Inversion Formulas for Riemann-Liouville Transform and Its Dual Associated with Singular Partial Differential Operators

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

t We deﬁne Riemann-Liouville transform ᏾α and its dual ᏾α associated with two singular 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- t t tors ᏾α, ᏾α and a Plancherel theorem for ᏾α.

1. Introduction The mean operator is defined for a continuous function f on R2, even with respect to the first variable by 1 2π R0( f )(r,x) = f (r sinθ,x + r cosθ)dθ, (1.1) 2π 0 which means that R0( f )(r,x)isthemeanvalueof f on the circle centered at (0,x)and t radius r. The dual of the mean operator R0 is defined by t 1 2 2 R0( f )(r,x) = f r +(x − y) , y dy. (1.2) π R

t ThemeanoperatorR0 and its dual R0 play an important role and have many applications, 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- t ize the operators R0 and R0. More precisely, we consider the following singular partial diﬀerential operators:

∂ Δ = , 1 ∂x (1.3) ∂2 2α +1 ∂ ∂2 Δ = + − ,(r,x) ∈]0,+∞[×R, α 0. 2 ∂r2 r ∂r ∂x2

Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 86238, Pages 1–26 DOI 10.1155/IJMMS/2006/86238 2 Inversion formulas for Riemann-Liouville transform

2 We associate to Δ1 and Δ2 the Riemann-Liouville transform Rα,deﬁnedonᏯ∗(R )(the space of continuous functions on R2, even with respect to the ﬁrst variable) by ⎧ ⎪ α 1 ⎪ − 2 ⎪ f rs 1 t ,x + rt ⎪ π −1 ⎨⎪ − − R = × − 2 α 1/2 − 2 α 1 α( f )(r,x) ⎪ 1 t 1 s dtds,ifα>0, (1.4) ⎪ ⎪ ⎪ 1 ⎪ 1 dt ⎩ f r 1 − t2,x + rt √ ,ifα = 0. π −1 1 − t2

t 2 The dual operator Rα is defined on the space ᏿∗(R ) (the space of infinitely differentiable functions on R2, rapidly decreasing together with all their derivatives, even with respect to the first variable) by ⎧ √ ⎪ +∞ u2−r2 ⎪ 2α 2 2 2 α−1 ⎨⎪ √ f (u,x + v) u − v − r ududv,ifα>0, π − 2− 2 tR ( )( , ) = r u r α f r x ⎪ ⎪ 1 ⎩ f r2 +(x − y)2, y dy,ifα = 0. π R (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]. t We establish for the operators Rα and Rα the same results given by Helgason, Ludwig, and Solmon for the classical Radon transform on R2 [10, 14, 17]andwefindtheresults given in [15] for the spherical mean operator. Especially (i) we give some harmonic analysis results related to the Fourier transform associated with the Riemann-Liouville transform Rα; t (ii) we define and characterize some spaces of the functions on which Rα and Rα are isomorphisms; t (iii) we give the following inversion formulas for Rα and Rα: = R 1tR = 1tR R f αKα α( f ), f Kα α α( f ), (1.6) = tR 2R = 2 R tR f α Kα α( f ), f Kα α α( f ),

1 2 ﬀ where Kα and Kα are integro-di erential operators; t (iv) we establish a Plancherel theorem for Rα; t (v) we show that Rα and Rα are transmutation operators. This paper is organized as follows. In Section 2, we show that for (μ,λ) ∈ C2,thedif- ferential system

Δ1u(r,x) =−iλu(r,x),

Δ ( , ) =− 2 ( , ), 2u r x μ u r x (1.7) ∂u u(0,0) = 1, (0,x) = 0, ∀x ∈ R, ∂r C. Baccar et al. 3 admits a unique solution ϕμ,λ given by 2 2 ϕμ,λ(r,x) = jα r μ + λ exp(−iλx), (1.8) where jα is the modified Bessel function defined by J (s) j (s) = 2αΓ(α +1) α , (1.9) α sα and Jα is the Bessel function of first kind and index α. Next, we prove a Mehler integral representation of ϕμ,λ and give some properties of Rα. In Section 3, we define the Fourier transform Fα connected with Rα, and we establish some harmonic analysis results (inversion formula, Plancherel theorem, Paley-Wiener t theorem) which lead to new properties of the operator Rα and its dual Rα. 2 t In Section 4, we characterize some subspaces of ᏿∗(R ) on which Rα and Rα are 1 isomorphisms, and we prove the inversion formulas cited below where the operators Kα 2 and Kα are given in terms of Fourier transforms. Next, we introduce fractional powers of the Bessel operator, ∂2 2α +1 ∂ = + , (1.10) α ∂r2 r ∂r and the Laplacian operator, ∂2 ∂2 Δ = + , (1.11) ∂r2 ∂x2 1 2 that we use to simplify Kα and Kα . t Finally, we prove the following Plancherel theorem for Rα: +∞ +∞ 2 2α+1 = 3 tR 2 f (r,x) r dr dx Kα α( f ) (r,x) dr dx, (1.12) R 0 R 0 3 ff where Kα is an integro-di erential operator. t In Section 5, we show that Rα and Rα satisfy the following relations of permutation: ∂2 tR Δ f = tR ( f ), tR Δ f = Δ tR ( f ), α 2 ∂r2 α α 1 1 α (1.13) ∂2 f Δ R ( f ) = R , Δ R ( f ) = R Δ f . 2 α α ∂r2 1 α α 1

2. Riemann-Liouville transform and its dual associated with the operators Δ1 and Δ2 t In this section, we deﬁne the Riemann-Liouville transform Rα and its dual Rα,andwe give some properties of these operators. It is well known [21]thatforeveryλ ∈ C,the system

2 αv(r) =−λ v(r); (2.1) v(0) = 1; v(0) = 0, 4 Inversion formulas for Riemann-Liouville transform where α is the Bessel operator, admits a unique solution, that is, the modiﬁed Bessel function r → jα(rλ). Thus, for all (μ,λ) ∈ C × C, the system

Δ1u(r,x) =−iλu(r,x),

Δ ( , ) =− 2 ( , ), 2u r x μ u r x (2.2) ∂u u(0,0) = 1, (0,x) = 0, ∀x ∈ R, ∂r admits the unique solution given by 2 2 ϕμ,λ(r,x) = jα r μ + λ exp(−iλx). (2.3)

The modiﬁed Bessel function jα has the Mehler integral representation, (we refer to [13, 21]) Γ 1 (α +1) 2 α−1/2 jα(s) = √ 1 − t exp(−ist)dt. (2.4) πΓ(α +1/2) −1

In particular,

∀ ∈ N ∀ ∈ R (k) k , s , jα (s) 1. (2.5)

