}_{=}*∂*^{2}

*∂r*^{2}

*t*R*α*(*f*), * ^{t}*R

*α*

Δ1*f*^{}* _{=}*Δ1

*t*R

*α*(

*f*),

Δ2R*α*(*f*)*=*R*α*

*∂*^{2}*f*

*∂r*^{2}

, Δ1R*α*(*f*)*=*R*α*

Δ1*f*^{}*.*

(1.13)

**2. Riemann-Liouville transform and its dual associated with the operators**Δ1**and**Δ2

In this section, we define the Riemann-Liouville transformR*α*and its dual* ^{t}*R

*α*, and we give some properties of these operators. It is well known [21] that for every

*λ*

*∈*C, the system

*α**v(r)**= −**λ*^{2}*v(r);*

*v(0)**=*1; *v** ^{}*(0)

*=*0, (2.1)

where*α* is the Bessel operator, admits a unique solution, that is, the modified Bessel
function*r**→**j**α*(rλ). Thus, for all (μ,λ)*∈*C*×*C, the system

Δ1*u(r,x)**= −**iλu(r,x),*
Δ2*u(r,x)**= −**μ*^{2}*u(r,x),*
*u(0, 0)**=*1, *∂u*

*∂r*(0,*x)**=*0, *∀**x**∈*R,

(2.2)

admits the unique solution given by
*ϕ**μ,λ*(r,x)*=**j**α*

*r*^{}*μ*^{2}+*λ*^{2}^{}exp(*−**iλx).* (2.3)

The modified Bessel function *j**α*has the Mehler integral representation, (we refer to
[13,21])

*j** _{α}*(s)

*=*Γ(α+ 1)

*√**π*Γ(α+ 1/2)
1

*−*1

1*−**t*^{2}^{}^{α}^{−}^{1/2}exp(*−**ist)dt.* (2.4)

In particular,

*∀**k**∈*N,*∀**s**∈*R, ^{}*j*_{α}^{(k)}(s)^{}1. (2.5)
On the other hand,

sup

*r**∈R*

*j** _{α}*(rλ)

^{}

*=*1 iﬀ

*λ*

*∈*R

*.*(2.6)

This involves that

sup

(r,x)*∈R*^{2}

*ϕ**μ,λ*(r,x)^{}*=*1 iﬀ(μ,λ)*∈*Γ, (2.7)

whereΓis the set defined by
Γ*=*R^{2}*∪*

(iμ,λ); (μ,λ)*∈*R^{2},*|**μ**|**|**λ**|*

*.* (2.8)

*Proposition 2.1. The eigenfunctionϕ*_{μ,λ}*given by (2.3) has the following Mehler integral*
*representation:*

*ϕ** _{μ,λ}*(r,

*x)*

*=*

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩
*α*
*π*

_{1}

*−*1cos^{}*μrs*^{}1*−**t*^{2}^{}exp^{}*−**iλ(x*+*rt)*^{}1*−**t*^{2}^{}^{α}^{−}^{1/2}^{}1*−**s*^{2}^{}^{α}^{−}^{1}*dt ds,*
*ifα >*0,

1
*π*

_{1}

*−*1cos^{}*rμ*^{}1*−**t*^{2}^{}exp^{}*−**iλ(x*+*rt)*^{}_{√}*dt*

1*−**t*^{2}, *ifα**=*0.

(2.9)

*Proof. From the following expansion of the function* *j**α*:

*j**α*(s)*=*2* ^{α}*Γ(α+ 1)

*J*

*(s)*

_{α}*s*^{α}* ^{=}*Γ(α+ 1)

+*∞*

*k**=*0

(*−*1)^{k}*k!Γ(α*+*k*+ 1)

*s*
2

2k

, (2.10)

we deduce that

*j**α*

*r*^{}*μ*^{2}+*λ*^{2}^{}*=*Γ(α+ 1)

+*∞*

*k**=*0

(*−*1)^{k}*k!Γ(k*+*α*+ 1)

*rμ*
2

2k

*j**α+k*(rλ), (2.11)

and from the equality (2.4), we obtain

*j**α*

*r*^{}*μ*^{2}+*λ*^{2}^{}*=* Γ(α+ 1)

