Generalized Besov type spaces on the Laguerre

In this paper we study generalized Besov type spaces on the La-guerre hypergroup and we give some characterizations using diﬀerent equivalent norms which allows to reach results of completeness, continuous embeddings and density of some subspaces. A generalized Calderón-Zygmund formula adapted to the harmonic analysis on the Laguerre Hypergroup is obtained inducing two more equivalent norms.


Introduction
Schwartz's theory of Fourier transform and the Lebesgue spaces has been exploited by many authors in the study of Besov spaces on R n ( [3], [24], [6]).This theory has been generalized to different spaces, and was applied further to investigate spaces analogous to the classical Besov spaces ( [4], [2]).
In the present work, we study Besov type spaces on the Laguerre hypergroup, so we fix α ≥0 and K = [0, +∞[×R and we define Besov type spaces using the harmonic analysis on the Laguerre hypergroup which can be seen as a deformation of the hypergroup of radial functions on the Heisenberg group (see [1]).
We consider the following system of partial differential operators: For α = n − 1; n ∈ N\{0}, the operator D 2 is the radial part of the sub-Laplacian on the Heisenberg group H n .We denote by ϕ λ,m , (λ, m) ∈ R × N, the unique solution of the following system: 2 )u; u(0, 0) = 1, ∂u ∂x (0, t) = 0 for all t ∈ R.
It has been proved in [19,Theorem II.1] that the Fourier-Laguerre transform is a topological isomorphism from S * (K) onto S(R × N) where • S * (K) is the Schwartz space of functions ψ : R 2 −→ C even with respect to the first variable, C ∞ on R 2 and rapidly decreasing together with all their derivatives; i.e. for all k, p, q ∈ N we have N k,p,q (ψ) = sup (x,t)∈K (1 + x 2 + t 2 ) k ∂ p+q ∂x p ∂t q ψ(x, t) < ∞. (1.2) • S(R × N) the space of functions Ψ : R × N −→ C satisfying : i) For all m, p, q, r, s ∈ N, the function is bounded and continuous on R, C ∞ on R * = R\{0} and such that the left and the right derivatives at zero exist.
ii ) For all k, p, q ∈ N, we have where We note that S * (K) (resp.S(R×N)) equipped with the semi-norms N k,p,q (resp.V k,p,q ), k, p, q ∈ N, is a Fréchet space ( [19]).
This paper deals with generalized Besov-Laguerre type spaces defined on K and it is organized as follows: in the second section, we collect some harmonic analysis properties of the Laguerre hypergroup which are given in [19] and [18].Next, we state a version of Schur lemma which will be useful for our purpose.In the third section we introduce the homogeneous Besov-Laguerre type spaces The definition of the so called spaces is given in terms of convolution f #ψ r with different kinds of smooth functions ψ.Next we characterize these spaces using discrete norms replacing the group R * + =]0, +∞[ by the 2-powers group D 2 = {2 j ; j ∈ Z} and we introduce some results and embeddings properties of these spaces with respect to their parameters p, q and γ.In the fourth section we establish some new harmonic analysis results on usual spaces on K, essentially we give a Delsarte type development and a Calderón-Zygmund type formula.Finally we study the non homogeneous Besov-Laguerre type spaces Λ γ p,q (K) (1 ≤ p, q ≤ ∞, 0 < γ < 2) introduced as intersection of the homogenous ones with L p -spaces and we give some characterizations with equivalent norms using the differences ∆ (x,t) f = T (α) (x,t) f − f .In proving these results, the main tool used is the harmonic analysis on the Laguerre hypergroup.
Finally, we mention that, C will be always used to denote a suitable positive constant that is not necessarily the same in each occurrence.

Preliminaries
Throughout this paper we fix α ≥ 0 and we denote by • C * (K) the space of continuous functions on R 2 even with respect to the first variable.
• C * ,c (K) the subspace of C * (K) consisting of functions with compact support.
• C ∞ * (K) the space of functions f : R 2 −→ C, even with respect to the first variable and These functions are known as generalized wavelets on K ( [19]).
ii) For all m ≤ m 0 , the function λ −→ ψ(λ, m) is C ∞ on R, with compact support and vanishes in a neighborhood of zero.
• L p (K) = L p (K, dµ α ), 1 ≤ p ≤ ∞, the space of Borel measurable functions on K such that f p < ∞, where dµ α being the positive measure defined on K given in the introduction.Each of these spaces is equipped with its usual topology.

