() CUBO A Mathematical Journal Vol.13, No¯ 02, (1–35). June 2011 Homogeneous Besov Spaces associated with the spherical mean operator L.T.Rachdi and A.Rouz Department of Mathematics, Faculty of Sciences of Tunis, 2092 El Manar 2 Tunis, Tunisia. email: ahlemrouz@yahoo.fr ABSTRACT We define and study homogeneous Besov spaces associated with the spherical mean operator. We establish some results of completeness, continuous embeddings and den- sity of subspaces. Next, we define a discrete equivalent norm on this space and we give other properties. RESUMEN Definimos y estudiamos los espacios homogneos Besov asociados con el operador esférico medio. Se establecen algunos resultados de la exhaustividad, de inclusiones continuas y de la densidad de subespacios. A continuación, se define una norma equivalente discreta en este espacio y se dan otras propiedades. Keywords and phrases:: Spherical mean operator, Besov space, Banach space, Fourier trans- form. Mathematics Subject Classification: 46E35 , 44A35. 2 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) 1 Introduction For a continuous function f on R × Rn, even with respect to the first variable, the spherical mean operator R is defined as R(f)(r,x) = ∫ Sn f(rη,x + rξ)dσn(η,ξ); (r,x) ∈ R × Rn, where Sn is the unit sphere, i.e. Sn = {(η,ξ) ∈ R × Rn ; η2 + |ξ|2 = 1} and σn is the surface measure on Sn normalized to have total measure one. The dual of the spherical mean operator tR is defined by t R(g)(r,x) = Γ ( n+1 2 ) π n+1 2 ∫ Rn g( √ r2 + |x − y|2,y)dy, where dy is the Lebesgue measure on Rn. The spherical mean operator R and its dual tR play an important role and have many applications, for example, in image processing of so-called synthetic aperture radar (SAR) data [14, 15], or in the linearized inverse scattering problem in acoustics [9]. Many aspects of such operator have been studied [1, 3, 6, 18, 21]. In particular, in [18] the first author with the others associated to the spherical mean operator the Fourier transform defined by ∀(µ,λ) ∈ Γ, F (f)(µ,λ) = ∫ Rn ∫ +∞ 0 f(r,x) ϕµ,λ(r,x) dνn(r,x), where • ϕµ,λ is the function defined by ∀(r,x) ∈ R × Rn, ϕµ,λ(r,x) = R ( cos(µ.)e−i〈λ|.〉 ) (r,x). • νn is the measure defined on [0, +∞[ × Rn, by dνn(r,x) = 1 2 n−1 2 Γ(n+1 2 )(2π) n 2 rn dr ⊗ dx. • Γ is the set given by Γ = R × Rn ∪ {(iµ,λ); (µ,λ) ∈ R × Rn, |µ| 6 |λ|} . They have constructed the harmonic analysis related to the Fourier transform F (Inversion for- mula, Schwartz theorem, Paley-Wiener theorem, Plancherel theorem). There are many ways to define Besov Spaces [4, 5, 13, 16, 20, 23]. It is well known that Besov spaces can be defined for instance in terms of convolutions f ∗ φt with different kinds of smooth functions φ and that can be also described by means of differences △xf [10, 11, 22]. CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 3 In this work, we define and study a class of homogeneous Besov spaces connected with the spherical mean operator R. More precisely, let φ be a smooth function on R×Rn, even with respect to the first variable. For all p,q ∈ [1, +∞] and γ ∈ R, we define the Besov space Bγ,φp,q ( [0, +∞[ × R n ) to be the space of tempered distributions f on R × Rn, even with respect to the first variable such that f = ∫ +∞ 0 f ∗ φt ∗ φt dt t , where ∗ is the convolution product associated with the spherical mean operator and φt; t > 0 is the dilated function of φ defined by ∀(r,x) ∈ [0, +∞[ × Rn, φt(r,x) = 1 t2n+1 φ( r t , x t ) (see Definition 10 below). The space B γ,φ p,q ( [0, +∞[ × Rn ) is equipped firstly with the norm Mγ,φp,q (f) =    (∫ +∞ 0 (‖f ∗ φt‖p,νn tγ )qdt t ) 1 q , if 1 6 q < +∞; esssup t>0 ‖f ∗ φt‖p,νn tγ , if q = +∞. with ‖f ∗ φt‖p,νn =    (∫ Rn ∫ +∞ 0 |f ∗ φt(r,x)|p dνn(r,x) ) 1 p , if p ∈ [1, +∞[ ; esssup (r,x)∈[0,+∞ [×Rn |f ∗ φt(r,x)|, if p = +∞. 