*√**π*Γ(α+ 1/2)
_{1}

*−*1*j**α**−*1/2

*rμ*^{}1*−**t*^{2}^{}exp(*−**irλt)*^{}1*−**t*^{2}^{}^{α}^{−}^{1/2}*dt.*

(2.12)
Then, the results follow by using again the relation (2.4) for*α >*0, and from the fact that

*j*_{−}_{1/2}(s)*=*coss, for*α**=*0. (2.13)

*Definition 2.2. The Riemann-Liouville transform*R*α* associated with the operatorsΔ1

andΔ2is the mapping defined onᏯ*∗*(R^{2}) by the following. For all (r,x)*∈*R^{2},

R*α*(*f*)(r,x)*=*

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎩
*α*
*π*

1

*−*1*f*^{}*rs*^{}1*−**t*^{2},*x*+*rt*^{}

*×*

1*−**t*^{2}^{}^{α}^{−}^{1/2}^{}1*−**s*^{2}^{}^{α}^{−}^{1}*dt ds,* if*α >*0,
1

*π*
1

*−*1*f*^{}*r*^{}1*−**t*^{2},*x*+*rt*^{}_{√}*dt*

1*−**t*^{2}, if*α**=*0.

(2.14)

*Remark 2.3. (i) From*Proposition 2.1andDefinition 2.2, we have
*ϕ**μ,λ*(r,*x)**=*R*α*

cos(μ.) exp(*−**iλ.)*^{}(r,x). (2.15)
(ii) We can easily see, as in [2], that the transformR*α*is continuous and injective from
Ᏹ*∗*(R^{2}) (the space of infinitely diﬀerentiable functions onR^{2}, even with respect to the
first variable) into itself.

*Lemma 2.4. For* *f* *∈*Ꮿ*∗*(R^{2}*),* *f* *bounded, andg**∈**∗*(R^{2}*),*

R

+*∞*

0 R*α*(*f*)(r,*x)g(r,x)r*^{2α+1}*dr dx**=*

R

+*∞*

0 *f*(r,x)* ^{t}*R

*α*(g)(r,x)dr dx, (2.16)

*where*

*R*

^{t}*α*

*is the dual transform defined by*

*t*R*α*(g)(r,x)*=*

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ 2α

*π*
+_{∞}

*r*

^{√}_{u}^{2}_{−}_{r}^{2}

*−**√*

*u*^{2}*−**r*^{2}*g(u,x*+*v)*^{}*u*^{2}*−**v*^{2}*−**r*^{2}^{}^{α}^{−}^{1}*u du dv,* *ifα >*0,
1

*π*

R*g*^{}*r*^{2}+ (x*−**y)*^{2},*y*^{}*d y,* *ifα**=*0.

(2.17) To obtain this lemma, we use Fubini’s theorem and an adequate change of variables.

*Remark 2.5. By a simple change of variables, we have*

R0(*f*)(r,*x)**=* 1
2π

_{2π}

0 *f*(rsinθ,x+*r*cosθ)dθ. (2.18)
**3. Fourier transform associated with Riemann-Liouville operator**

In this section, we define the Fourier transform associated with the operatorR*α*, and we
give some results of harmonic analysis that we use in the next sections.

We denote by

(i)*dν(r,x) the measure defined on [0, +**∞*[*×R*by
*dν(r,x)**=** _{√}* 1

2π2* ^{α}*Γ(α+ 1)

*r*

^{2α+1}

*dr*

*⊗*

*dx,*(3.1) (ii)

*L*

^{1}(d

*ν*) the space of measurable functions

*f*on [0, +

*∞*[

*×R*satisfying

*f*1,ν*=*

R

+* _{∞}*
0

*f*(r,x)^{}*dν*(r,x)*<*+*∞**.* (3.2)

*Definition 3.1. (i) The translation operator associated with Riemann-Liouville transform*
is defined on*L*^{1}(d*ν*) by the following. For all (r,x), (s,*y)**∈*[0, +*∞*[*×R*,

᐀(r,x)*f*(s,*y)**=* Γ(α+ 1)

*√**π*Γ(α+ 1/2)
_{π}

0 *f*^{}*r*^{2}+*s*^{2}+ 2rscosθ,x+*y*^{}sin^{2α}*θdθ.* (3.3)
(ii) The convolution product associated with the Riemann-Liouville transform of *f*,
*g**∈**L*^{1}(dν) is defined by the following. For all (r,x)*∈*[0, +*∞*[*×R*,

*f*#g(r,x)*=*

R

_{+}_{∞}

0 ᐀(r,*−**x)**f*ˇ(s,*y)g*(s,*y)dν(s,y),* (3.4)
where ˇ*f*(s,*y)**=* *f*(s,*−**y).*

We have the following properties.

