SPACES OF DLp TYPE AND A CONVOLUTION PRODUCT ASSOCIATED WITH THE SPHERICAL MEAN OPERATOR

We define and study the spaces ℳp(ℝ×ℝn), 1≤p≤∞, that are of DLp type. Using the harmonic analysis associated with the spherical mean operator, we give a new characterization of the dual space ℳ′p(ℝ×ℝn) and describe its bounded subsets. Next, we define a convolution product in ℳ′p(ℝ×ℝn)×Mr(ℝ×ℝn), 1≤r≤p<∞, and prove some new results.


Introduction
The spherical mean operator is defined, for a function f on R n+1 , even with respect to the first variable, by ( f )(r,x) = S n f (rη,x + rξ)dσ n (η,ξ), (r,x) ∈ R × R n , (1.1) where S n is the unit sphere {(η, ξ) ∈ R × R n : η 2 + ξ 2 = 1} in R n+1 and σ n is the surface measure on S n normalized to have total measure one.This operator plays an important role and has many applications, for example, in image processing of so-called synthetic aperture radar (SAR) data (see [7,8]), or in the linearized inverse scattering problem in acoustics [6].In [10], the authors associate to the operator a Fourier transform and a convolution product and have established many results of harmonic analysis (inversion formula, Paley-Wiener and Plancherel theorems, etc.).
In [11], the authors define and study Weyl transforms related to the mean operator and have proved that these operators are compact.The spaces D L p , 1 ≤ p ≤ ∞, have been studied by many authors [1,2,4,5,12,13].In this work, we introduce the function spaces ᏹ p (R × R n ), 1 ≤ p ≤ ∞, similar to D L p , but replace the usual derivatives by the operator where l is the Bessel operator defined on ]0,+∞[ by (1. 3) The main result of this paper gives a new characterization of the dual space ᏹ p (R × R n ) of the space ᏹ p (R × R n ) and a description of its bounded subsets.More precisely, in Section 2, we recall some harmonic results related to a convolution product and the Fourier transform connected with the spherical mean operator, that we use in the following sections.
In the Section 3, we define the space ᏹ p (R × R n ), 1 ≤ p ≤ ∞, to be the space of measurable functions f on ]0,+∞[ × R n+1 such that for all k ∈ N, L k f belongs to the space L p (dν) (the space of functions of pth power integrable on [0,+∞[ × R n+1 with respect to the measure r n dr ⊗ dx).We give some properties of this space, in particular we prove that it is a Frechet space.
Section 4 is consecrated to the study of the dual space ᏹ p (R × R n ).We give a nice description of the elements of this space and we characterize its bounded subsets.
In the last section, we define and study a convolution product in

Spherical mean operator
In this section, we define and recall some properties of the spherical mean operator.For more details see [3,6,10,11].We denote by (A) Ᏹ * (R × R n ) the space of infinitely differentiable functions on R × R n , even with respect to the first variable, (B) S n the unit sphere in R × R n , where for ξ = (ξ 1 ,...,ξ n ), we have ξ 2 = ξ 2 1 + ••• + ξ 2 n , (C) dσ the normalized surface measure on S n .Definition 2.1.The spherical mean operator is defined on (2.2) We have where j (n−1)/2 is the normalized Bessel function defined by with J (n−1)/2 the Bessel function of first kind and index (n − 1)/2 [9,15], and if The normalized Bessel function j (n−1)/2 has the following Mehler integral representation: and therefore Moreover, for all λ ∈ C, the function is the unique solution of the differential equation where l is the Bessel operator defined on ]0,+∞[ by (1.3).On the other hand, the function ϕ µ,λ is the unique solution of the system where D j = ∂/∂x j , and ∆ is the Laplacien operator on R n : (2.11) Now let Γ be the set (2.12) We have for all (µ,λ) ∈ Γ, In the following, we will define a convolution product and the Fourier transform associated with the spherical mean operator.For this, we use the product formula for the functions ϕ µ,λ .For all (r,x),(s, y We denote by (see [11]) with (2.17) (C) dγ(µ,λ) the measure defined on the set Γ by (2.18) (D) L p (dγ), 1 ≤ p ≤ +∞, the space of measurable functions on Γ, satisfying The translation operator associated with the spherical mean operator is defined on L 1 (dν) by for all (r,x),(s, y) (ii) A convolution product associated with the spherical mean operator of f ,g ∈ L 1 (dν) is defined by for all (r,x) (2.21) where g(r,x) = g(r,−x). (2.22) We have the following properties.