Definition 2.1:
• The generalized translation operators T (α) (x,t) on the Laguerre hypergroup are given for a suitable function f by: • The generalized convolution product on the Laguerre hypergroup is defined for a pair of functions f and g in C * ,c (K) by: f or all (x, t) ∈ K.
We recall that (K, * , i) is an hypergroup in the sense of Jewett ([15], [5]) where i denotes the involution defined on K by i(x, t) = (x, −t).This hypergroup is the Laguerre hypergroup which can be seen as a deformation of the hypergroup of radial functions on the Heisenberg group (see [1]).
• f r (x, t) = r −(2α+4) f ((x, t) r ) the dilated of the function f defined on K preserving the mean of f with respect to the measure dµ α , in the sense that

Proposition 2.3:
The following properties hold 1) For all f ∈ C * ,c (K), we have (see [19]) 2) (i) For all f ∈ C * (K) and (x, t), (y, s) ∈ K, we have (see [18]) where W α ((x, t), (y, s), (z, v)) is given by ), (y, s)) and W α ((x, t), (y, s), (z, v)) equals 0 otherwise.S α ((x, t), (y, s)) is given, for α = 0, by and we have 3) For f in L p (K) and g in L q (K), 1 ≤ p, q ≤ ∞, the function f #g belongs to L r (K); 1 p + 1 q = 1 + 1 r , and we have (ii) Let f and g in L 1 (K), then we have (ii) For all k, p, q ∈ N we have Proof: We obtain the desired result by using Proposition II.7 in [19] and the fact that In the sequel we equip S * (K) with the semi-norms N k,p,q which define the standard topology on S * (K).
We finish this preliminary section by giving a version of Schur lemma that will be useful for our purposes.

Generalized homogeneous Besov-Laguerre type spaces
In what follows we equip the spaces R * + and D 2 by the invariant measure dr r and the counting measure respectively.
We define the generalized homogeneous Besov-Laguerre type spaces

Remarks 3.2: 1)
We begin by mentioning that the definition of the generalized homogeneous Besov-Laguerre type spaces given here is the same than that introduced by Chemin in the classical case (see [8]) and generalized by Bahouri, Gérard and Xu on the Heisemberg group (see [2]).We do not choose the classical definition introduced by Peetre (see [20]) in which • Λ γ,ψ p,q (K) is defined as a set of distributions modulo polynomials.In fact in the case γ < 2α+4 p , the condition dr r in the sense of distribution and not only in the sense of distribution modulo polynomials, thus the two points of view are equivalent.We note finally that, similarly to the classical case, for γ ≥ 2α+4 p , the space • Λ γ,ψ p,q (K), as we define, is not a Banach space.
2) We note here that the expression (3.1) is independent, in S * (K), of the choice of ψ in S 1 * ,0 and it corresponds to the analogous one given in [2, Definition 3.1, p.12] replacing the diadic decomposition by the continuous decomposition.
3) If f belongs to L 2 (K), then (3.1) holds in L 2 (K).Which is a consequence of Plancherel's formula (see [18]).Hence one can write And, using Lebesgue theorem, the right hand side of the above equality tends to zero as ε tends to +∞.Indeed and the right hand side of the above inequality is in L 1 (R, |λ| α+1 dλ).
4) The expression (3.1) is not true in S * (K) if f is a polynomial function on K. Indeed in this case, for all r > 0, we have f #ψ r = 0.