(i) Since

*∀**r,s*0, *j** _{α}*(rλ)

*j*

*(sλ)*

_{α}*=*Γ(α+ 1)

*√**π*Γ(α+ 1/2)
_{π}

0 *j*_{α}^{}*λ*^{}*r*^{2}+*s*^{2}+ 2rscosθ^{}sin^{2α}*θ dθ,* (3.5)
(we refer to [21]) we deduce that the eigenfunction*ϕ** _{μ,λ}* defined by the relation (2.3)
satisfies the product formula

᐀(r,x)*ϕ**μ,λ*(s,*y)**=**ϕ**μ,λ*(r,x)ϕ*μ,λ*(s,*y).* (3.6)
(ii) If *f* *∈**L*^{1}(dν), then for all (r,*x)**∈*[0, +*∞*[*×R*,᐀(r,x)*f* belongs to*L*^{1}(dν), and we
have

᐀(r,x)*f*^{}_{1,ν}*f*1,ν*.* (3.7)

(iii) For*f*,g*∈**L*^{1}(d*ν*), *f*#gbelongs to*L*^{1}(d*ν*), and the convolution product is commu-
tative and associative.

(iv) For *f*,g*∈**L*^{1}(d*ν),*

*f*#g1,ν*f*1,ν*g*1,ν*.* (3.8)

*Definition 3.2. The Fourier transform associated with the Riemann-Liouville operator is*
defined by

*∀*(μ,λ)*∈*Γ, F*α*(*f*)(μ,*λ)**=*

R

+_{∞}

0 *f*(r,*x)ϕ** _{μ,λ}*(r,x)d

*ν*(r,x), (3.9) whereΓis the set defined by the relation (2.8).

We have the following properties.

(i) Let *f* be in*L*^{1}(dν). For all (r,x)*∈*[0, +*∞*[*×R*, we have

*∀*(μ,λ)*∈*Γ, F*α*

᐀(r,*−**x)**f*^{}(μ,λ)*=**ϕ**μ,λ*(r,x)F*α*(*f*)(μ,λ). (3.10)
(ii) For *f*,g_{∈}*L*^{1}(d*ν), we have*

*∀*(μ,λ)*∈*Γ, F*α*(*f*#g)(μ,*λ)**=*F*α*(*f*)(μ,λ)F*α*(g)(μ,λ). (3.11)
(iii) For *f* *∈**L*^{1}(d*ν*), we have

*∀*(μ,*λ)**∈*Γ, F*α*(*f*)(μ,λ)*=**B**◦*F*α*(*f*)(μ,λ), (3.12)
where

*∀*(μ,λ)*∈*R^{2}, F^{}*α*(*f*)(μ,*λ)**=*

R

+_{∞}

0 *f*(r,x)*j** _{α}*(rμ) exp(

*−*

*iλx)dν(r,x),*

*∀*(μ,*λ)**∈*Γ, *B f*(μ,*λ)**=**f*^{}*μ*^{2}+*λ*^{2},λ^{}*.*

(3.13)

**3.1. Inversion formula and Plancherel theorem for**F*α***. We denote by (see [15])**
(i) *∗*(R^{2}) the space of infinitely diﬀerentiable functions on R^{2} rapidly decreasing
together with all their derivatives, even with respect to the first variable;

(ii)*∗*(Γ) the space of functions *f* :Γ*→*Cinfinitely diﬀerentiable, even with respect
to the first variable and rapidly decreasing together with all their derivatives, that is, for
all*k*1,k2,*k*3*∈*N,

sup

(μ,λ)*∈*Γ

1 +*|**μ**|*^{2}+*|**λ**|*^{2}*k*1
*∂*

*∂μ*
*k*2*∂*

*∂λ*
*k*3

*f*(μ,λ)^{}_{}*<*+*∞*, (3.14)
where

*∂ f*

*∂μ*(μ,*λ)**=*

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

*∂*

*∂r*

*f*(r,λ)^{}, if*μ**=**r**∈*R,
1

*i*

*∂*

*∂t*

*f*(it,λ)^{}, if*μ**=**it,**|**t**|**|**λ**|**.*

(3.15)

