Generalized Besov type spaces on the Laguerre hypergroup

BLAISE PASCAL

Miloud Assal, Hacen Ben Abdallah

Generalized Besov type spaces on the Laguerre hypergroup

Volume 12, n^o1 (2005), p. 117-145.

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Generalized Besov type spaces on the Laguerre hypergroup

Miloud Assal Hacen Ben Abdallah

Abstract

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, con- tinuous 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.

1 Introduction

Schwartz’s theory of Fourier transform and the Lebesgue spaces has been exploited by many authors in the study of Besov spaces onRⁿ([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 hyper- group, so we fixα≥0andK= [0,+∞[×Rand 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:







D₁ = ∂

∂t, D₂ = ∂²

∂x² +2α+ 1 x

∂

∂x+x²∂²

∂t²; (x, t)∈]0,∞[×R. For α=n−1; n∈N\{0}, the operatorD₂ is the radial part of the sub- Laplacian on the Heisenberg groupHⁿ. We denote byϕ_λ,m, (λ, m)∈R×N,

the unique solution of the following system:







D₁u=iλu,

D₂u=−4|λ|(m+^α+1₂ )u;

u(0,0) = 1, ^∂u

∂x(0, t) = 0 for all t∈R.

One knows that ϕ_λ,m(x, t) = e^iλtL^α_m(|λ|x²), where L^α_m is the Laguerre functions defined on R+ by L^α_m(x) = e^−x2Lαm(x)

L^αm(0) and L^α_m is the Laguerre polynomial of degreemand orderα(see [17], [11], [13], [16]).

We recall that for(λ, m)∈R×Nand for a suitable functionf :K−→C the Fourier-Laguerre transformF(f)(λ, m)off at(λ, m)is defined by ([19], [22, 23], [12]):

F(f)(λ, m) = Z

K

ϕ−λ,m(x, t)f(x, t)dµ_α(x, t), (1.1) where dµ_α(x, t) = x^2α+1dxdt

πΓ(α+ 1).

It has been proved in [19, Theorem II.1] that the Fourier-Laguerre transform is a topological isomorphism fromS∗(K)onto S(R×N) where

• S∗(K)is the Schwartz space of functionsψ:R²−→Ceven with respect to the first variable,C^∞ onR²and rapidly decreasing together with all their derivatives; i.e. for all k, p, q∈Nwe have

Ne_k,p,q(ψ) = sup

(x,t)∈K

n

(1 +x²+t²)^k

∂^p+q

∂x^p∂t^qψ(x, t) o

<∞. (1.2)

• S(R×N)the space of functions Ψ :R×N−→Csatisfying : i)For allm, p, q, r, s∈N,the function

λ7−→λ^p

|λ|(m+ α+ 1 2 )q

Λ^r₁

Λ₂+ ∂

∂λ s

Ψ(λ, m)

is bounded and continuous onR,C^∞ onR^∗=R\{0}and such that the left and the right derivatives at zero exist.

ii ) For allk, p, q∈N, we have V_k,p,q(Ψ) = sup

(λ,m)∈R^∗×N

1 +λ²(1 +m²)k Λ^p₁

Λ₂+ ∂

∂λ q

Ψ(λ, m)

<∞.

where

• Λ₁Ψ(λ, m) = _|λ|¹

m∆₊∆₋Ψ(λ, m) + (α+ 1)∆₊Ψ(λ, m) .

• Λ₂Ψ(λ, m) = ⁻¹

2λ

(α+m+ 1)∆₊Ψ(λ, m) +m∆₋Ψ(λ, m) .

• ∆₊Ψ(λ, m) = Ψ(λ, m+ 1)−Ψ(λ, m).

• ∆₋Ψ(λ, m) = Ψ(λ, m)−Ψ(λ, m−1), ifm≥1and∆₋Ψ(λ,0) = Ψ(λ,0).

We note thatS_∗(K)(resp. S(R×N))equipped with the semi-normsNe_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 Kand 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 Λ^•γ_p,q(K) (1≤p, q ≤ ∞, γ ∈R). The definition of the so called spaces is given in terms of convolutionf#ψ_r with different kinds of smooth functionsψ. Next we characterize these spaces using discrete norms replacing the groupR^∗+=]0,+∞[by the 2-powers groupD2={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 spacesΛ^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_(x,t)^(α)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.

2 Preliminaries

Throughout this paper we fix α≥0and we denote by

• R^∗=R\{0} and R^∗+=]0,+∞[

• K= [0,∞[×R.

• C_∗(K)the space of continuous functions onR²even 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 : R² −→ C, even with respect to the first variable and C^∞ onR².

• S_∗,0(K)the subset of functionsψ in S_∗(K) such thatFψ ∈ D(R^∗×N).

• S_∗,0¹ (K) the subset of functionsψ inS_∗,0(K)such that Z ∞

0

Fψ(r²λ, m)2dr

r = 1, f or (λ, m)∈R^∗×N. These functions are known as generalized wavelets on K([19]).