On the other hand,

sup jα(rλ) = 1iﬀ λ ∈ R. (2.6) r∈R

This involves that

sup ϕμ,λ(r,x) = 1iﬀ (μ,λ) ∈ Γ, (2.7) (r,x)∈R2 where Γ is the set deﬁned by Γ = R2 ∪ (iμ,λ); (μ,λ) ∈ R2, |μ| |λ| . (2.8)

Proposition 2.1. The eigenfunction ϕμ,λ given by (2.3) has the following Mehler integral representation: ⎧ ⎪ 1 ⎪ α 2 α−1/2 2 α−1 ⎪ cos μrs 1 − t2 exp − iλ(x + rt) 1 − t 1 − s dtds, ⎪ − ⎨⎪ π 1 ( , ) = if α>0, ϕμ,λ r x ⎪ ⎪ ⎪ 1 ⎪ 1 dt ⎩ cos rμ 1 − t2 exp − iλ(x + rt) √ , if α = 0. π −1 1 − t2 (2.9) C. Baccar et al. 5

Proof. From the following expansion of the function jα:

∞ J (s) + (−1)k s 2k j (s) = 2αΓ(α +1) α = Γ(α +1) , (2.10) α α !Γ( + +1) 2 s k=0 k α k we deduce that

∞ + (−1)k rμ 2k j r μ2 + λ2 = Γ(α +1) j (rλ), (2.11) α !Γ( + +1) 2 α+k k=0 k k α and from the equality (2.4), we obtain

Γ 1 − 2 2 (α +1) 2 2 α 1/2 jα r μ + λ = √ jα−1/2 rμ 1 − t exp(−irλt) 1 − t dt. π Γ(α +1/2) −1 (2.12)

Then, the results follow by using again the relation (2.4)forα>0, and from the fact that

j−1/2(s) = coss,forα = 0. (2.13)

Deﬁnition 2.2. The Riemann-Liouville transform Rα associated with the operators Δ1 2 2 and Δ2 is the mapping deﬁned on Ꮿ∗(R ) by the following. For all (r,x) ∈ R ,

⎧ ⎪ α 1 ⎪ − 2 ⎪ f rs 1 t ,x + rt ⎪ π −1 ⎨⎪ − − R = × − 2 α 1/2 − 2 α 1 α( f )(r,x) ⎪ 1 t 1 s dtds,ifα>0, (2.14) ⎪ ⎪ ⎪ 1 ⎪ 1 dt ⎩ f r 1 − t2,x + rt √ ,ifα = 0. π −1 1 − t2

Remark 2.3. (i) From Proposition 2.1 and Deﬁnition 2.2,wehave

ϕμ,λ(r,x) = Rα cos(μ.)exp(−iλ.) (r,x). (2.15)

(ii) We can easily see, as in [2], that the transform Rα is continuous and injective from 2 2 Ᏹ∗(R ) (the space of infinitely differentiable functions on R , even with respect to the first variable) into itself. 6 Inversion formulas for Riemann-Liouville transform

2 2 Lemma 2.4. For f ∈ Ꮿ∗(R ), f bounded, and g ∈ ᏿∗(R ), +∞ +∞ 2α+1 t Rα( f )(r,x)g(r,x)r dr dx = f (r,x) Rα(g)(r,x)dr dx, (2.16) R 0 R 0

t where Rα is the dual transform deﬁned by ⎧ √ ⎪ +∞ u2−r2 ⎪ 2α 2 2 2 α−1 ⎨⎪ √ g(u,x + v) u − v − r ududv, if α>0, π − 2− 2 tR ( )( , ) = r u r α g r x ⎪ ⎪ 1 ⎩ g r2 +(x − y)2, y dy, if α = 0. π R (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 1 2π R0( f )(r,x) = f (r sinθ,x + r cosθ)dθ. (2.18) 2π 0

3. Fourier transform associated with Riemann-Liouville operator

In this section, we deﬁne the Fourier transform associated with the operator Rα,andwe 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

1 dν(r,x) = √ r2α+1dr ⊗ dx, (3.1) 2π2αΓ(α +1)

(ii) L1(dν) the space of measurable functions f on [0,+∞[×R satisfying +∞

f 1,ν = f (r,x) dν(r,x) < +∞. (3.2) R 0

Deﬁnition 3.1. (i) The translation operator associated with Riemann-Liouville transform is deﬁned on L1(dν) by the following. For all (r,x),(s, y) ∈ [0,+∞[×R, Γ π (α +1) 2 2 2α ᐀(r,x) f (s, y) = √ f r + s +2rscosθ,x + y sin θdθ. (3.3) πΓ(α +1/2) 0

(ii) The convolution product associated with the Riemann-Liouville transform of f , g ∈ L1(dν)isdeﬁnedbythefollowing.Forall(r,x) ∈ [0,+∞[×R, +∞ ˇ f #g(r,x) = ᐀(r,−x) f (s, y)g(s, y)dν(s, y), (3.4) R 0 where fˇ(s, y) = f (s,−y). C. Baccar et al. 7

We have the following properties. (i) Since Γ π (α +1) 2 2 2α ∀r,s 0, jα(rλ)jα(sλ) = √ jα λ r + s +2rscosθ sin θdθ, (3.5) πΓ(α +1/2) 0

(we refer to [21]) we deduce that the eigenfunction ϕμ,λ deﬁned by the relation (2.3) satisﬁes the product formula

᐀(r,x)ϕμ,λ(s, y) = ϕμ,λ(r,x)ϕμ,λ(s, y). (3.6)

1 1 (ii) If f ∈ L (dν), then for all (r,x) ∈ [0,+∞[×R, ᐀(r,x) f belongs to L (dν), and we have ᐀ (r,x) f 1,ν f 1,ν. (3.7)

(iii) For f ,g ∈ L1(dν), f #g belongs to L1(dν), and the convolution product is commu- tative and associative. (iv) For f ,g ∈ L1(dν),

f #g 1,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, Fα( f )(μ,λ) = f (r,x)ϕμ,λ(r,x)dν(r,x), (3.9) R 0 where Γ isthesetdefinedbytherelation(2.8). We have the following properties. (i) Let f be in L1(dν). For all (r,x) ∈ [0,+∞[×R,wehave

∀(μ,λ) ∈ Γ, Fα ᐀(r,−x) f (μ,λ) = ϕμ,λ(r,x)Fα( f )(μ,λ). (3.10)

(ii) For f ,g ∈ L1(dν), we have

∀(μ,λ) ∈ Γ, Fα( f #g)(μ,λ) = Fα( f )(μ,λ)Fα(g)(μ,λ). (3.11)

(iii) For f ∈ L1(dν), we have

∀(μ,λ) ∈ Γ, Fα( f )(μ,λ) = B ◦ Fα( f )(μ,λ), (3.12) where +∞ 2 ∀(μ,λ) ∈ R , Fα( f )(μ,λ) = f (r,x)jα(rμ)exp(−iλx)dν(r,x), R 0 (3.13) ∀(μ,λ) ∈ Γ, Bf(μ,λ) = f μ2 + λ2,λ . 8 Inversion formulas for Riemann-Liouville transform