Each of these spaces is equipped with its usual topology:

(i)*L*^{2}(d*ν*) the space of measurable functions on [0, +*∞*[*×R*such that
*f*_{}_{2,ν}_{=}

R

+* _{∞}*
0

*f*(r,*x)*^{}^{2}*dν(r,x)*^{}^{1/2}*<*+*∞*; (3.16)

(ii)*dγ(μ,λ) the measure defined on*Γby

Γ*f*(μ,λ)dγ(μ,λ)

*=** _{√}* 1
2π2

*Γ(α+ 1)*

^{α}

R

+*∞*

0 *f*(μ,λ)^{}*μ*^{2}+*λ*^{2}^{}^{α}*μ dμ dλ*+

R

_{|}_{λ}_{|}

0 *f*(iμ,λ)^{}*λ*^{2}*−**μ*^{2}^{}^{α}*μ dμ dλ*

;
(3.17)
(iii)*L** ^{p}*(dγ),

*p*

*=*1,

*p*

*=*2, the space of measurable functions onΓsatisfying

*f**p,γ**=*

Γ

*f*(μ,λ)^{}^{p}*dγ(μ,λ)*
1/ p

*<*+*∞**.* (3.18)

*Remark 3.3. It is clear that a functionf* belongs to*L*^{1}(d*ν) if, and only if, the functionB f*
belongs to*L*^{1}(dγ), and we have

Γ*B f*(μ,λ)dγ(μ,λ)*=*

R

+_{∞}

0 *f*(r,x)d*ν(r,x).* (3.19)

Proposition 3.4 (inversion formula forF*α**). Let* *f* *∈**L*^{1}(d*ν) such that*F*α*(*f) belongs to*
*L*^{1}(dγ), then for almost every (r,x)*∈*[0, +*∞*[*×R**,*

*f*(r,*x)**=*

ΓF*α*(*f*)(μ,*λ)ϕ** _{μ,λ}*(r,x)dγ(μ,λ). (3.20)

*Proof. From [9,*19], one can see that if *f* *∈**L*^{1}(dν) is such thatF^{}*α*(*f*)*∈**L*^{1}(dν), then for
almost every (r,x)*∈*[0, +*∞*[*×R*,

*f*(r,x)*=*

R

+*∞*
0

F*α*(*f*)(μ,λ)j*α*(rμ) exp(iλx)d*ν*(μ,λ). (3.21)
Then, the result follows from the relation (3.12) andRemark 3.3.

*Theorem 3.5. (i) The Fourier transform*F*α**is an isomorphism from**∗*(R^{2}*) onto**∗*(Γ).

*(ii) (Plancherel formula) for* *f* *∈**∗*(R^{2}*),*

F*α*(*f*)^{}_{2,γ}*= **f*2,*ν**.* (3.22)

*(iii) (Plancherel theorem) the transform*F*α**can be extended to an isometric isomorphism*
*fromL*^{2}(dν) onto*L*^{2}(dγ).

*Proof. This theorem follows from the relation (3.12),*Remark 3.3, and the fact thatF^{}*α*is
an isomorphism from*∗*(R^{2}) onto itself, satisfying that for all*f* *∈**∗*(R^{2}),

F*α*(*f*)^{}_{2,ν}*= **f*2,*ν**.* (3.23)

*Lemma 3.6. For* *f* *∈**∗*(R^{2}*),*

*∀*(μ,*λ)**∈*R^{2}, F*α*(*f*)(μ,λ)*=*Λ*α**◦** ^{t}*R

*α*(

*f*)(μ,

*λ),*(3.24)

*where*

*R*

^{t}*α*

*is the dual transform of the Riemann-Liouville operator, and*Λ

*α*

*is a constant*

*multiple of the classical Fourier transform on*R

^{2}

*defined by*

Λ*α*(*f*)(μ,λ)*=*

R

+*∞*

0 *f*(r,x) cos(rμ) exp(*−**iλx)dm(r,x),* (3.25)
*wheredm(r,x) is the measure defined on [0, +**∞*[*×R**by*

*dm(r,x)**=** _{√}* 1

2π2* ^{α}*Γ(α+ 1)

*dr*

*⊗*

*dx.*(3.26)

This lemma follows from the relation (2.15) andLemma 2.4.