The space ᏹ
We denote by (A) L the partial differential operator defined by From Proposition 2.4 and Remark 2.5, we deduce that for all f ∈ L p (dν), 1 ≤ p ≤ 2, Ᏺ f belongs to the space L p (dγ) and we have even with respect to the first variable, and such that for all k ∈ N there exists g k ∈ L p (dν) satisfying The space ᏹ p (R × R n ) is equipped with the topology generated by the family of norms where g k ,k ∈ N, is the function given by the relation (3.5).Let (3.8) In the following, we will give some properties of the space Then for all k ∈ N, (g m,k ) m∈N is a Cauchy sequence in L p (dν).We put From relations (3.9) and (3.11), we deduce that belongs to the space L p (dγ) with p = p/(p − 1); is the space of continuous functions on R × R n even with respect to the first variable. Proof.
From relation (3.4), we have which gives On the other hand hence This equality, together with the fact that the function Ᏺ(g k ) belongs to the space L p (dν) implies (i). (ii From the assertion (i) and relations (2.26) and (2.31), we deduce that for all k ∈ N, the function On the other hand, the transform Ᏺ is an isometric isomorphism from L 2 (dν) onto itself, then from the inversion formula for Ᏺ and using the continuity of the function f , we have for all (r,x Consequently, (ii) follows from relation (2.7) and the fact that for all k ∈ N, α ∈ N n , the function belongs to the space L 1 (dν).
, r ≥ 2, and r = r/(r − 1).From Proposition 3.3, we deduce that f ∈ Ᏹ * (R × R n ) and for all k ∈ N, the function (3.21) belongs to the space L p (dν).By applying Holder's inequality, it follows that this last function belongs to the space L r (dν).On the other hand, for all (r,x (3.25) From Proposition 2.4 and the fact that we deduce that, for all k ∈ N, the function L k f belongs to the space L r (dν).

The dual space ᏹ
In this section, we will give a new characterization of the dual space In addition, T ∈ ᏹ p (R × R n ) if and only if there exist m ∈ N and c > 0 such that This proves that for all f ∈ L p (dν) and k ∈ N, the functional L k T f defined by the relation (4.3) belongs to the space ᏹ p (R × R n ).
In the following, we will prove that every element of ᏹ p (R × R n ) is also of this type.
where L k T fk is given by relation (4.3).
Proof.It is clear that if then T belongs to the space equipped with the norm We consider the mappings where From relation (4.2) we deduce that This means that Ꮾ is a continuous functional on the subspace Im(Ꮽ) of the space (L p (dν)) m+1 .From Hahn-Banach theorems, there exists a continuous extension of Ꮾ to (L p (dν)) m+1 , denoted again by Ꮾ.
We denote by ) the space of infinitely differentiable functions on R × R n , even with respect to the first variable and with compact support, equipped with its usual topology; even with respect to the first variable and with support in B(0,a), normed by Proof.Let p ≥ n + 1 and g p the function defined by Using relation (2.7), we deduce that there exists p o ∈ N such that for all p ≥ p o the function g p is of class C 2m on R × R n (e.g., we can choose p o = 3n + 1 + 2m).Now, we prove that the function g p is infinitely differentiable on R × R n \ {(0, ...,0)}.The function g p can be written as By relation (2.6) and Fubini's theorem we get where On the other hand, we have e isu 1 + s 2 p ds , (4.30) then, we get where Q p is a real polynomial.Since h p is an even function on R, then we deduce that where k p is the infinitely differentiable function defined on R by Now, the function is infinitely differentiable on R and we have M. Dziri et al. 371 This shows that the function g p is infinitely differentiable on R × R n \ {(0, ...,0)}, even with respect to the first variable.