• D(R×N) the subspace of S(R×N) of functions ψ satisfying the fol- lowing:

i) There existsm₀∈Nsatisfyingψ(λ, m) = 0, for all(λ, m)∈R×N such thatm > m₀.

ii) For all m ≤ m₀, the function λ 7−→ 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 func- tions on Ksuch thatkfk_p<∞, where

kfk_p = Z

K

|f(x, t)|^pdµ_α(x, t) ¹_p

, if p∈[1,∞[, kfk_∞ = esssup

(x,t)∈K

|f(x, t)|,

dµ_α being the positive measure defined onKgiven in the introduction.

Each of these spaces is equipped with its usual topology.

Definition 2.1:

• The generalized translation operatorsT_(x,t)^(α) on the Laguerre hypergroup are given for a suitable functionf by:

T_(x,t)^(α)f(y, s) =

















 1 2π

Z 2π 0

f((x²+y²+ 2xycosθ)¹², t+s+xysinθ)dθ, if α= 0, α

π Z 1

0

Z 2π 0

f((x²+y²+ 2xyρcosθ)¹², t+s+xyρsinθ)×

ρ(1−ρ²)^α−1dθdρ, if α >0.

• The generalized convolution product on the Laguerre hypergroup is defined for a pair of functionsf and ginC∗,c(K) by:

f#g(x, t) = Z

K

T_(x,t)^(α) f(y, s)g(y,−s)dµ_α(y, s) f or all (x, t)∈K.

We recall that (K,∗, i) is an hypergroup in the sense of Jewett ([15], [5]) whereidenotes the involution defined onKbyi(x, t) = (x,−t). This hyper- group is the Laguerre hypergroup which can be seen as a deformation of the hypergroup of radial functions on the Heisenberg group (see [1]).

Notation 2.2: Let r >0. We will denote by

• (x, t)_r = (x r, t

r²) the dilated of(x, t)∈K.

• f_r(x, t) = r^−(2α+4)f((x, t)_r) the dilated of the function f defined on K preserving the mean off with respect to the measure dµ_α, in the sense that

Z

K

f_r(x, t)dµ_α(x, t) = Z

K

f(x, t)dµ_α(x, t), ∀r >0and f ∈L¹(K). (2.1)

• ∆_(x,t)f =T_(x,t)^(α)f−f, for all (x, t)∈K.

Proposition 2.3: The following properties hold 1) For all f ∈ C_∗,c(K), we have (see [19])

(i) T_(0,0)^(α) f(y, s) =f(y, s), ∀(y, s)∈K.

(ii) T_(0,t)^(α)f(y, s) =f(y, s+t), ∀(y, s)∈K, t∈R. (iii) T_(x,t)^(α) f(y, s) =T_(y,s)^(α) f(x, t) ∀(x, t),(y, s)∈K.

2) (i) For allf ∈ C∗(K) and (x, t),(y, s)∈K, we have (see [18]) T_(x,t)^(α)f(y, s) =

Z

K

W_α((x, t),(y, s),(z, v))f(z, v)z^2α+1dzdv

where W_α((x, t),(y, s),(z, v)) is given by α

π(xyz)^2α h

x²y²−z²−(x²+y²) 2

2

−(v−(s+t))²iα−1

,

if (z, v) ∈ S_α((x, t),(y, s)) and W_α((x, t),(y, s),(z, v)) equals 0 otherwise.

S_α((x, t),(y, s)) is given, forα6= 0, by S_α((x, t),(y, s)) =n

(z, v)∈K; z²−(x²+y²) 2

2

+ (v−(s+t))²≤x²y²o and

S₀((x, t),(y, s)) = n

(z, v)∈K;

z²−(x²+y²) 2

2

+ (v−(s+t))²=x²y² o

. (ii) Let f be in L^p(K), 1 ≤ p ≤ ∞. Then for all (x, t) ∈ K, the

function T_(x,t)^(α)f belongs to L^p(K) and we have kT_(x,t)^(α) fk_p≤ kfk_p.

3) Forf inL^p(K)andginL^q(K),1≤p, q≤ ∞, the function f#gbelongs to L^r(K); ¹_p+ ¹_q = 1 +¹_r, and we have

kf#gk_r≤ kfk_pkgk_q.

4) (i) Let f be in L¹(K). Then the functionF(f) is bounded on R×N and we have

kF(f)k_L∞(R×N) ≤ kfk₁ where kF(f)k_L^∞_(R×N) =esssup

(λ,m)∈R×N

|F(f)(λ, m)|.

(ii) Let f and g inL¹(K), then we have F(f#g) =F(f)F(g).

(iii) Letf be in L¹(K). Then for all (x, t) inK and(λ, m) inR×N, we have

F(T_(x,t)^(α)f)(λ, m) =ϕ_λ,m(x, t)F(f)(λ, m).