Using the relation (3.12) and the fact that the mapping*B*is continuous from*∗*(R^{2})
into itself, we deduce that the Fourier transformF*α*is continuous from*∗*(R^{2}) into itself.

On the other hand,Λ*α*is an isomorphism from *∗*(R^{2}) onto itself. Then, Lemma 3.6
implies that the dual transform* ^{t}*R

*α*maps continuously

*∗*(R

^{2}) into itself.

Proposition 3.7. (i)* ^{t}*R

*α*

*is not injective when applied to*

*∗*(R

^{2}

*).*

(ii)* ^{t}*R

*α*(

*∗*(R

^{2}))

*=*

*∗*(R

^{2}

*).*

*Proof. (i) Letg**∈**∗*(R^{2}) such that supp*g**⊂ {*(r,x)*∈*R^{2},*|**r**|**|**x**|}*,*g**=*0.

SinceF^{}*α* is an isomorphism from*∗*(R^{2}) onto itself, there exists *f* *∈**∗*(R^{2}) such
thatF^{}*α*(*f*)*=**g. From the relation (3.12) and*Lemma 3.6, we deduce that* ^{t}*R

*α*(

*f*)

*=*0.

(ii) We obtain the result by the same way as in [1].

**3.2. Paley-Wiener theorem. We denote by**

(i)Ᏸ*∗*(R^{2}) the space of infinitely diﬀerentiable functions onR^{2}, even with respect to
the first variable, and with compact support;

(ii) H*∗*(C^{2}) the space of entire functions *f* :C^{2}*→*C, even with respect to the first
variable rapidly decreasing of exponential type, that is, there exists a positive constant*M,*
such that for all*k**∈*N,

sup

(μ,λ)*∈C*^{2}

1 +*|**μ*_{|}^{2}+*|**λ*_{|}^{2}^{}^{k}^{}*f*(μ,*λ)*^{}exp^{}*−**M*^{}* _{|}*Imμ

*+*

_{|}*|*Imλ

_{|}^{}

*<*+

*∞*; (3.27)

(iii)H*∗*,0(C^{2}) the subspace ofH*∗*(C^{2}), consisting of functions *f* :C^{2}*→*C, such that
for all*k**∈*N,

sup

(μ,λ)*∈R*^{2}

*|**μ**|**|**λ**|*

1*−**μ*^{2}+ 2λ^{2}^{}^{k}^{}*f*(iμ,*λ)*^{}*<*+*∞*; (3.28)

(iv)Ᏹ^{}* _{∗}*(R

^{2}) the space of distributions onR

^{2}, even with respect to the first variable, and with compact support;

(v) Ᏼ*∗*(C^{2}) the space of entire functions *f* :C^{2}*→*C, even with respect to the first
variable, slowly increasing of exponential type, that is, there exist a positive constant*M*
and an integer*k, such that*

sup

(μ,λ)*∈C*^{2}

1 +*|**μ**|*^{2}+*|**λ**|*^{2}*−**k**f*(μ,λ)^{}exp^{}*−**M*^{}*|*Imμ*|*+*|*Imλ*|*

*<*+*∞*; (3.29)

(vi)Ᏼ*∗*,0(C^{2}) the subspace ofᏴ*∗*(C^{2}), consisting of functions *f* :C^{2}*→*C, such that
there exists an integer*k, satisfying*

sup

(μ,λ)*∈R*^{2}

*|**μ**|**|**λ**|*

1*−**μ*^{2}+ 2λ^{2}^{}^{−}^{k}^{}*f*(iμ,λ)^{}*<*+*∞**.* (3.30)

Each of these spaces is equipped with its usual topology.