3.1. Inversion formula and Plancherel theorem for Fα. We denote by (see [15]) 2 2 (i) ᏿∗(R ) the space of infinitely differentiable functions on R rapidly decreasing together with all their derivatives, even with respect to the first variable; (ii) ᏿∗(Γ) the space of functions f : Γ → C infinitely differentiable, even with respect to the first variable and rapidly decreasing together with all their derivatives, that is, for all k1,k2,k3 ∈ N, k2 k3 k1 ∂ ∂ sup 1+|μ|2 + |λ|2 f (μ,λ) < +∞, (3.14) (μ,λ)∈Γ ∂μ ∂λ where ⎧ ⎪ ∂ ⎨⎪ f (r,λ) ,ifμ = r ∈ R, ∂f = ∂r (μ,λ) ⎪ (3.15) ∂μ ⎪ 1 ∂ ⎩ f (it,λ) ,ifμ = it, |t| |λ|. i ∂t Each of these spaces is equipped with its usual topology: (i) L2(dν) the space of measurable functions on [0,+∞[×R such that +∞ 1/2 2 f 2,ν = f (r,x) dν(r,x) < +∞; (3.16) R 0

(ii) dγ(μ,λ) the measure deﬁned on Γ by f (μ,λ)dγ(μ,λ) Γ 1 +∞ |λ| = √ f (μ,λ) μ2 + λ2 αμdμdλ+ f (iμ,λ) λ2 − μ2 αμdμdλ ; 2π2αΓ(α +1) R 0 R 0 (3.17)

(iii) Lp(dγ), p = 1, p = 2, the space of measurable functions on Γ satisfying 1/p p f p,γ = f (μ,λ) dγ(μ,λ) < +∞. (3.18) Γ

Remark 3.3. It is clear that a function f belongs to L1(dν) if, and only if, the function Bf belongs to L1(dγ), and we have +∞ Bf(μ,λ)dγ(μ,λ) = f (r,x)dν(r,x). (3.19) Γ R 0

1 Proposition 3.4 (inversion formula for Fα). Let f ∈ L (dν) such that Fα( f ) belongs to L1(dγ), then for almost every (r,x) ∈ [0,+∞[×R,

f (r,x) = Fα( f )(μ,λ)ϕ (r,x)dγ(μ,λ). (3.20) Γ μ,λ C. Baccar et al. 9

1 1 Proof. From [9, 19], one can see that if f ∈ L (dν)issuchthatFα( f ) ∈ L (dν), then for almost every (r,x) ∈ [0,+∞[×R, +∞ f (r,x) = Fα( f )(μ,λ)jα(rμ)exp(iλx)dν(μ,λ). (3.21) R 0

Then, the result follows from the relation (3.12)andRemark 3.3.

2 Theorem 3.5. (i) The Fourier transform Fα is an isomorphism from ᏿∗(R ) onto ᏿∗(Γ). 2 (ii) (Plancherel formula) for f ∈ ᏿∗(R ), F = α( f ) 2,γ f 2,ν. (3.22)

(iii) (Plancherel theorem) the transform Fα can be extended to an isometric isomorphism from L2(dν) onto L2(dγ). Proof. This theorem follows from the relation (3.12), Remark 3.3, and the fact that Fα is 2 2 an isomorphism from ᏿∗(R ) onto itself, satisfying that for all f ∈ ᏿∗(R ), F = α( f ) 2,ν f 2,ν. (3.23)

2 Lemma 3.6. For f ∈ ᏿∗(R ),

2 t ∀(μ,λ) ∈ R , Fα( f )(μ,λ) = Λα ◦ Rα( f )(μ,λ), (3.24)

t where Rα is the dual transform of the Riemann-Liouville operator, and Λα is a constant multiple of the classical Fourier transform on R2 deﬁned by +∞ Λα( f )(μ,λ) = f (r,x)cos(rμ)exp(−iλx)dm(r,x), (3.25) R 0 where dm(r,x) is the measure deﬁned on [0,+∞[×R by

1 dm(r,x) = √ dr ⊗ dx. (3.26) 2π2αΓ(α +1)

This lemma follows from the relation (2.15)andLemma 2.4. 2 Using the relation (3.12) and the fact that the mapping B is continuous from ᏿∗(R ) 2 into itself, we deduce that the Fourier transform Fα is continuous from ᏿∗(R ) into itself. 2 On the other hand, Λα is an isomorphism from ᏿∗(R ) onto itself. Then, Lemma 3.6 t 2 implies that the dual transform Rα maps continuously ᏿∗(R ) into itself. t 2 Proposition 3.7. (i) Rα is not injective when applied to ᏿∗(R ). t 2 2 (ii) Rα(᏿∗(R )) = ᏿∗(R ). 10 Inversion formulas for Riemann-Liouville transform

2 2 Proof. (i) Let g ∈ ᏿∗(R ) such that suppg ⊂{(r,x) ∈ R , |r| |x|}, g = 0. 2 2 Since Fα is an isomorphism from ᏿∗(R ) onto itself, there exists f ∈ ᏿∗(R )such t that Fα( f ) = g.Fromtherelation(3.12)andLemma 3.6,wededucethat Rα( f ) = 0. (ii) We obtain the result by the same way as in [1].

3.2. Paley-Wiener theorem. We denote by 2 2 (i) Ᏸ∗(R ) the space of infinitely differentiable functions on R , even with respect to the first variable, and with compact support; 2 2 (ii) H∗(C ) the space of entire functions f : C → 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 1+|μ|2 + |λ|2 k f (μ,λ) exp − M |Imμ| + |Imλ| < +∞; (3.27) (μ,λ)∈C2

2 2 2 (iii) H∗,0(C )thesubspaceofH∗(C ), consisting of functions f : C → C,suchthat for all k ∈ N,

sup 1 − μ2 +2λ2 k f (iμ,λ) < +∞; (3.28) (μ,λ)∈R2 |μ||λ|

2 2 (iv) Ᏹ∗(R ) the space of distributions on R , even with respect to the ﬁrst variable, and with compact support; 2 2 (v) Ᏼ∗(C ) the space of entire functions f : C → 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,suchthat

− sup 1+|μ|2 + |λ|2 k f (μ,λ) exp − M |Imμ| + |Imλ| < +∞; (3.29) (μ,λ)∈C2

2 2 2 (vi) Ᏼ∗,0(C )thesubspaceofᏴ∗(C ), consisting of functions f : C → C,suchthat thereexistsanintegerk, satisfying

− sup 1 − μ2 +2λ2 k f (iμ,λ) < +∞. (3.30) (μ,λ)∈R2 |μ||λ|

Each of these spaces is equipped with its usual topology. Deﬁnition 3.8. The Fourier transform associated with the Riemann-Liouville operator is 2 deﬁned on Ᏹ∗(R )by

2 ∀(μ,λ) ∈ C , Fα(T)(μ,λ) = T,ϕμ,λ . (3.31) C. Baccar et al. 11

2 Proposition 3.9. For every T ∈ Ᏹ∗(R ),

2 ∀(μ,λ) ∈ C , Fα(T)(μ,λ) = B ◦ Fα(T)(μ,λ), (3.32) where 2 ∀(μ,λ) ∈ C , Fα(T)(μ,λ) = T, jα(μ.)exp(−iλ.) , (3.33) and B is the transform deﬁned by the relation (3.12). Using [7, Lemma 2] (see also [15]) and the fact that Fα is an isomorphism from 2 2 2 2 Ᏸ∗(R )(resp.,Ᏹ∗(R )) onto H∗(C )(resp.,Ᏼ∗(C )), we deduce the following theorem.