Proposition 2.4: (See [1]) Let ψ in S∗(K) and (x, t) ∈ K. Then T_(x,t)^(α)ψ belongs to S∗(K)and we have for all p, q∈N

D^p₁D^q₂

T_(x,t)^(α)ψ

=T_(x,t)^(α)

D₁^pD₂^qψ

.

Proposition 2.5: Let ψ inC∗(K). Thenψ belongs to S∗(K) if and only if (i) For allp, q∈Nthe function(x, t)7−→D^p₁D^q₂ψ(x, t)is of classC²onR². (ii) For allk, p, q∈N we have

N_k,p,q(ψ) = sup

(x,t)∈K

n

(1 +N²(x, t))^k

D^p₁D^q₂ψ(x, t) o

<∞,

whereN(x, t) = (x²+|t|)^1/2 is the norm of (x, t)∈K.

Proof: We obtain the desired result by using Proposition II.7 in [19] and the fact that

(1+x²+t²)^k≤(1+N²(x, t))^2k ≤2^2k(1+x²+t²)^2k, ∀(x, t)∈Kand k∈N.

In the sequel we equip S∗(K) with the semi-norms N_k,p,q which define the standard topology onS_∗(K).

We finish this preliminary section by giving a version of Schur lemma that will be useful for our purposes.

Lemma A. (Schur lemma). Let 1 < q < ∞ and q⁰ its conjugate expo- nent. Let(Ω₁,M₁, µ₁) and(Ω₂,M₂, µ₂) be a pair ofσ-finite measure spaces and let F : Ω₁×Ω₂ −→ R+ be a measurable function. Define T_Ff for all measurable positive functionf onΩ₁ by

T_Ff(ω₂) = Z

Ω1

F(ω₁, ω₂)f(ω₁)dµ₁(ω₁), for all ω₂∈Ω₂.

If there exist C > 0 and measurable functions h_i : Ω_i −→]0,+∞[ (i = 1,2) such that

Z

Ω1

F(ω₁, ω₂)h^q₁⁰(ω₁)dµ₁(ω₁)≤Ch^q₂⁰(ω₂) µ₂−a.e.

Z

Ω2

F(ω₁, ω₂)h^q₂(ω₂)dµ₂(ω₂)≤Ch^q₁(ω₁) µ₁−a.e.

ThenT_F can be extended as a bounded operator fromL^q(Ω₁, µ₁)intoL^q(Ω₂, µ₂).

3 Generalized homogeneous Besov-Laguerre type spaces

In what follows we equip the spaces R^∗₊ and D2 by the invariant measure ^dr_r and the counting measure respectively.

Definition 3.1: Let1 ≤p, q ≤ ∞, γ ∈Rand ψ ∈S_∗,0¹ (K). We define the generalized homogeneous Besov-Laguerre type spaces Λ^•γ,ψ_p,q(K) as the set of tempered distributionsf such that

f = Z ∞

0

f#ψ_r#ψ_rdr

r (3.1)

andkfk^•

Λ^γ,ψp,q(K) <∞, where

kfk^•

Λ^γ,ψp,q(K) =









 Z ∞

0

kf#ψ_rk_p r^γ

qdr r

¹_q

, if 1≤q <∞, esssup

r>0

kf#ψ_rk_p r^γ

, if q=∞.

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 whichΛ^•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 kfk^•

Λ^γ,ψp,q(K) < ∞ implies the convergence of the integral

Z ∞ 0

f^#ψ_r^#ψ_rdr 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, forγ ≥ ^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_∗⁰(K), of the choice of ψ in S_∗,0¹ and it corresponds to the analogous one given in [2, Definition 3.1, p.12] replacing the diadic decomposition by the continuous decomposition.

3) Iff belongs toL²(K), then (3.1) holds in L²(K). Which is a conse- quence of Plancherel’s formula (see [18]). Hence one can write

f− Z ε

1/ε

f^#ψ_r^#ψ_rdr r

2

2

= Z +∞

−∞

(_+∞

X

m=0

L^α_m(0)

Ff(λ, m)

2

×

1− Z ε

1/ε

Fψ_r(λ, m) 2dr

r

2)

|λ|^α+1dλ.

And, using Lebesgue theorem, the right hand side of the above equality tends to zero asεtends to +∞. Indeed

1−

Z ε 1/ε

Fψ_r(λ, m)2dr r

2−→0 as ε−→+∞,

+∞

X

m=0

L^α_m(0)

Ff(λ, m)

2 1−

Z ε 1/ε

Fψ_r(λ, m)2dr r

2

≤

+∞

X

m=0

L^α_m(0)

Ff(λ, m)

2

. and the right hand side of the above inequality is inL¹(R,|λ|^α+1dλ).

4)The expression (3.1) is not true inS_∗⁰(K)iff is a polynomial function onK. Indeed in this case, for allr >0, we havef^#ψ_r = 0.