Let On the other hand, by using the fact that the function g p is infinitely differentiable on R × R n \ {(0, ...,0)}, we deduce that the function Moreover, from relation (4.37), we have and this implies by using relation (4.38) that and this completes the proof of the proposition by taking ψ p = γg p .
To prove the main result of this section, that is, Theorem 4.7, we will define some new families of norms on the space Ᏸ * ,a (R × R n ).We use these norms to prove that the elements of all bounded subset where l is defined by relation (1.3).where P s is a real polynomial.On the other hand, and also by induction, we deduce that for all s ≥ 1, and by Holder's inequality, we get 1/p 2 m+m1 N p,m+m1 (ϕ), (4.49) which implies that and the proof of the lemma is complete.
Theorem 4.6.Let a > 0 and B a weakly * bounded set of Ᏸ * ,a (R × R n ).Then, there exists m ∈ N such that the elements of B can be continuously extended to ᐃ m a (R × R n ).Moreover, the family of these extensions is equicontinuous.[14] and Lemma 4.5 there exist a positive constant c and m ∈ N such that for all T ∈ B , for all (4.51) We consider the mappings and for all T ∈ B , From relation (4.51), we deduce that for all This means that L T is a continuous functional on the subspace A(D * ,a (R × R n )) of the space (L p (dν)) m+1 and that for all T ∈ B , From the Hahn-Banach theorems, L T can be continuously extended on (L p (dν)) m+1 , denoted again by L T .Furthermore, for all T ∈ B , Now, from the Riez theorem, there exists ( Thus, from (4.56) it follows that for all Using Holder's inequality and relation (4.59), we get for all This shows that the mapping L T oA is a continuous extension of T on ᐃ m a (R × R n ) and that the family {L T oA} T∈B is equicontinuous, when applied to ᐃ m a (R × R n ).This completes the proof of Theorem 4.6.
In the following, we will give a new characterization of the space , the function T * ϕ belongs to the space L p (dν), where From Theorem 4.1, there exist m∈N and f 0 ,..., f m ∈ L p (dν) such that , then from inequality (2.24), we deduce that f k * L k ϕ ∈ L p (dν).This implies that the function T * ϕ belongs to the space L p (dν).Conversely, let From Holder's inequality and using the hypothesis, we obtain M. Dziri et al. 375 from which we deduce that the set Now, using Theorem 4.6, it follows that for all a > 0 there exists m ∈ N such that for all ϕ ∈ Ᏸ * (R × R n ), ϕ p,ν ≤ 1, the mapping T T * ϕ can be continuously extended on the space ᐃ m a (R × R n ) and the family of these extensions is equicontinuous, which means that there exists c > 0 such that for all This involves that for all On the other hand, we have for all where for all ϕ ∈ S * (R × R n ), This last inequality shows that the functional T * T ψ can be continuously extended on the space L p (dν) and from Riez's theorem, there exists g ∈ L p (dν) such that We complete the proof by using the hypothesis, relation (4.73), and Theorem 4.1.
In the following, we will give a characterization of the bounded sets in ᏹ p (R × R n ).(i) B is weakly bounded in ᏹ p (R × R n ), (ii) there exist c > 0 and m ∈ N such that for every T ∈ B , it is possible to find f 0,T ,..., f m,T ⊂ L p (dν) satisfying Proof.
(1) Suppose that B is weakly * bounded in ᏹ p (R × R n ), then from [14] B is equicontinuous.There exist c > 0 and m ∈ N such that As in the proof of Theorem 4.6, we consider the mappings with and for all T ∈ B , Then, relation (4.77) implies that for all Using Hahn-Banach's theorem and Riez's theorem, we deduce that L T can be continuously extended on (L p (dν)) m+1 , denoted again by L T , and that there exists ( f T,k ) 0≤k≤m ⊂ L p (dν) verifying for all ψ = (ψ 0 ,...,ψ m ) ∈ (L p (dν)) m+1 , In particular, if This proves that (i)⇒(ii).