Remark 3.4:
In view of their independence with respect to ψ the spaces • Λ γ,ψ p,q (K) (1 ≤ p, q ≤ ∞ and γ ∈ R) will be denoted indifferently with or without ψ, which will be chosen adequately in S 1 * ,0 (K).In what follows we give some properties of the generalized Besov-Laguerre type spaces.
K), then for all 1 ≤ p, q ≤ ∞ and γ ∈ R we have .
The proposition is proved.
K). Then for ε > 1, the function is obviously C ∞ and belongs to • Λ γ p,q (K).Moreover the same reasoning given in Proposition 3.3 leads to where 1 1 ε is the characteristic function of the set R \ [1/ε, ε].And the right hand side of the above inequality tends to zero as ε tends to ∞.
Proof: Let us first prove that, for g ∈ L q (R * + , L p (K), dr r ), Φ(g) defines an element of S * (K), that is for all h ∈ S * (K), Take ψ ∈ S * (K) such that F(ψ) = 1 on SuppFφ.Then, using Hölder's and Young's inequalities, we obtain On the other hand, using Lemma 3.8, we get where q is the conjugate exponent of q.Then, for k sufficiently large it holds Now, let ψ in S 1 * ,0 .We proceed as in Proposition 3.3 to obtain ) .The lemma is proved.
Proof: (Proposition 3.7) Let ψ in S 1 * ,0 and take φ = ψ in Lemma 3.9.Then Φ defined by (3.5) is a continuous linear mapping from L q (R * + , L p (K), dr r ) to • Λ γ p,q (K).On the other hand the operator Ψ associating to f in • Λ γ p,q (K) the function Ψ(f ) defined on R * + by: is obviously a linear isometry from • Λ γ p,q (K) to L q (R * + , L p (K), dr r ) and using the decomposition (3.1), we obtain Φ • Ψ = Id • Λ γ p,q (K) .This implies is a closed subspace of K) can be identified with a closed subspace of L q (R * + , L p (K), dr r ).The completeness of Remark 3.10: From the Proof of Lemma 3.9, the result of Proposition 3.7 remains valid, for q = 1, if γ = 2α+4 p .
To introduce some embedding results of the spaces • Λ γ p,q (K) with respect to their parameters p, q and γ we begin by the following lemma which will be useful.
Then, for all 1 < p < ∞, we have Moreover, there exists C p > 0 such that Proof: The proof of the above lemma is the same as in [21] p. 46.
To obtain more general inclusion properties we introduce a discrete norm on , where For f in Then D γ,θ p,q is a norm on Remarks 3.14: 1) An immediate consequence of the above theorem is the independence of the norm D γ,θ p,q with respect to θ that will be denoted with or without θ.
2) The case 1 < q < ∞ could be proved by interpolation with the extreme cases (q =1 and q = ∞), but a direct proof is presented in this paper.
Proof: Taking into account the fact that Fθ = 0 on C λ1,λ2 for λ 1 , λ 2 ∈ R; λ 2 > 4λ 1 , where C λ1,λ2 is defined as in (3.7), then there exists Fσ in So, using the fact that And, by the same reasoning giving in Proposition 3.3, we obtain for Conversely, let Fθ supported on C λ1,λ2 and let The above reasoning leads to K) .Now we consider the case q = ∞.Let us assume that D γ p,q (f ) < ∞ and let r > 0 and j ∈ Z be such that 2 j ≤ r ≤ 2 j+1 , then from (3.8) we get This completes the proof.Remark 3.15: Equipped with the norm D γ p,q , the space Let us first prove the following lemmas that will be useful in the sequel.
Lemma 4.4: Let f : R + −→ C be a measurable function.Then it holds and we have Proof: We obtain the equality (4.1) by a polar decomposition formula.
In the following lemma we give a Delsarte type development (see [10]) on the Laguerre hypergroup of the function T (α) (x,t) ψ using the differential operators D 1 and D 2 .Lemma 4.5: Let ψ in S * (K) and (x, t) ∈ K.Then, for all (y, s) ∈ K, there exist 0 ≤ η, µ ≤ 1 such that with 0 ≤ µ, η ≤ 1.On the other hand we have (see [19]) On the other hand, using Hölder's inequality and the fact that g belongs to

•
Λ γ p,q (K) one can prove easily that the right hand side of the above inequality tends to 0 as ε, ε tend to ∞.This implies that the family (∆ (x,t) g ε ) ε is a Cauchy net in L p (dµ α ).We get the desired result using (4.7).

=
C T F f #ψ r p r γ (x, t)