*Definition 3.8. The Fourier transform associated with the Riemann-Liouville operator is*
defined onᏱ^{}* _{∗}*(R

^{2}) by

*∀*(μ,λ)*∈*C^{2}, F*α*(T)(μ,*λ)**=*

*T,ϕ*_{μ,λ}^{}*.* (3.31)

*Proposition 3.9. For everyT**∈*Ᏹ^{}* _{∗}*(R

^{2}

*),*

*∀*(μ,λ)*∈*C^{2}, F*α*(T)(μ,*λ)**=**B**◦*F*α*(T)(μ,λ), (3.32)
*where*

*∀*(μ,λ)*∈*C^{2}, F^{}*α*(T)(μ,λ)*=*

*T*,*j**α*(μ.) exp(*−**iλ.)*^{}, (3.33)
*andBis the transform defined by the relation (3.12).*

Using [7, Lemma 2] (see also [15]) and the fact that F^{}*α* is an isomorphism from
Ᏸ*∗*(R^{2}) (resp.,Ᏹ^{}* _{∗}*(R

^{2})) ontoH

*∗*(C

^{2}) (resp.,Ᏼ

*∗*(C

^{2})), we deduce the following theorem.

*Theorem 3.10 (of Paley-Wiener). The Fourier transform*F*α**is an isomorphism*
*(i) from*Ᏸ*∗*(R^{2}*) onto*H*∗*,0(C^{2}*);*

*(ii) from*Ᏹ^{}*∗*(R^{2}*) onto*Ᏼ*∗*,0(C^{2}*).*

FromLemma 3.6,Theorem 3.10, and the fact thatΛ*α*is an isomorphism fromᏰ*∗*(R^{2})
ontoH*∗*(C^{2}), we have the following corollary.

Corollary 3.11. (i)* ^{t}*R

*α*

*maps injectively*Ᏸ

*∗*(R

^{2}

*) into itself.*

(ii)* ^{t}*R

*α*(Ᏸ

*∗*(R

^{2}))

*=*Ᏸ

*∗*(R

^{2}

*).*

**4. Inversion formulas for**R*α***and*** ^{t}*R

*α*

**and Plancherel theorem for**

*R*

^{t}*α*

In this section, we will define some subspaces of *∗*(R^{2}) on which R*α* and * ^{t}*R

*α*are isomorphisms, and we give their inverse transforms in terms of integro-diﬀerential oper- ators. Next, we establish Plancherel theorem for

*R*

^{t}*α*.

We denote by

(i)ᏺthe subspace of*∗*(R^{2}), consisting of functions *f* satisfying

*∀**k**∈*N,*∀**x**∈*R,
*∂*

*∂r*^{2}
*k*

*f*(0,*x)**=*0, (4.1)

where

*∂*

*∂r*^{2} * ^{=}*
1

*r*

*∂*

*∂r*; (4.2)

(ii)*∗*,0(R^{2}) the subspace of*∗*(R^{2}), consisting of functions *f*, such that

*∀**k**∈*N,*∀**x**∈*R,
_{+}_{∞}

0 *f*(r,x)r^{2k}*dr**=*0; (4.3)

(iii)^{0}* _{∗}*(R

^{2}) the subspace of

*∗*(R

^{2}), consisting of functions

*f*, such that suppF

^{}

*α*(

*f*)

*⊂*

(μ,*λ)**∈*R^{2};*|**μ**|**|**λ**|*

*.* (4.4)

*Lemma 4.1. (i) The mapping*Λ*α**is an isomorphism from**∗*,0(R^{2}*) onto*ᏺ.

*(ii) The subspace*ᏺ*can be written as*
ᏺ*=*

*f* *∈**∗*(R^{2});*∀**k**∈*N,*∀**x**∈*R;
*∂*

*∂r*
2k

*f*(0,*x)**=*0

*.* (4.5)

*Proof. Let* *f* *∈**∗*,0(R^{2}).

(i) For*ν>**−*1, we have
*∂*

*∂μ*^{2}
*k*

*j** _{ν}*(rμ)

^{}

*=*Γ(

*ν*+ 1) 2

*Γ(ν+*

^{k}*k*+ 1)

*−**r*^{2}^{}^{k}*j** _{ν}*+k(rμ), (4.6)

thus, from the expression ofΛ*α*, given inLemma 3.6, and the fact that *j** _{−}*1/2(s)

*=*coss, we obtain

*∂*

*∂μ*^{2}
*k*

Λ*α*(*f*)^{}(0,λ)*=*

*√**π*

2* ^{k}*Γ(k+ 1/2)(

*−*1)

^{k}R

_{+}_{∞}

0 *f*(r,x)r^{2k}exp(*−**iλx)dm(r,x),*
(4.7)
which gives the result.