Theorem 3.10 (of Paley-Wiener). The Fourier transform Fα is an isomorphism 2 2 (i) from Ᏸ∗(R ) onto H∗,0(C ); 2 2 (ii) from Ᏹ∗(R ) onto Ᏼ∗,0(C ). 2 From Lemma 3.6, Theorem 3.10, and the fact that Λα is an isomorphism from Ᏸ∗(R ) 2 onto H∗(C ), we have the following corollary.

t 2 Corollary 3.11. (i) Rα maps injectively Ᏸ∗(R ) into itself. t 2 2 (ii) Rα(Ᏸ∗(R )) = Ᏸ∗(R ).

t t 4. Inversion formulas for Rα and Rα and Plancherel theorem for Rα 2 t In this section, we will deﬁne some subspaces of ᏿∗(R ) on which Rα and Rα are isomorphisms, and we give their inverse transforms in terms of integro-diﬀerential oper- t ators. Next, we establish Plancherel theorem for Rα. We denote by 2 (i) ᏺ the subspace of ᏿∗(R ), consisting of functions f satisfying ∂ k ∀k ∈ N, ∀x ∈ R, f (0,x) = 0, (4.1) ∂r2 where

∂ 1 ∂ = ; (4.2) ∂r2 r ∂r

2 2 (ii) ᏿∗,0(R )thesubspaceof᏿∗(R ), consisting of functions f ,suchthat +∞ ∀k ∈ N, ∀x ∈ R, f (r,x)r2kdr = 0; (4.3) 0

0 2 2 (iii) ᏿∗(R )thesubspaceof᏿∗(R ), consisting of functions f ,suchthat 2 suppFα( f ) ⊂ (μ,λ) ∈ R ; |μ| |λ| . (4.4) 12 Inversion formulas for Riemann-Liouville transform

2 Lemma 4.1. (i) The mapping Λα is an isomorphism from ᏿∗,0(R ) onto ᏺ. (ii) The subspace ᏺ can be written as

2k 2 ∂ ᏺ = f ∈ ᏿∗(R ); ∀k ∈ N,∀x ∈ R; f (0,x) = 0 . (4.5) ∂r

2 Proof. Let f ∈ ᏿∗,0(R ). (i) For ν > −1, we have

k ∂ Γ(ν +1) k jν(rμ) = − r2 jν (rμ), (4.6) ∂μ2 2kΓ(ν + k +1) +k thus, from the expression of Λα,giveninLemma 3.6, and the fact that j−1/2(s) = coss,we obtain

√ k +∞ ∂ Λ = π − k 2k − 2 α( f ) (0,λ) ( 1) f (r,x)r exp( iλx)dm(r,x), ∂μ 2kΓ(k +1/2) R 0 (4.7) which gives the result. (ii) The proof of (ii) is immediate. Theorem 4.2. (i) For all real numbers γ, the mappings (i) f → (r2 + x2)γ f (ii) f →|r|γ f are isomorphisms from ᏺ onto itself. (ii) For f ∈ ᏺ,thefunctiong deﬁned by

⎧ √ ⎨ f r2 − x2,x if |r| |x|, = g(r,x) ⎩ (4.8) 0 otherwise,

2 belongs to ᏿∗(R ). Proof. (i) Let f ∈ ᏺ,byLeibnitzformula,wehave

k1 k2 ∂ ∂ γ r2 + x2 f (r,x) ∂r ∂x (4.9) k1 k2 k1+k2−i− j = j i 2 2 γ−i− j ∂ C C Pj(r)Pi(x) r + x − f (r,x), k1 k2 k1 j k2−i j=0 i=0 ∂r ∂x

where Pi and Pj are real polynomials. C. Baccar et al. 13

Let n ∈ N such that γ − k1 − k2 + n>0. By Taylor formula and the fact that f ∈ ᏺ,we have

− − k1 j 2n 1 k1 j+2n ∂ r − ∂ ( f )(r,x) = (1 − t)2n 1 ( f )(rt,x)dt ∂r (2n − 1)! 0 ∂r (4.10) − 2n +∞ k1 j+2n r − ∂ =− (1 − t)2n 1 ( f )(rt,x)dt, (2n − 1)! 1 ∂r

− − − − k2 i k1 j 2n 1 k2 i k1 j+2n ∂ ∂ r − ∂ ∂ f (r,x) = (1 − t)2n 1 f (rt,x)dt ∂x ∂r (2n − 1)! 0 ∂x ∂r

− − 2n +∞ k2 i k1 j+2n r − ∂ ∂ =− (1 − t)2n 1 f (rt,x)dt. (2n − 1)! 1 ∂x ∂r (4.11)

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

(r,x) −→ r2 + x2 γ f (r,x) (4.12) belongs to ᏺ and that the mapping

f −→ r2 + x2 γ f (4.13) is continuous from ᏺ onto itself. The inverse mapping is given by

− f −→ r2 + x2 γ f. (4.14)

By the same way, we show that the mapping

f −→ | r|γ f (4.15) is an isomorphism from ᏺ onto itself. (ii) Let f ∈ ᏺ,and ⎧ √ ⎨ f r2 − x2,x if |r| |x|, g(r,x) = (4.16) ⎩0if|r| |x|, we have ⎛ ⎞ k2 k1 k1 k2 p q+j ∂ ∂ = ⎝ ∂ ∂ 2 − 2 ⎠ (g)(r,x) Pj(r) Qp,q(x) 2 ( f ) r x ,x , ∂x ∂r j=0 p,q=0 ∂x ∂r (4.17) where Pj and Qp,q are real polynomials. This equality, together with the fact that f be- 2 longs to ᏺ, implies that g belongs to ᏿∗(R ). 14 Inversion formulas for Riemann-Liouville transform

Theorem 4.3. The Fourier transform Fα associated with Riemann-Liouville transform is 0 2 an isomorphism from ᏿∗(R ) onto ᏺ.