Proposition 3.3: Let 1 ≤ p, q ≤ ∞ and γ ∈ R. Then the space Λ^•γ,ψ_p,q(K) is independent of the choice of the function ψ in S_∗,0¹ (K).

Proof: Assumeψandφbe a pair of functions belonging toS_∗,0¹ (K). To get the desired result it suffices to prove thatkfk^•

Λ^γ,φp,q(K) ≤Ckfk^•

Λ^γ,ψp,q(K), for all f ∈ Λ^•γ,ψ_p,q(K). Since Fψ and Fφ belong to D(R^∗×N), then there exist α, β >0such that(SuppFψ_r)∩(SuppFφ_ρ) =∅, for all(r/ρ)∈/[α, β]. This implies that ψ_r^#φ_ρ = 0, for all (r/ρ)∈/ [α, β]. And so, using (3.1), one can write

f^#φ_ρ= Z ∞

0

f^#φ_ρ^#ψ_r^#ψ_rdr r =

Z β α

f^#ψ_rρ^#φ_ρ^#ψ_rρdr r . And by Minkowski’s inequality

kf#φ_ρk_p

ρ^γ ≤ C

Z β α

kf#ψ_rρk_p ρ^γ

dr r

= C Z ∞

0

r^γ11_[α,β](r)kf^#ψ_rρk_p (rρ)^γ

dr r

= C(H ? G)(ρ) with H(s) = s^γ11_[α,β](s), G(s) = kf^#ψ_sk_p

s^γ and H ? G is the convolution of H andG on the group(R^∗+,^dr_r). Now, by Young’s inequality, it holds

kfk^•_Λγ,φ p,q(K) =

Z ∞ 0

kf^#φ_ρk_p ρ^γ

qdρ ρ

¹_q

≤ C H ? G

L^q(R+,B(R+),^dρ_ρ)

≤ CkHk_L1(R+,B(R+),^dρ_ρ)kGk_Lq(R+,B(R+),^dρ_ρ)

= Ckfk^•_Λγ,ψ p,q(K). This completes the proof of the proposition.

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 withoutψ, which will be chosen adequately in S_∗,0¹ (K).

In what follows we give some properties of the generalized Besov-Laguerre type spaces.

Proposition 3.5: Let1≤p, q ≤ ∞,γ ∈R. The Besov-Laguerre type space

•

Λ^γ_p,q(K)is homogeneous of degreed(p, γ) = ^2α+4_p −γ in the sense that, for all f ∈ Λ^•γ_p,q(K)

kd_rfk ^•_Λγ

p,q(K) =r^{2α+4p −γ}kfk_Λ^•γ

p,q(K), for all r >0 whered_rf(x, t) =f((x, t)_r), for all (x, t)∈K.

Proof: Assumef in Λ^•γ_p,q(K), then for all1≤p, q≤ ∞andγ ∈Rwe have kd_rfk^•

Λ^γp,q(K) =

k(d_rf)#ψ_ρk_p ρ^γ

L^q(R+,B(R+),^dρ_ρ)

= r^2α+4p

kf#ψ^ρ

rk_p ρ^γ

L^q(R+,B(R+),^dρ_ρ)

= r^{2α+4p −γ}

kf#ψ_ρk_p ρ^γ

L^q(R+,B(R+),^dρ_ρ)

.

The proposition is proved.

Proposition 3.6: Let 1 ≤ p ≤ ∞, 1 ≤q < ∞ and γ ∈ R. The subspace

•

Λ^γ_p,q(K)∩ C_∗^∞(K) is dense in Λ^•γ_p,q(K).

Proof: Let φ∈S_∗,0¹ (K) andf ∈ Λ^•γ_p,q(K). Then forε >1, the function f_ε=

Z ε 1/ε

f^#φ_r^#φ_rdr r

is obviouslyC^∞ and belongs to Λ^•γ_p,q(K). Moreover the same reasoning given in Proposition 3.3 leads to

kf_ε−fk^•

Λ^γp,q(K)≤C

kf^#φ_rk_p r^γ 11_ε(r)

L^q(R+,B(R+),^dr_r)

where 11_ε is the characteristic function of the set R\[1/ε, ε]. And the right hand side of the above inequality tends to zero asεtends to ∞.

Proposition 3.7: Let 1 ≤ p, q ≤ ∞ and γ < ^2α+4_p . Then Λ^•γ_p,q(K) is a Banach space.

Let us first establish the following lemmas.