(ii) The proof of (ii) is immediate.

*Theorem 4.2. (i) For all real numbersγ, the mappings*
(i) *f* *→*(r^{2}+*x*^{2})^{γ}*f*

(ii) *f* *→ |**r**|*^{γ}*f*

*are isomorphisms from*ᏺ*onto itself.*

*(ii) For* *f* *∈*ᏺ, the function*gdefined by*

*g*(r,x)*=*

⎧⎨

⎩

*f*^{}^{√}*r*^{2}*−**x*^{2},x^{} *if**|**r**|**|**x**|*,

0 *otherwise,* (4.8)

*belongs to**∗*(R^{2}*).*

*Proof. (i) Let* *f* *∈*ᏺ, by Leibnitz formula, we have

*∂*

*∂r*
*k*1 *∂*

*∂x*
*k*2

*r*^{2}+*x*^{2}^{}^{γ}*f*^{}(r,x)

*=*

*k*1

*j**=*0
*k*2

*i**=*0

*C*_{k}^{j}_{1}*C*^{i}_{k}_{2}*P**j*(r)P*i*(x)^{}*r*^{2}+*x*^{2}^{}^{γ}^{−}^{i}^{−}^{j}*∂*^{k}^{1}^{+k}^{2}^{−}^{i}^{−}^{j}

*∂r*^{k}^{1}^{−}^{j}*∂x*^{k}^{2}^{−}^{i}*f*(r,*x),*

(4.9)

where*P**i*and*P**j*are real polynomials.

Let*n**∈*Nsuch that*γ**−**k*1*−**k*2+*n >*0. By Taylor formula and the fact that*f* *∈*ᏺ, we
have

*∂*

*∂r*
*k*1*−**j*

(*f*)(r,*x)**=* *r*^{2n}
(2n*−*1)!

_{1}

0(1*−**t)*^{2n}^{−}^{1}
*∂*

*∂r*

*k*1*−**j+2n*

(*f*)(rt,x)dt

*= −* *r*^{2n}
(2n*−*1)!

+_{∞}

1 (1*−**t)*^{2n}^{−}^{1}
*∂*

*∂r*

*k*1*−**j+2n*

(*f*)(rt,*x)dt,*

(4.10)

*∂*

*∂x*

*k*2*−**i**∂*

*∂r*
*k*1*−**j*

*f*(r,x)*=* *r*^{2n}
(2n*−*1)!

_{1}

0(1*−**t)*^{2n}^{−}^{1}
*∂*

*∂x*

*k*2*−**i**∂*

*∂r*

*k*1*−**j+2n*

*f*(rt,x)dt

*= −* *r*^{2n}
(2n*−*1)!

+*∞*

1 (1*−**t)*^{2n}^{−}^{1}
*∂*

*∂x*

*k*2*−**i**∂*

*∂r*

*k*1*−**j+2n*

*f*(rt,x)dt.

(4.11) The relations (4.9) and (4.11) imply that the function

(r,x)*−→*

*r*^{2}+*x*^{2}^{}^{γ}*f*(r,x) (4.12)

belongs toᏺand that the mapping
*f* *−→*

*r*^{2}+*x*^{2}^{}^{γ}*f* (4.13)

is continuous fromᏺonto itself. The inverse mapping is given by
*f* *−→*

*r*^{2}+*x*^{2}^{}^{−}^{γ}*f .* (4.14)

By the same way, we show that the mapping

*f* _{−→ |}*r*_{|}^{γ}*f* (4.15)

is an isomorphism fromᏺonto itself.

(ii) Let *f* *∈*ᏺ, and

*g*(r,x)*=*

⎧⎨

⎩

*f*^{√}*r*^{2}*−**x*^{2},x^{} if*|**r**|**|**x**|*,

0 if*|**r**|**|**x**|*, (4.16)

we have
*∂*

*∂x*
*k*2*∂*

*∂r*
*k*1

(g)(r,*x)**=*

*k*1

*j**=*0

*P**j*(r)

⎛

⎝ ^{k}^{2}

*p,q**=*0

*Q**p,q*(x)
*∂*

*∂x*
*p* *∂*

*∂r*^{2}
*q+j*

(*f*)^{}*r*^{2}*−**x*^{2},x^{}

⎞

⎠,
(4.17)
where*P**j* and*Q**p,q* are real polynomials. This equality, together with the fact that *f* be-

longs toᏺ, implies that*g*belongs to*∗*(R^{2}).