0 2 Proof. Let f ∈ ᏿∗(R ). From the relation (3.12), we get ∂ k ∂ k F ( f )(0,λ) = B ◦ F ( f ) (0,λ) ∂μ2 α ∂μ2 α ∂ k = B F ( f ) (0,λ) (4.18) ∂μ2 α ∂ k = F ( f )(λ,λ) = 0, ∂μ2 α

2 because suppFα( f ) ⊂{ (μ,λ) ∈ R , |μ| |λ|}, this shows that Fα maps injectively 0 2 ᏿∗(R )intoᏺ. On the other hand, let h ∈ ᏺ and ⎧ √ ⎨ h r2 − x2,x if |r| |x|, g(r,x) = (4.19) ⎩0if|r| |x|.

2 2 From Theorem 4.2(ii), g belongs to ᏿∗(R ), so there exists f ∈ ᏿∗(R ) satisfying 0 2 Fα( f ) = g. Consequently, f ∈ ᏿∗(R )andFα( f ) = h. From Lemmas 3.6, 4.1,andTheorem 4.3, we deduce the following result.

t 0 2 2 Corollary 4.4. The dual transform Rα is an isomorphism from ᏿∗(R ) onto ᏿∗,0(R ).

t 4.1. Inversion formula for Rα and Rα 1 Theorem 4.5. (i) The operator Kα deﬁned by

− π α K 1( f )(r,x) = Λ 1 μ2 + λ2 |μ|Λ ( f ) (r,x) (4.20) α α 22α+1Γ2(α +1) α

2 is an isomorphism from ᏿∗,0(R ) onto itself. 2 (ii) The operator Kα deﬁned by

− π α K 2(g)(r,x) = F 1 μ2 + λ2 |μ|F (g) (r,x) (4.21) α α 22α+1Γ2(α +1) α

0 2 is an isomorphism from ᏿∗(R ) onto itself. This theorem follows from Lemma 4.1,Theorems4.2 and 4.3.

2 0 2 Theorem 4.6. (i) For f ∈ ᏿∗,0(R ) and g ∈ ᏿∗(R ), there exists the inversion formula for Rα: = R 1 tR = 1 tR R g αKα α(g), f Kα α α( f ). (4.22)

2 0 2 t (ii) For f ∈ ᏿∗,0(R ) and g ∈ ᏿∗(R ), there exists the inversion formula for Rα: = tR 2R = 2R tR f αKα α( f ), g Kα α α(g). (4.23) C. Baccar et al. 15

0 2 Proof. (i) Let g ∈ ᏿∗(R ). From the relation (2.15), Proposition 3.4, Lemma 3.6,and Theorem 4.3,wehave +∞ 2 2 α t g(r,x) = μ + λ μΛα ◦ Rα(g)(μ,λ)Rα cos(μ.)exp(iλ.) (r,x)dm(μ,λ) R 0 +∞ 2 2 α t = Rα μ + λ μΛα ◦ Rα(g)(μ,λ)cos(μ.)exp(iλ.)dm(μ,λ) (r,x) R 0

− π α = R Λ 1 μ2 + λ2 |μ|Λ ◦ tR (g) (r,x) α α 22α+1Γ2(α +1) α α = R 1 tR αKα α(g)(r,x). (4.24)

This relation, together with Corollary 4.4 and Theorem 4.5(i), implies that Rα is an iso- ᏿ R2 ᏿0 R2 1tR morphism from ∗,0( )onto ∗( ), and that Kα α is its inverse; in particular for 2 f ∈ ᏿∗,0(R ), we have

1tR R = Kα α α( f ) f. (4.25)

2 (ii) Let f ∈ ᏿∗,0(R ). From (i), we have

1tR R = Kα α α( f ) f. (4.26)

0 2 Let us put g = Rα( f ), then g ∈ ᏿∗(R ), and we have

R−1 = 1tR α (g) Kα α(g), (4.27) and from Lemma 3.6, it follows that

− − π α R 1( ) = Λ 1 2 + 2 | |F ( ) , α g α 2α+1Γ2 μ λ μ α g 2 (α +1) (4.28) − − − π α tR 1R 1(g) = F 1 μ2 + λ2 |μ|F (g) = K 2(g), α α α 22α+1Γ2(α +1) α α which gives

= tR 2R f αKα α( f ). (4.29)

1 2 4.2. The expressions of the operators Kα and Kα . In the previous subsection, we have 1 2 Λ F defined the operators Kα and Kα in terms of Fourier transforms α and α.Here,we will give nice expressions of these operators using fractional powers of partial differential operators. For this, we need the following inevitable notations. (i) Ᏹ∗(R) is the space of even infinitely differentiable functions on R. (ii) ᏿∗(R)isthesubspaceofᏱ∗(R), consisting of functions rapidly decreasing together with all their derivatives. (iii) ᏿∗(R) is the space of even tempered distributions on R. 16 Inversion formulas for Riemann-Liouville transform

2 2 (iv) ᏿∗(R ) is the space of tempered distributions on R , even with respect to the ﬁrst variable. Each of these spaces is equipped with its usual topology. (i) For a ∈ R, a −1/2, dωa(r) is the measure deﬁned on [0,+∞[by 1 dω (r) = r2a+1dr. (4.30) a 2aΓ(a +1)

(ii) a is the Bessel operator deﬁned on ]0,+∞[by

d2 2a +1 d 1 = + , a − . (4.31) a dr2 r dr 2

R ωa ᏿ R (iii) For an even measurable function f on , T f is the element of ∗( ), defined by +∞ ωa = ∈ ᏿ R T f ,ϕ f (r)ϕ(r)dωa(r), ϕ ∗( ). (4.32) 0 R2 ν (iv) For a measurable function g on , even with respect to the first variable, Tg (resp., m ᏿ R2 Tg ) is the element of ∗( ), defined by +∞ ν = ν Tg ,ϕ g(r,x)ϕ(r,x)d (r,x), R 0 (4.33) +∞ m = ∈ ᏿ R2 resp., Tg ,ϕ g(r,x)ϕ(r,x)dm(r,x) , ϕ ∗ , R 0 where dν and dm are the measures defined by the relations (3.1)and(3.26). a ∈ R Definition 4.7. (i) The translation operator τr , r , associated with Bessel operator a is defined on ᏿∗(R) by the following. For all s ∈ R, ⎧ Γ π ⎪ (a +1) 2a 1 ⎪√ f r2 + s2 +2rscosθ sin θdθ if a>− , ⎨ πΓ(a +1/2) 0 2 a = τr f (s) ⎪ (4.34) ⎪ ⎩⎪ f (r + s)+ f |r − s| 1 if a =− . 2 2 (ii) The convolution product of f ∈ ᏿∗(R)andT ∈ ᏿∗(R)isdefinedby ∀ ∈ R ∗ = a r , T a f (r) T,τr f . (4.35)