Lemma 3.8: Let h ∈S∗(K) and ψ ∈ S_∗,0¹ (K). Then, for all k ∈ N, there existsψ_[k]∈S_∗,0(K) such that

h#ψ_r =r^2k(D^k₂h)#(ψ_[k])_r (3.2) whereD₂ is the differential operator given in the introduction part. Further- more there existsC >0such that

kh#ψ_rk_L¹_(K) ≤Cr^2k for all0≤r≤1 (3.3) and

kh^#ψ_rk_L¹_(K) ≤C for all r≥1. (3.4)

Proof: For k= 1one can write F(h^#ψ_r)(λ, m) = 4|λ|(m+α+ 1

2 )r²Fh(λ, m) Fψ(r²λ, m) 4r²|λ|(m+ ^α+1

2 )

= r²F(D₂h)(λ, m)Fψ_[1](r²λ, m) where Fψ_[1](λ, m) = ^Fψ(λ,m)

4|λ|(m+^α+1₂ ), which leads to h^#ψ_r=r²(D₂h)^#(ψ_[1])_r,

and hence we obtain (3.2) by induction onk. The inequalities (3.3) and (3.4) follow from (3.2) and Proposition 2.3.

Lemma 3.9: Let1≤p, q ≤ ∞and γ < ^2α+4_p . For φ inS_∗,0¹ , put Φ(g) =

Z ∞ 0

r^γg(r)^#φ_rdr

r . (3.5)

Then Φ defines a linear and continuous mapping from L^q(R^∗+, L^p(K),^dr_r) to Λ^•γ_p,q(K).

Proof: Let us first prove that, for g ∈ L^q(R^∗+, L^p(K),^dr_r), Φ(g) defines an element ofS_∗⁰(K), that is for allh∈S_∗(K),

Z ∞ 0

r^γ < g(r)^#φ_r, h >

dr r <∞.

Takeψ ∈S∗(K) such thatF(ψ) = 1onSuppFφ. Then, using Hölder’s and Young’s inequalities, we obtain

|< g(r)^#φ_r, h >| = |< g(r)^#φ_r, h^#ψ_r >|

≤ kg(r)#φ_rk_L∞(K)kh#ψ_rk_L¹_(K)

≤ Cr^−2α+4p kg(r)k_L^p_(K)kh#ψ_rk_L¹_(K). On the other hand, using Lemma 3.8, we get

Z ∞ 0

r^γ < g(r)^#φ_r, h >

dr r

≤C Z 1

0

r^{2k+γ−2α+4p} kg(r)k_L^p_(K)dr r +

Z ∞ 1

r^γ−2α+4p kg(r)k_L^p_(K)

dr r

≤C Z 1

0

kg(r)k^q_Lp(K)

dr r

1/qZ 1 0

r^{(2k+γ−2α+4p )¯q}dr r

1/¯q

+C Z ∞

1

kg(r)k^q_Lp(K)

dr r

1/qZ ∞ 1

r^{(γ−2α+4p )¯q}dr r

1/¯q

. whereq¯is the conjugate exponent ofq.Then, fork sufficiently large it holds

Z ∞ 0

r^γ < g(r)#φ_r, h >

dr

r ≤Ckgk_Lq(R^∗+,L^p(K),^dr_r)<∞.

Now, letψ inS_∗,0¹ . We proceed as in Proposition 3.3 to obtain kΦ(g)^#ψ_ρk_L^p_(K)

ρ^γ ≤C

Z β α

r^γkg(rρ)k_L^p_(K)dr r =C

Z ∞ 0

r^γ11_[α,β](r)kg(rρ)k_L^p_(K)dr r which leads to

kΦ(g)k^•

Λ^γp,q(K) ≤Ckgk_Lq(R^∗+,L^p(K),^dr_r). The lemma is proved.

Proof: (Proposition 3.7) LetψinS_∗,0¹ and takeφ=ψin 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 functionΨ(f)defined on R^∗+ by:

Ψ(f)(r) = f#ψ_r r^γ

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 obtainΦ◦Ψ =Id_Λ^•γ

p,q(K). This implies (Ψ◦Φ)◦Ψ = Ψ on

•

Λ^γ_p,q(K).

So Ψ •

Λ^γ_p,q(K)

= ker

Ψ◦Φ−Id_Lq(R^∗+,L^p(K),^dr_r)

is a closed subspace of L^q(R^∗+, L^p(K),^dr_r). Since Ψ is an isometry, then Λ^•γ_p,q(K) can be identified with a closed subspace of L^q(R^∗+, L^p(K),^dr_r). The completeness of Λ^•γ_p,q(K) follows.

Remark 3.10: From the Proof of Lemma 3.9, the result of Proposition 3.7 remains valid, for q= 1, ifγ = ^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.

Lemma 3.11: Letf ∈S⁰(K) satisfying (3.1) and let f˜=

Z ∞ 0

f#ψ_r

2dr r

1/2

. (3.6)

Then, for all 1< p <∞, we have

f ∈L^p(K)⇐⇒f˜∈L^p(K).

Moreover, there exists C_p>0such that 1

C_pkf˜k_L^p_(K) ≤ kfk_L^p_(K) ≤C_pkfk˜ _L^p_(K).

Proof: The proof of the above lemma is the same as in [21] p. 46.