*Theorem 4.3. The Fourier transform*F*α* *associated with Riemann-Liouville transform is*
*an isomorphism from*^{0}* _{∗}*(R

^{2}

*) onto*ᏺ.

*Proof. Let* *f* *∈*^{0}* _{∗}*(R

^{2}). From the relation (3.12), we get

*∂*

*∂μ*^{2}
*k*

F*α*(*f*)(0,*λ)**=*
*∂*

*∂μ*^{2}
*k*

*B**◦*F*α*(*f*)^{}(0,*λ)*

*=**B*

! *∂*

*∂μ*^{2}
*k*

F*α*(*f*)

"

(0,λ)

*=*
*∂*

*∂μ*^{2}
*k*

F*α*(*f*)(λ,λ)*=*0,

(4.18)

because suppF^{}*α*(*f*)*⊂ {* (μ,λ)*∈*R^{2}, *|**μ**|**|**λ**|}*, this shows that F*α* maps injectively

^{0}* _{∗}*(R

^{2}) intoᏺ. On the other hand, let

*h*

*∈*ᏺand

*g*(r,x)

*=*

⎧⎨

⎩

*h*^{√}*r*^{2}*−**x*^{2},x^{} if*|**r**|**|**x**|*,

0 if*|**r**|**|**x**|**.* (4.19)

From Theorem 4.2(ii), *g* belongs to *∗*(R^{2}), so there exists *f* *∈**∗*(R^{2}) satisfying
F*α*(*f*)*=**g. Consequently,* *f* *∈*^{0}* _{∗}*(R

^{2}) andF

*α*(

*f*)

*=*

*h.*

From Lemmas3.6,4.1, andTheorem 4.3, we deduce the following result.

*Corollary 4.4. The dual transform** ^{t}*R

*α*

*is an isomorphism from*

^{0}

*(R*

_{∗}^{2}

*) onto*

*∗*,0(R

^{2}

*).*

**4.1. Inversion formula for**R*α***and*** ^{t}*R

*α*

*Theorem 4.5. (i) The operatorK*_{α}^{1}*defined by*

*K*_{α}^{1}(*f*)(r,x)*=*Λ^{−}*α*^{1}

*π*
2^{2α+1}Γ^{2}(α+ 1)

*μ*^{2}+*λ*^{2}^{}^{α}*|**μ**|*Λ*α*(*f*)

(r,x) (4.20)

*is an isomorphism from**∗*,0(R^{2}*) onto itself.*

*(ii) The operatorK*_{α}^{2}*defined by*

*K*_{α}^{2}(g)(r,x)*=*F^{−}_{α}^{1}^{} *π*
2^{2α+1}Γ^{2}(α+ 1)

*μ*^{2}+*λ*^{2}^{}^{α}*|**μ**|*F*α*(g)

(r,x) (4.21)

*is an isomorphism from*^{0}* _{∗}*(R

^{2}

*) onto itself.*

This theorem follows fromLemma 4.1, Theorems4.2and4.3.

*Theorem 4.6. (i) For* *f* *∈**∗*,0(R^{2}*) andg**∈*_{∗}^{0}(R^{2}*), there exists the inversion formula for*
R*α**:*

*g**=*R*α**K*_{α}^{1}* ^{t}*R

*α*(g),

*f*

*=*

*K*

_{α}^{1}

*R*

^{t}*α*R

*α*(

*f*). (4.22)

*(ii) For*

*f*

*∈*

*∗*,0(R

^{2}

*) andg*

*∈*

^{0}

*(R*

_{∗}^{2}

*), there exists the inversion formula for*

*R*

^{t}*α*

*:*

*f* *=** ^{t}*R

*α*

*K*

_{α}^{2}R

*α*(

*f*),

*g*

*=*

*K*

_{α}^{2}R

*α*

*t*R

*α*(g). (4.23)