(iii) The Fourier Bessel transform is deﬁned on ᏿∗(R)by +∞ ∀μ ∈ R, Fa( f )(μ) = f (r)ja(rμ)dωa(r), (4.36) 0

and on ᏿∗(R)by ∀ϕ ∈ ᏿∗(R), Fa(T),ϕ = T,Fa(ϕ) . (4.37) C. Baccar et al. 17

We have the following properties (we refer to [19]). (i) Fa is an isomorphism from ᏿∗(R)(resp.,᏿∗(R)) onto itself, and we have

−1 = Fa Fa. (4.38)

∈ ᏿ R ∈ R a ᏿ R (ii) For f ∗( ), and r , τr f belongs to ∗( ), and we have

a = Fa τr f (μ) ja(rμ)Fa( f )(μ). (4.39)

(iii) For f ∈ ᏿∗(R)andT ∈ ᏿∗(R), the function T ∗a f belongs to Ᏹ∗(R), and is slowly increasing, moreover ωa = ( ) ( ) (4.40) Fa TT∗a f Fa f Fa T . In the following, we will deﬁne the fractional powers of Bessel operator and the Lapla- cian operator deﬁned on R2 by

∂2 ∂2 Δ = + (4.41) ∂r2 ∂x2 1 2 that we use to give simple expressions of Kα and Kα . In [16], the author has proved that the mappings

−→ ωa −→ ωa z T|r|z , z T(2z+a+1Γ(z/2+a+1)/Γ(−z/2))|r|−z−2a−2 , (4.42) deﬁned initially for −2(a +1)< e(z) < 0, can be extended to a valued functions on ∗ ᏿∗(R), analytic on C\{−2(k + a),k ∈ N },andwehave ωa = ωa T|r|z Fa T(2z+a+1Γ(z/2+a+1)/Γ(−z/2))|r|−z−2a−2 . (4.43)

∗ Deﬁnition 4.8. For z ∈ C\{−(k + a), k ∈ N }, the fractional power of Bessel operator a is deﬁned on ᏿∗(R)by − z = ωa ∗ a f (r) T(22z+a+1Γ(z+a+1)/Γ(−z))|s|−2z−2a−2 a f (r). (4.44)

From the relations (4.40)and(4.43), we deduce that for f ∈ ᏿∗(R)andz ∈ C\{−(k + a), k ∈ N∗},wehave ωa ωa z = ( ) 2z (4.45) Fa T(−a) f Fa f T|r| . On the other hand, from [8, 10], we deduce that the mappings

−→ m √m z T(r2+x2)z , T 2/π(22z+α+1Γ(z+1)Γ(α+1)/Γ(−z))(r2+x2)−z−1 , (4.46)

2 deﬁned initially for −1 < e(z) < 0, can be extended to a valued functions in ᏿∗(R ), analytic on C\{−k, k ∈ N∗},andwehave m = Λ √m T(r2+x2)z α T 2/π(22z+α+1Γ(z+1)Γ(α+1)/Γ(−z))(r2+x2)−z−1 , (4.47) 18 Inversion formulas for Riemann-Liouville transform

2 where Λα is defined on ᏿∗(R )by 2 Λα(T),ϕ = T,Λα(ϕ) , ϕ ∈ ᏿∗ R , (4.48) and Λα(ϕ)isgiveninLemma 3.6. Definition 4.9. For z ∈ C\{−k, k ∈ N∗}, the fractional power of the Laplacian operator 2 Δ is defined on ᏿∗(R )by −Δ z = m ( ) f (r,x) T(1/π)(22z+1Γ(z+1)/Γ(−z))(s2+y2)−z−1∗ f (r,x), (4.49) where (i) ∗ is the usual convolution product defined by ˇ 2 2 T ∗ f (r,x) = T,σ(r,x) f , T ∈ ᏿∗ R , f ∈ ᏿∗ R ; (4.50)

(ii) 1 2 σ f (s, y) = f (r + s, y − x)+ f (r − s, y − x) , f ∈ ᏿∗ R . (4.51) (r,x) 2

2 2 It is well known that for f ∈ ᏿∗(R )andT ∈ ᏿∗(R ), the function T ∗ f belongs to 2 Ᏹ∗(R ) and is slowly increasing, and we have

Λ m = Λ Λ α TT∗ f α( f ) α(T), (4.52)

2 thus from the relations (4.47)and(4.52), we deduce that for f ∈ ᏿∗(R )andz ∈ C\ {−k, k ∈ N∗}, Λ √m = Λ m α T 2π2αΓ(α+1)(−Δ)z f α( f )T(r2+x2)z . (4.53)

1 Theorem 4.10. The operator Kα deﬁned in Theorem 4.5 can be written as π ∂2 1/2 K 1( f ) = − (−Δ)α f , (4.54) α 22α+1Γ2(α +1) ∂r2 where 2 1/2 ∂ 1/2 − f (r,x) = − − f (·,x) (r). (4.55) ∂r2 1/2

2 2 Proof. Let f ∈ ᏿∗,0(R ). Using Fubini’s theorem, we get for every ϕ ∈ ᏿∗(R )thefol- lowing: Λ m α T(−∂2/∂r2)1/2 f ,ϕ +∞ 1 ω−1/2 = T − 1/2 · ,F− ϕ(·, y) × exp(−ixy)dxdy 2 +2 2 ( −1/2) ( f ( ,x)) 1/2 2 α Γ (α +1) R 0 (4.56) C. Baccar et al. 19 and by the relation (4.45), we obtain Λ m α T(−∂2/∂r2)1/2 f ,ϕ +∞ 1 ω−1/2 = − · · × − 2 +2 2 F 1/2 f ( ,x) T|r| ,ϕ( , y) exp( ixy)dxdy, 2 α Γ (α +1) R 0 (4.57) which involves that +∞ Λ m = Λ α T(−∂2/∂r2)1/2 f ,ϕ r α( f )(r, y)ϕ(r, y)dm(r, y), (4.58) R 0 this shows that m m Λ 2 2 1/2 = (4.59) α T(−∂ /∂r ) f T|r|Λα( f ).