Proposition 3.12: 1) Let 1≤q ≤ ∞, γ₁, γ₂ ∈ R and 1≤ p₁ ≤p₂ ≤ ∞ such thatd(p₁, γ₁) =d(p₂, γ₂) whered(p, γ) = ^2α+4

p −γ. Then we have

•

Λ^γ_p¹_1,q(K) ⊆ Λ^•γ_p²_2,q(K) (with continuous embedding).

2) Letp≥2. Then

•

Λ⁰_p,2(K) ⊆L^p(K) (with continuous embedding).

3) Let1≤p≤ ∞. Then we have

•

Λ⁰_p,1(K) ⊆L^p(K) (with continuous embedding).

Proof: 1) Letf ∈Λ^•γ_p1,q(K)and let1≤p₃≤ ∞such that_p¹

1+_p¹

3 = 1+_p¹

2. We consider φ ∈ S∗(K) satisfying Fφ = 1 on SuppFψ. Then it holds for allr >0

kf^#ψ_rk_L^p2(K) = kf^#ψ_r^#φ_rk_L^p2(K)

≤ kf#ψ_rk_L^p1(K)kφ_rk_L^p3(K) =Ckf#ψ_rk_L^p1(K)r^{(2α+4)(p13−1)}. Hence, we obtain

kf#ψ_rk_L^p2(K)

r^γ2 ≤ Ckf#ψ_rk_L^p1(K)

r^γ1 . 2) Letf ∈S⁰(K)and let f˜defined in (3.6). Then,

kfk˜ _L^p_(K) =

Z ∞ 0

f#ψ_r

2dr r

1/2 L^p(K)

=

Z ∞ 0

f#ψ_r

2dr r

L^p/2(K)

≤

Z ∞ 0

(f^#ψ_r)² L^p/2(K)

dr r

1/2

=

Z ∞ 0

f^#ψ_r

2 L^p(K)

dr r

1/2

= kfk^•_Λ0 p,2(K).

The desired continuous embedding holds using Lemma 3.11.

3) Let f ∈S⁰(K) satisfying (3.1). Then kfk_L^p_(K)=

Z ∞ 0

f#ψ_r#ψ_rdr r

L^p(K)

≤C Z ∞

0

kf#ψ_rk_L^p_(K)dr

r =Ckfk^•

Λ⁰_p,1(K). The proof is finished.

To obtain more general inclusion properties we introduce a discrete norm on Λ^•γ_p,q(K) replacing the group R^∗+ =]0,+∞[ by the 2-powers group D2= {2^j;j∈Z}.

Theorem 3.13: Let 1 ≤ p, q ≤ ∞, γ ∈ R and θ ∈ S∗(K) such that Fθ ∈ D(R^∗×N) and, for fixed λ₁, λ₂∈R; λ₂>4λ₁>0, Fθ(λ, m)6= 0 on C_λ1,λ2, where

C_λ1,λ2 =n

(λ, m)∈R×N; λ₁≤λ≤λ₂o

. (3.7)

Forf in Λ^•γ_p,q(K) put

D^γ,θ_p,q(f) =









 X

j∈Z

kf^#θ₂^jk_p 2^γj

q¹_q

, if 1≤q <∞, sup

j∈Z

kf#θ₂jk_p 2^γj

, if q=∞.

ThenD^γ,θ_p,q is a norm on Λ^•γ_p,q(K) equivalent tok.k^•_Λγ p,q(K).

Remarks 3.14: 1)An immediate consequence of the above theorem is the independence of the normD^γ,θ_p,q with respect to θ that will be denoted with or withoutθ.

2) The case 1< q <∞ could be proved by interpolation with the extreme cases (q=1and q=∞), but a direct proof is presented in this paper.

Proof: Taking into account the fact thatFθ6= 0onC_λ1,λ2 forλ₁, λ₂∈ R; λ₂ > 4λ₁, where C_λ1,λ2 is defined as in (3.7), then there exists Fσ in D(R^∗×N)such thatFθ(λ, m)Fσ(λ, m) = 1onC_λ1,λ2. Letψ∈S_∗,0¹ satisfying SuppFψ⊂ C_4λ1,λ2. This gives, for all1≤r≤2and (λ, m)∈R×N

Fψ(2^2jr²λ, m) =Fψ(2^2jr²λ, m)Fθ(2^2jλ, m)Fσ(2^2jλ, m). (3.8)

So, using the fact thatF(ψ_r)(λ, m) =F(ψ)(r²λ, m) it holds f^#ψ₂^j_r =f^#ψ₂^j_r^#θ₂^j#σ₂^j.

And, by the same reasoning giving in Proposition 3.3, we obtain for 1≤q <∞

kfk_Λ^•γ

p,q(K) = X

j∈Z

2^−jγq Z 2

1

kf^#ψ₂^j_rk_p r^γ

qdr r

¹_q

≤ C X

j∈Z

kf^#θ₂^jk_p 2^jγ

q¹_q

= CD^γ_p,q(f).