(2) Suppose that there exist c > 0 and m ∈ N such that for every T ∈ B we can find f 0,T ,..., f m,T ⊂ L p (dν) satisfying which means that the set B is weakly * bounded in ᏹ p (R × R n ) and proves that (ii)⇒(i).
(3) Suppose that (ii) holds.Let ϕ ∈ Ᏸ * (R × R n ), then from Theorem 4.7 we know that for all T ∈ B , the function T * ϕ belongs to the space L p (dν).But from which we deduce that the set Now, using Theorem 4.6, it follows that for all a > 0, there exists m ∈ N such that for all ϕ ∈ Ᏸ * (R × R n ), ϕ p,ν ≤ 1, and T ∈ B , the mapping T T * ϕ can be continuously extended on the space ᐃ m a (R × R n ) and the family of these extensions is equicontinuous, which means that there exists c > 0 satisfying for all T ∈ B , for all ϕ ∈ Ᏸ * (R × R n ); for all ψ ∈ ᐃ m a (R × R n ), (4.69) holds.On the other hand, for every T ∈ B , we have for all .70) holds.From relations (4.69) and (4.70), we deduce that the functional T * T ψ can be continuously extended on the space L p (dν) and from Riez's theorem, there exist g T,ψ ∈ L p (dν) such that This completes the proof.

Convolution product on the space ᏹ
In this section, we define and study a convolution product on the space ᏹ Since the operator τ (r,x) is continuous from L p (dν) into itself, we deduce that for all f ∈ ᏹ p (R × R n ) and (r,x) ∈ [0,∞[ × R n , the function τ (r,x) f belongs to the space ᏹ p (R × R n ).Moreover, γ m,p τ (r,x) f = max 0≤k≤m τ (r,−x) gk p,ν ≤ max 0≤k≤m g k p,ν = γ m,p ( f ), (5.3) which shows that the operator τ (r,x) is continuous from ᏹ p (R × R n ) into itself.Let T ∈ ᏹ p (R × R n ); T = m k=0 L k T fk with { f k } 0≤k≤m ⊂ L p (dν) and φ ∈ M r (R × R n ), 1 ≤ r ≤ p, then for all k ∈ N, there exists φ k ∈ L r (dν) such that T φk = L k T φ .From inequality (2.24), it follows that for 0 ≤ k ≤ m, the function f k * φ k belongs to the space L q (dν) with 1/q = 1/r + 1/ p − 1 = 1/r − 1/ p and by using the density of S * (R × R n ) in M r (R × R n ), we deduce that the expression m k=0 f k * φ k is independent of the sequence { f k } 0≤k≤m .Then, we put (5.5) This allows us to say that (5.9) Then for every T ∈ ᏹ p (R × R n ), the mapping φ −→ T * φ (5.10) Proof.Let T ∈ ᏹ p (R × R n ); T = m k=0 L k T fk with { f k } 0≤k≤m ⊂ L p (dν), then for φ ∈ M r (R × R n ), 1 ≤ r ≤ p, and by using relation (5.5), we get T * φ = m k=0 f k * φ k , where φ k ∈ L r (dν) and (5.11) From Lemma 5.3, we have for all s ∈ N, for all φ ∈ M r (R × R n ), L s T T * φ = T T * φs . (5.12) Using relation (5.6), we deduce that the function T * φ belongs to the space ᏹ q (R × R n ).
( Definition 5.5.Let 1 ≤ p, q,r < ∞ such that (5.9) holds.A convolution product of T ∈ ᏹ p (R × R n ) and S ∈ ᏹ q (R × R n ) is defined by for all φ ∈ M r (R × R n ), S * T,φ = S,T * φ . (5.17) From this definition and Proposition 5.4 we deduce the following result.
Proposition 5.6.Let 1 ≤ p, q,r < f ∞ such that (5.9) holds.Then, for all T ∈ ᏹ p (R × R n ) and S ∈ ᏹ q (R × R n ), the functional S * T is continuous on M r (R × R n ).