Now, from Lemma 4.1, we deduce that the function

(μ,λ) −→ | μ|Λα( f )(μ,λ) (4.60) belongs to the subspace ᏺ. Then, from the relation (4.59), it follows that the function 2 2 1/2 2 (−∂ /∂r ) f belongs to the subspace ᏿∗,0(R ), and we have ∂2 1/2 ∀(μ,λ) ∈ R2, Λ − f (μ,λ) =|μ|Λ (μ,λ). (4.61) α ∂r2 α

2 By the same way, and using the relation (4.53), we deduce that for every f ∈ ᏿∗,0(R ), the α 2 2 function (−Δ) f belongs to the subspace ᏿∗,0(R ), and we have that for all (μ,λ) ∈ R , √ α α 2 2 α Λα 2π2 Γ(α +1)(−Δ) f (μ,λ) = μ + λ Λα( f )(μ,λ). (4.62)

Hence, the theorem follows from the relations (4.61)and(4.62). Deﬁnition 4.11. Let a,b ∈ R, b a −1/2. (i) The Sonine transform is the mapping deﬁned on Ᏹ∗(R) by the following. For all r ∈ R, ⎧ 1 ⎪ 2Γ(b +1) b−a−1 ⎨⎪ 1 − t2 f (rt)t2a+1dt if b>a, = Γ(b − a)Γ(a +1) 0 Sb,a( f )(r) ⎪ (4.63) ⎩⎪ f (r)ifb = a.

t (ii) The dual transform Sb,a is the mapping deﬁned on ᏿∗(R) by the following. For all r ∈ R, ⎧ +∞ ⎪ 2Γ(b +1) b−a−1 ⎨⎪ t2 − r2 f (t)tdt if b>a, t = Γ(b − a)Γ(a +1) r Sb,a( f )(r) ⎪ (4.64) ⎩⎪ f (r)ifb = a. 20 Inversion formulas for Riemann-Liouville transform

Then, we have the following. (i) The Sonine transform is an isomorphism from Ᏹ∗(R) onto itself. (ii) The dual Sonine transform is an isomorphism from ᏿∗(R) onto itself. (iii) For f ∈ Ᏹ∗(R), f bounded, and g ∈ ᏿∗(R), we have +∞ +∞ 2b+1 t 2a+1 Sb,a( f )(r)g(r)r dr = f (r) Sb,a(g)(r)r dr. (4.65) 0 o

(iv) jb = Sb,a(ja). (v)

Γ(a +1) F = F ◦ tS . (4.66) b 2b−aΓ(b +1) a b,a

Formoredetails,wereferto[18, 20, 21]. We denote the following. 2 2 (i) For T ∈ ᏿∗(R ), ϕ ∈ ᏿∗(R ), Sa,0(T),ϕ = T,ψ , (4.67)

t with ψ(r,x) = Sa,0(ϕ(·,x))(r). (ii) For all (r,x) ∈ R2, T#ϕ(r,x) = T,᐀(r,−x)ϕˇ , (4.68) where ᐀(r,x) is the translation operator given by Deﬁnition 3.1. 2 (iii) Fα is the mapping deﬁned on ᏿ ∗(R )by 2 ∀ϕ ∈ ᏿∗(R ), Fα(T),ϕ = T,Fα(ϕ) . (4.69)

2 (iv) Lα is the operator deﬁned on ᏿∗(R )by

2α Lα f (r,x) = − α f (·,x) (r), (4.70)

z where (−α) is the fractional power of Bessel given by Deﬁnition 4.8. 2 Theorem 4.12. The operator Kα ,deﬁnedinTheorem 4.5,isgivenby

2 π 0 2 K ( f )(r,x) = S (T)# − Δ L ( fˇ)(r,−x), f ∈ ᏿∗ R , (4.71) α 24α+2Γ4(α +1) α,0 2 α where (i) T is the distribution deﬁned by T,ϕ = ϕ(y, y)dy; (4.72) R

(ii) Δ2 is the operator deﬁned in Section 2. C. Baccar et al. 21

2 ∈ ᏿0 R2 Proof. By the deﬁnition of Kα , and the relation (3.12), we have that for f ∗( ),

2 Kα ( f )(r,x) √ +∞ = π/2 2 2 2 2αF 2 2 2 2 3 +1 3 μ μ + λ α( f ) μ + λ ,λ jα r μ + λ exp(iλx)dμdλ. 2 α Γ (α+1) R 0 (4.73)

By a change of variables, and using Fubini’s theorem, we get √ +∞ ν 2 = π/2 ν4α ν2 − 2 F ν √exp(iλx) ν ν ν Kα ( f )(r,x) 3 +1 3 λ α( f )( ,λ) jα(r ) d dλ. 2 α Γ (α +1) 0 −ν ν2 − λ2 (4.74)

0 2 2 On the other hand, for f ∈ ᏿∗(R ), the function Lα f belongs to Ᏹ∗(R ), and is slowly increasing. Moreover, we have F ν = ν α T T 4α . (4.75) Lα f |μ| Fα( f )

0 2 But, for f ∈ ᏿∗(R ), the function Fα( f )belongstothesubspaceᏺ;accordingto 2 Theorem 4.2, we deduce that the function Lα f belongs to ᏿∗(R ), and we have

2 4α ∀(μ,λ) ∈ R , Fα Lα f (μ,λ) =|μ| Fα( f )(μ,λ). (4.76)

This involves that √ +∞ ν 2 = π/2 ν2 − 2 F ν √exp(iλx) ν ν ν Kα ( f )(r,x) 3 +1 3 λ α Lα f ( ,λ) jα(r ) d dλ 2 α Γ (α +1) 0 −ν ν2 − λ2 √ +∞ ν = π/2 F − Δ ν √exp(iλx) ν ν ν 3 +1 3 α 2 Lα f ( ,λ) jα(r ) d dλ. 2 α Γ (α +1) 0 −ν ν2 − λ2 (4.77)

2 Since for every f ∈ ᏿∗(R ), we have that

2 ∀(r,x),(μ,λ) ∈ R , Fα ᐀(r,x) f (ν,λ) = jα(rν)exp(iλx)Fα( f )(ν,λ), (4.78) we get √ +∞ ν ν ν 2 = π/2 F ᐀ − Δ ν √ d dλ Kα ( f )(r,x) 3 +1 3 α (r,x) 2 Lα f ( ,λ) . (4.79) 2 α Γ (α +1) 0 −ν ν2 − λ2 Using the expression of Fα,weobtain +∞ ν +∞ 2 = 1 ᐀ − Δ Kα ( f )(r,x) (r,x) 2 Lα f (s, y) 4α+2Γ4 −ν R 2 (α +1) 0 0 dλ × j (sν)exp(−iλy)s2α+1dsdy √ νdν. α ν2 − λ2 (4.80) 22 Inversion formulas for Riemann-Liouville transform

From the fact that ν − exp(√ iλy) dλ = πj0(νy), (4.81) −ν ν2 − λ2 and using Fubini’s theorem, we deduce that

2 Kα ( f )(r,x) +∞ = π ᐀ − Δ ν × ν 2α+1 ν ν 4 +2 4 (r,x) 2 Lα f (s, y)jα(s ) j0( y)s ds d dy 2 α Γ (α +1) R 0 +∞ = π ᐀ − Δ · ν ν ν ν 3 +2 3 Fα (r,x) 2 Lα f ( , y) ( )j0( y) d dy, 2 α Γ (α +1) R 0 (4.82) and from the relation (4.66), we have

