WEYL TRANSFORMS ASSOCIATED WITH THE RIEMANN-LIOUVILLE OPERATOR

In his book [14], Wong studies the properties of pseudodifferential operators arising in quantum mechanics, first envisaged by Weyl [13], as bounded linear operators on L2(Rn) (the space of square integrable functions on Rn with respect to the Lebesgue measure). For this reason, M. W. Wong calls the operators treated in his book Weyl transforms. Here, we consider the singular partial differential operators


Introduction
In his book [14], Wong studies the properties of pseudodifferential operators arising in quantum mechanics, first envisaged by Weyl [13], as bounded linear operators on L 2 (R n ) (the space of square integrable functions on R n with respect to the Lebesgue measure).For this reason, M. W. Wong calls the operators treated in his book Weyl transforms.

Weyl transforms
The transform α generalizes the mean operator defined by 0 ( f )(r,x) = 1 2π 2π 0 f (r sinθ,x + r cos θ)dθ. (1. 3) The mean operator 0 and its dual play an important role and have many applications, for example, in image processing of the so-called synthetic aperture radar (SAR) data [5,6], or in the linearized inverse scattering problem in acoustics [3].
In [1], we have defined a convolution product and a Fourier transform Ᏺ α associated with α , and, we have established many harmonic analysis results (inversion formula, Paley-Wiener, and Plancherel theorems, etc.).
Using these results, we define and study, in this paper the Weyl transforms associated with α , we give criteria in terms of symbols to prove the boundedness and compactness of these transforms.To obtain these results, we have first defined the Fourier-Wigner transform associated with the operator α , and we have established for it an inversion formula.
More precisely, in Section 2, we recall some properties of harmonic analysis for the operator α .In Section 3, we define the Fourier-Wigner transform associated with α , study some of its properties, and prove an inversion formula.
In Section 4, we introduce the Weyl transform W σ associated with α , with σ a symbol in class S m , for m ∈ R, and we give its connection with the Fourier-Wigner transform.We prove that for σ sufficiently smooth, W σ is a compact operator from L 2 (dν), the space of square integrable functions on [0,+∞[×R, with respect to the measure In Section 5, we define W σ for σ in a certain space L p (dν ⊗ dγ), with p ∈ [1,2], and we establish that W σ is again a compact operator.
In Section 6, we define W σ for σ in another function space, and use this to prove in Section 7 that for p > 2, there exists a function σ ∈ L p (dν ⊗ dγ), with the property that the Weyl transform W σ is not bounded on L 2 (dν).

Riemann-Liouville transform associated with the operators Δ 1 and Δ 2
In this section, we recall some properties of the Riemann-Liouville transform that we use in the next sections.For more details, see [1].For all (μ,λ) ∈ C × C, the system N. B. Hamadi and L. T. Rachdi 3 admits a unique solution given by where j α is the modified Bessel function defined by and J α is the Bessel function of first kind and index α (see [7,12]).Moreover, we have where Γ is the set defined by Proposition 2.1.The eigenfunction ϕ μ,λ given by (2.2) has the following Mehler integral representation: This result shows that where α is the Riemann-Liouville transform associated with the operators Δ 1 and Δ 2 , given in the introduction.We denote by (i) Ꮿ * ,c (R 2 ) the subspace of Ꮿ * (R 2 ) consisting of functions with compact support; (ii) dν(r,x) the measure defined on [0,+∞[×R by p dν(r,x) (2.9) (iv) dγ(μ,λ) the measure defined on Γ by (2.10) (v) L p (dγ), p ∈ [1,+∞], the space of measurable functions on Γ satisfying (2.12) (ii) The convolution product associated with the Riemann-Liouville transform of f ,g ∈ L 1 (dν) is defined by where f (s, y) = f (s,−y).
We have the following properties.
(i) We have the following product formula: (ii) Let f be in L 1 (dν).Then, for all (s, y) ∈ [0,+∞[×R, we have where Γ is the set defined by the relation (2.5).
We have the following properties.
(i) Let f be in L 1 (dν).For all (r,x) ∈ [0,+∞[×R, we have where, for every (μ,λ Proof.The mapping Ᏺ α given by the relation (2.23) is an isometric isomorphism from On the other hand, we have Ᏺ α ( f ) ∞,ν f 1,ν .Thus, from these relations and the Riesz-Thorin theorem [10,11], we deduce that for all f ∈ L p (dν), with p ∈ [1,2], the function Ᏺ α ( f ) belongs to L p (dν), with p = p/(p − 1), and we have We complete the proof by using the fact that which is a consequence of the relation (2.22).
We denote by (see [1,9]) (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 → C infinitely differentiable, even with respect to the first variable and rapidly decreasing together with all their derivatives, that is, for all where (2.29) Each of these spaces is equipped with its usual topology.