Conversely, letFθsupported onC_λ1,λ2 and letψ∈S_∗,0¹ (K)satisfyingFψ = 1 onC_λ1,4λ2. Then it holds, for 1≤r≤2and (λ, m)∈R×N, that

Fθ(2^2jλ, m) =Fψ(2^2jr²λ, m)Fθ(2^2jλ, m). (3.9) The above reasoning leads to

D^γ_p,q(f)≤Ckfk^•

Λ^γp,q(K).

Now we consider the caseq =∞. Let us assume that D^γ_p,q(f)< ∞ and let r >0andj∈Zbe such that2^j≤r≤2^j+1, then from (3.8) we get

kf^#ψ_rk_p≤Ckf^#θ₂^jk_p≤C2^γj ≤Cr^γ which implies thatkfk ^•

Λ^γp,∞(K)<∞.

Conversely let us take kfk^•

Λ^γp,∞(K) < ∞, then it holds from (3.9) that for 1≤r≤2the following estimation

kf^#θ₂^jk_p≤Ckf^#ψ₂^j_rk_p≤C(2^jr)^γ≤(2^γC)2^γj. This completes the proof.

Remark 3.15: Equipped with the normD^γ_p,q, the spaceΛ^•γ_p,q(K)(1≤p, q≤

∞, γ ∈ R) is homogeneous in a weaker sense: there exists c₁, c₂ > 0 such that for allf ∈ Λ^•γ_p,q(K)

c₁r^{2α+4p −γ}D^γ_p,q(f)≤D^γ_p,q(d_rf)≤c₂r^{2α+4p −γ}D^γ_p,q(f), for all r >0.

Proposition 3.16: Let1≤p≤ ∞ and γ ∈R. Then, for 1≤q₁≤q₂≤ ∞ we have

•

Λ^γ_p,q1(K) ⊆ Λ^•γ_p,q2(K) (with continuous embedding).

Proof: The result holds using the discrete norm and the fact thatl^q1 ⊂l^q2 for allq₁≤q₂ and we have

X|u_j|^q21/q2

≤ X

|u_j|^q11/q1

for all(u_j)∈l^q1.

4 Generalized non homogeneous Besov- Laguerre type spaces

In this section we study the non homogeneous Besov-Laguerre type spaces defined asL^p(K)subspaces and we give some characterizations using equiva- lent norms. The main tool used here is the Calderón formula on the Laguerre hypergroup introduced in Lemma 4.7. In what follows we equip the spaces Kby the invariant measure _N^dxdt3(x,t).

Definition 4.1:Let 1≤p, q ≤ ∞and γ >0. The non homogeneous Besov- Laguerre type space isΛ^γ_p,q(K) = Λ^•γ_p,q(K)∩L^p(K) endowed with the norm k.k ^•

Λ^γp,q(K)

Theorem 4.2: Let 1≤p, q≤ ∞ and 0< γ <2. For f inΛ^γ_p,q(K) put

B^γ_p,q(f) =









 Z

K

k∆_(x,t)fk_p N^γ(x, t)

q dxdt N³(x, t)

¹_q

, if 1≤q <∞, esssup

(x,t)∈K

k∆_(x,t)fk_p N^γ(x, t)

, if q =∞.

ThenB^γ_p,q is a norm onΛ^γ_p,q(K) equivalent tok.k^•_Λγ p,q(K).

Remark 4.3: This characterization is similar to the results obtained by T. Coulhon, E. Russ and V. Tardivel-Nachef in [9].

Let us first prove the following lemmas that will be useful in the sequel.

Lemma 4.4: Letf :R+−→Cbe a measurable function. Then it holds f ∈L¹(R+,dr

r ) if and only if f◦N∈L¹

K, dxdt N³(x, t)

and we have

Z

K

f(N(x, t)) dxdt N³(x, t) = 4

Z ∞ 0

f(r)dr

r . (4.1)

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 functionT_(x,t)^(α)ψ using the differential operators D₁and D₂.

Lemma 4.5: Letψ in S∗(K)and (x, t)∈K. Then, for all (y, s)∈K, there exist0≤η, µ≤1 such that

T_(x,t)^(α)ψ(y, s) =ψ(x, t) +sT_(x,t)^(α) (D₁ψ)(0, µs) +y²

h

T_(x,t)^(α)(D₂ψ)(ηy, s)−(2α+ 1) Z 1

0

T_(x,t)^(α)(D₂ψ)(ηuy, s)u^2α+1du i

−y⁴h

η²T_(x,t)^(α)(D²₁ψ)(ηy, s)−(2α+ 1)η² Z 1

0

u²T_(x,t)^(α)(D₁²ψ)(ηuy, s)u^2α+1du i

. (4.2) Furthermore we have

k∆_(y,s)ψk₁≤C(N²(y, s) +N⁴(y, s)), for all(y, s)∈K. (4.3)

Proof: For ψ ∈S∗(K) we have (see [1, Proposition 2.2])T_(y,s)^(α) ψ ∈S∗(K).