2 Kα ( f )(r,x) +∞ = π ◦ t ᐀ − Δ · ν × ν ν ν 4 +2 4 F0 Sα,0 (r,x) 2 Lα f ( , y) ( ) j0( y) d dy, 2 α Γ (α +1) R 0 (4.83) and the relation (4.38) implies that

2 = π t ᐀ − Δ · Kα ( f )(r,x) Sα,0 (r,x) 2 Lα f ( , y) (y)dy. (4.84) 24α+2Γ4(α +1) R

t 4.3. Plancherel theorem for Rα 3 Proposition 4.13. The operator Kα deﬁned by ∂2 1/4 K 3( f ) = π − (−)α/2 f (4.85) α ∂r2

2 is an isomorphism from ᏿∗,0(R ) onto itself, where 2 1/4 ∂ 1/4 − f (r,x) = − − f (·,x) (r). (4.86) ∂r2 1/2

2 Proof. Let f ∈ ᏿∗,0(R ). From the relations (4.45)and(4.53), we deduce that for all (μ,λ) ∈ R2, √ 2 1/4 α/2 ∂ |μ| μ2 + λ2 Λ ( f )(μ,λ) = Λ 2π2αΓ(α +1) − (−Δ)α/2 f (μ,λ), α α ∂r2 (4.87) C. Baccar et al. 23 which implies that for all (μ,λ) ∈ R2, π 1 α/2 Λ K 3 ( f )(μ,λ) = |μ| μ2 + λ2 Λ ( f )(μ,λ). (4.88) α α 2 2αΓ(α +1) α

Then, the result follows from Lemma 4.1 and Theorem 4.2.

0 2 Proposition 4.14. For g ∈ ᏿∗(R ), there exists the Plancherel formula +∞ +∞ 2 ν = 3 tR 2 g(r,x) d (r,x) Kα α(g) (r,x) dm(r,x). (4.89) R 0 R 0

0 2 Proof. Let g ∈ ᏿∗(R ), from Theorem 3.5 (Plancherel formula), we have +∞ 2 2 g(r,x) dν(r,x) = Fα(g)(μ,λ) dγ(μ,λ). (4.90) R 0 Γ

Fromtherelation(3.12), Lemma 3.6, and the fact that 2 suppFα(g) ⊂ (μ,λ) ∈ R /|μ| |λ| , (4.91) we get +∞ +∞

2 2 2 α/2 t 2 g(r,x) dν(r,x) = μ(μ + λ ) Λα ◦ Rα(g)(μ,λ) dm(μ,λ). R 0 R 0 (4.92)

We complete the proof by using the formula (4.88), and the fact that for every f ∈ 2 ᏿∗(R ), +∞ +∞ Λ 2 = π 2 α( f )(μ,λ) dm(μ,λ) 2 +1 2 f (μ,λ) dm(μ,λ). (4.93) R 0 2 α Γ (α +1) R 0

We denote by 2 ν 2 ν (i) L0(d )thesubspaceofL (d ) consisting of functions g such that 2 suppFα(g) ⊂ (μ,λ) ∈ R /|μ| |λ| ; (4.94)

(ii) L2(dm) the space of square integrable functions on [0,+∞[×R with respect to the measure dm(r,x). 3 ◦ tR Theorem 4.15. The operator Kα α can be extended to an isometric isomorphism from 2 ν 2 L0(d ) onto L (dm). 24 Inversion formulas for Riemann-Liouville transform

2 Proof. The theorem follows from Propositions 4.13, 4.14, and the density of ᏿∗,0(R ) ᏿0 R2 2 2 ν (resp., ∗( )) in L (dm)(resp.,L0(d )).

5. Transmutation operators Proposition 5.1. The Riemann-Liouville transform and its dual satisfy the following permutation properties. 2 (i) For all f ∈ ᏿∗(R ),

∂2 tR Δ f = tR ( f ), tR Δ f = Δ tR ( f ). (5.1) α 2 ∂r2 α α 1 1 α

2 (ii) For all f ∈ Ᏹ∗(R ),

∂2 f Δ R ( f ) = R , Δ R ( f ) = R Δ f . (5.2) 2 α α ∂r2 1 α α 1

2 2 t Proof. (i) We know that the operators Δ1, Δ2, ∂ /∂r ,and Rα are continuous mappings 2 from ᏿∗(R ) into itself. Then, by applying the usual Fourier transform Λα,wehave

2 Λ tR Δ =− 2Λ ◦ tR = Λ ∂ tR α α 2 f (μ,λ) μ α α( f )(μ,λ) α 2 α( f ) (μ,λ), ∂r (5.3)

t t t Λα Δ1 Rα f (μ,λ) = iλΛα Rα( f ) (μ,λ) = Λα Rα Δ1 f (μ,λ).

2 Consequently, (i) follows from the fact that Λα is an isomorphism from ᏿∗(R )onto itself. 2 (ii) We obtain the result from (i), Lemma 2.4, and the fact that for f ∈ Ᏹ∗(R ), and 2 g ∈ Ᏸ∗(R ),

+∞ +∞ Δ2 f (r,x)g(r,x)dν(r,x) = f (r,x)Δ2g(r,x)dν(r,x). (5.4) R 0 R 0

Theorem 5.2. (i) The Riemann-Liouville transform Rα is a transmutation operator of

∂2 ,Δ into Δ , Δ (5.5) ∂r2 1 2 1 from

2 0 2 ᏿∗,0 R onto ᏿∗ R . (5.6) C. Baccar et al. 25

t (ii) The dual transform Rα is a transmutation operator of ∂2 Δ , Δ into , Δ (5.7) 2 1 ∂r2 1 from

0 2 2 ᏿∗(R ) onto ᏿∗,0 R . (5.8)

This theorem follows from Proposition 5.1 and the fact that Rα is an isomorphism 2 0 2 t 0 2 2 from ᏿∗,0(R )onto᏿∗(R )and Rα is an isomorphism from ᏿∗(R )onto᏿∗,0(R ).

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with a singular diﬀerential operator on (0,∞)], Journal de Mathematiques´ Pures et Appliquees.´ Neuvieme` Serie´ 60 (1981), no. 1, 51–98. [19] , Inversion of the Lions transmutation operators using generalized wavelets,Appliedand Computational Harmonic Analysis 4 (1997), no. 1, 97–112. [20] , Generalized Harmonic Analysis and Wavelet Packets: An Elementary Treatment of Theory &Applications, Gordon and Breach Science, Amsterdam, 2001. [21] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, London, 1966.

C. Baccar: Department of Mathematics, Faculty of Sciences of Tunis, University Tunis El Manar, 2092 Tunis, Tunisia

N. B. Hamadi: Department of Mathematics, Faculty of Sciences of Tunis, University Tunis El Manar, 2092 Tunis, Tunisia

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