Fourier-Wigner transform associated with Riemann-Liouville operator
Defintion 3.1.The Fourier-Wigner transform associated with the Riemann-Liouville operator is the mapping Remark 3.2.The transform V can also be written in the forms , where ǧ(s, y) = g(s,−y) and * is the convolution product given in Definition 2.2.
The transform V can be extended to a continuous bilinear operator, denoted also by V , from L p (dν) × L p (dν) into L p (dν ⊗ dγ), where p = p/(p − 1) is the conjugate exponent of p.
Using the previous theorem and the relation (2.25), we get the following result.

Weyl transform associated with Riemann-Liouville operator
In this section, we introduce and study the Weyl transform and give its connection with the Fourier-Wigner transform.To do this, we must define the class of pseudodifferential operators [14].
Proposition 4.3.Let σ be the symbol given by where (4.6) Proof.From relations (3.1), (4.2) and Fubini's theorem we get, for all (r,x The result follows from relation (2.25) and the fact that  Proof.Let f ∈ * (R 2 ), since Ᏺ α is an isomorphism from * (R 2 ) onto itself, and we deduce that for all (r,x) ∈ [0,+∞[×R, the function (s, y) → -(r,x) f (s, y) belongs to * (R 2 ).Then, by the inversion formula for Ᏺ α , we get, for all (s, y) ∈ R 2 ; By Definition 4.4 and Fubini's theorem, we obtain, for all (r,x) ∈ R 2 , Now, the function On the other hand, the mapping f → G f , given for all ((t,z),(μ,λ) , and for all (r,x) ∈ R 2 , we have Proof.The function k can be written in the form Since the Fourier transform Ᏺ α is an isomorphism from * (R 2 ) onto * (Γ), we deduce that the function . Then, the lemma follows from the fact that for all g ∈ * (R 2 × R 2 ), the function is a Hilbert-Schmidt operator, and consequently it is compact.
(iii) From (ii) and the fact that the space * (R 2 ) is dense in L p (dν), p ∈ [1,+∞[, we deduce that W σ can be extended to a continuous mapping from L p (dν) into L p (dν).
By Lemma 4.6, the kernel k belongs to L 2 (dν ⊗ dν), hence W σ is a Hilbert-Schmidt operator.In particular, it is compact.
(ii) The case p = 1 can be obtained by the same way.

Weyl transform with symbol in
We denote by (i) * (R 2 ) the space of tempered distributions on R 2 , even with respect to the first variable.It is the topological dual of * (R 2 ); (ii) * (R 2 × Γ) the space of tempered distributions on R 2 × Γ, even with respect to the first variables of R 2 and Γ.It is the topological dual of * (R 2 × Γ).
Proposition 6.3.Let σ 1 ∈ S * (R 2 × Γ), given by the function equal to 1.One has where c = R ∞ 0 g(r,x)dν(r,x) and δ is the Dirac distribution at (0,0).Proof.By relation (6.1), we have for all f in * (R 2 ), (6.4) and by Theorem 3.6 We break down the proof into two lemmas, of which the theorem is an immediate consequence.
Lemma 7.2.Let 2 < p < ∞.Suppose that for all σ ∈ L p (dν ⊗ dγ), the Weyl transform W σ given by relation (6.1) is a bounded linear operator on L 2 (dν).Then, there exists a positive constant M such that Proof.Under the assumption of the lemma, there exists for each σ Let f ,g ∈ * (R 2 ) such that f 2,ν = g 2,ν = 1, and let us define the operator By the Banach-Steinhauss theorem, the operator Q f ,g is bounded on L p (dν ⊗ dγ), then there exists a positive constant M such that From this, we deduce that for all f ,g ∈ * (R 2 ), and σ ∈ L p (dν ⊗ dγ), we have which implies (7.1).
Proof.Suppose that there exists M > 0 such that relation (7.1) holds.