And using Proposition 2.3 together with Taylor’s formula we get T_(y,s)^(α) ψ(x, t) =ψ(x, t) +s ∂

∂s

T_(x,t)^(α) ψ

(0, µs) +y² ∂²

∂y²

T_(x,t)^(α) ψ (ηy, s) with0≤µ, η ≤1. On the other hand we have (see [19])











∂ψ

∂s(y, s) = D₁ψ(y, s)

∂²ψ

∂y²(y, s) = D₃ψ(y, s)−(2α+ 1) Z 1

0

D₃ψ(yu, s)u^2α+1du

where the operator D₃ is given by D₃ψ(y, s) = (D₂−y²D²₁)ψ(y, s). So we obtain the development (4.2) from Propositions 2.3 and 2.4. Also from (4.2) we deduce that

|∆_(y,s)ψ(x, t)| ≤ |s|

T_(0,µs)^(α) (D₁ψ)(x, t) +y²

T_(ηy,s)^(α) (D₂ψ)(x, t)

+ (2α+ 1) Z 1

0

T_(ηuy,s)^(α) (D₂ψ)(x, t) du

+y⁴

T_(ηy,s)^(α) (D₁²ψ)(x, t)

+ (2α+ 1) Z 1

0

T_(ηuy,s)^(α) (D₁²ψ)(x, t) du

. (4.4) So by integration of (4.4) overKwith respect to the measuredµ_α we obtain (4.3).

Lemma 4.6: Letψ∈S∗(K). Then, for all(x, t)∈Kand r >0 we have

k∆_(x,t)ψ_rk₁≤Cmin

1,N(x, t) r

2

. (4.5)

Proof: From the expression of the kernelW_α and using (4.3), one can see easily that

k∆_(x,t)ψ_rk₁=k∆_(x,t)rψk₁≤C

N(x, t) r

2

+

N(x, t) r

4

. (4.6)

The contraction property of the translation operators T_(y,s)^(α) on L¹(dµ_α) (see Proposition 2.3, 2), (ii)) leads to

k∆_(x,t)ψ_rk₁≤Cmin

2,

N(x, t) r

2

+

N(x, t) r

4

≤2Cmin

1,

N(x, t) r

2 .

The following lemma gives a version of Calderón-Zygmund formula ([14], [7]) on the Laguerre hypergroup.

Lemma 4.7: Let g in Λ^•γ_p,q(K)∩L^p(dµ_α), 1 ≤ p, q ≤ ∞, 0 < γ < 2 and ψ∈S_∗,0¹ (K). For 1<ε <∞, put

g_ε(y, s) = Z ε

1/ε

(g#ψ_r#ψ_r)(y, s)dr

r , for(y, s)∈K.

Then, for all(x, t)∈K,∆_(x,t)g_ε converges to∆_(x,t)ginL^p(dµ_α)asε→+∞.

Proof: Using the fact thatR∞

0 (Fψ_r(λ, m))^2dr_r =R∞

0 (Fψ(r²λ, m))^2dr_r = 1, we deduce easily that, forg∈S_∗(K),(x, t)∈Kand (λ, m)∈R×N,

F(∆_(x,t)g_ε)(λ, m)− F(∆_(x,t)g)(λ, m)

=

ϕ_λ,m(x, t)−1

Fg(λ, m) Z ε

1/ε

(Fψ_r(λ, m))²dr r −

Z ∞ 0

(Fψ_r(λ, m))²dr r

=F(∆_(x,t)g)(λ, m) Z ε

1/ε

(Fψ_r(λ, m))²dr r −

Z ∞ 0

(Fψ_r(λ, m))²dr r

. Using the factF(∆_(x,t)g)∈S(R×N), we obtain, for all k, p, q∈N

V_k,p,q

F(∆_(x,t)g_ε)− F(∆_(x,t)g)

−−−−−−−−−→ε−→ ∞ 0.

And so, using the fact that ([19, Theorem II.1]) the Fourier-Laguerre trans- form is a topological isomorphism fromS_∗(K)ontoS(R×N), one can con- clude that ∆_(x,t)g_ε tends to ∆_(x,t)g in S∗(K) as ε tends to ∞. Let us now takeg ∈L^p(dµ_α) considered as an element of S_∗⁰(K) the topological dual of S∗(K)then an elementary calculation leads to

<∆_(x,t)g_ε, f >=< g,∆_(x,−t)f_ε> . This leads to, for allg∈L^p(dµ_α)

∆_(x,t)g_ε −−−−−−−−−→ε−→ ∞ ∆_(x,t)g in S_∗⁰(K). (4.7) So, using Lemma (4.6) it holds forε < ε⁰ large that

∆_(x,t)g_ε−∆_(x,t)g_ε0

p≤Z 1/ε 1/ε⁰

+ Z ε⁰

ε

min

1,N(x, t) r

2

kg∗ψ_rk_pdr r .