4 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) Then we have established the coming results • The Besov space Bγ,φp,q ( [0, +∞[ × Rn ) is independent of the choice of the function φ and will be denoted by B γ p,q ( [0, +∞[ × Rn ) . This means that for all smooth functions φ and ψ, there exists a positive constant Cφ,ψ such that ∀f ∈ Bγ,φp,q ( [0, +∞[ × Rn ) , Mγ,φp,q (f) 6 Cφ,ψ M γ,ψ p,q (f). • The space Bγp,q ( [0, +∞[ × Rn ) is homogeneous with degree equal to (2n + 1)/p − γ − 2n − 1, that is for all f ∈ Bγp,q ( [0, +∞[ × Rn ) and t > 0, the distribution ft belongs to the space B γ p,q ( [0, +∞[×Rn ) and we have Mγ,φp,q (ft) = t 2n+1 p −γ−2n−1 Mγ,φp,q (f). • The Besov space is a Banach one when γ < (2n + 1)/p. We have also proved some continuous embeddings and density of subspaces. Next, we define the following discrete norm on the space B γ p,q ( [0, +∞[ × Rn ) by setting Nγ,φp,q (f) =    ( ∑ k∈Z (‖f ∗ φ2k‖p,νn 2kγ )q ) 1 q , if 1 6 q < +∞; esssup k∈Z ‖f ∗ φ2k‖p,νn 2kγ , if q = +∞. We show that this norm defines the same topology as the norm M γ,φ p,q . We prove that this space is homogeneous in a weaker sense when equipped with the norm N γ,φ p,q , that is there exist two positive constants C1 and C2 such that for all f ∈ Bγp,q ( [0, +∞[ × Rn ) and t > 0 C1 t 2n+1 p −2n−1−γ Nγ,φp,q (f) 6 N γ,φ p,q (ft) 6 C2 t 2n+1 p −2n−1−γ Nγ,φp,q (f). Finally, we establish some new continuous embedding. 2 Fourier transform associated with the spherical mean op- erator In this section, we recall some harmonic analysis results related to the Fourier transform associated with the spherical mean operator. Let ϕµ,λ, (µ,λ) ∈ C × Cn, be the function defined by ∀(r,x) ∈ R × Rn, ϕµ,λ(r,x) = R ( cos(µ.)e−i〈λ|.〉 ) (r,x). It’s well known ([18, 21]) that CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 5 i. The function ϕµ,λ is given by ∀(r,x) ∈ R × Rn, ϕµ,λ(r,x)e−i〈λ|x〉j(n−1)/2(r √ µ2 + λ2 1 + . . . + λ2n), where j(n−1)/2 is the modified Bessel function defined by jn−1 2 (s) = 2 n−1 2 Γ ( n + 1 2 )Jn−1 2 (s) s n−1 2 = Γ ( n + 1 2 ) +∞∑ k=0 (−1)k k!Γ (α + k + 1) ( s 2 )2k , and J(n−1)/2 is the Bessel function of first kind and index (n − 1)/2 [7, 8, 17, 27]. ii. For all (µ,λ) ∈ C × Cn, ϕµ,λ is the unique infinitely differentiable function on R × Rn, even with respect to the first variable, satisfying    Dju(r,x1, ...,xn) = −iλju(r,x1, ...,xn), 1 6 j 6 n, Ξu(r,x1, ...,xn) = −µ 2u(r,x1, ...,xn), u(0,...,0) = 1, ∂u ∂r (0,x1, ...,xn) = 0, ∀(x1, ...,xn) ∈ Rn. where Dj = ∂ ∂xj ; 1 6 j 6 n, and Ξ = ∂2 ∂r2 + n r ∂ ∂r − n∑ j=1 D2j . (2.1) iii. The function ϕµ,λ is bounded on R × Rn if, and only if (µ,λ) belongs to the set Γ given by Γ = R × Rn ∪ {(iµ,λ); (µ,λ) ∈ R × Rn, |µ| 6 |λ|} . (2.2) In this case, we have sup (r,x)∈R×Rn |ϕµ,λ(r,x)| = 1. We denote by • Lp(dνn), p ∈ [1, +∞] , the space of measurable functions f on [0, +∞[ × Rn, such that ‖f‖ p,νn =    (∫ Rn ∫ +∞ 0 |f(r,x)| p dνn(r,x) ) 1 p < +∞, if p ∈ [1, +∞[ ; esssup (r,x)∈[0,+∞ [×Rn |f(r,x)| < +∞, if p = +∞, where νn is the measure defined in the introduction. • Γ+ the subset of Γ given by Γ+ = [0, +∞[ × Rn ∪ {(iµ,λ); (µ,λ) ∈ R × Rn, 0 6 µ 6 |λ|} . 6 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) • BΓ+ the σ-algebra on Γ+ defined by BΓ+ = θ−1(B[0,+∞ [×Rn ), where θ is the bijective function defined on Γ+ by θ(µ,λ) = ( √ µ2 + λ2,λ). (2.3) • γn the measure defined on Γ+ by γn(A) = νn(θ(A)); A ∈ BΓ+. • Lp(dγn), p ∈ [1, +∞] , the space of measurable functions on Γ+ satisfying ‖f‖ p,γn < +∞. Then we have the coming properties Proposition 1. i) For all non negative measurable function f on Γ+ (respectively integrable on Γ+ with respect to the measure dγn), we have ∫ ∫ Γ+ f(µ,λ)dγn(µ,λ) 1 2 n−1 2 Γ(n+1 2 )(2π) n 2 { ∫ Rn ∫ +∞ 0 f(µ,λ) ( µ2 + |λ|2 )n−1 2 µdµdλ + ∫ Rn ∫ |λ| 0 f(iµ,λ) ( |λ|2 − µ2 )n−1 2 µdµdλ } . ii) For all non negative measurable function g on [0, +∞[×Rn (respectively integrable on [0, +∞[× R n with respect to the measure dνn), the function g◦θ is measurable positive on Γ+ (respectively integrable on Γ+ with respect to the measure dγn) and we have ∫ Rn ∫ ∞ 0 g(r,x)dνn(r,x) = ∫ ∫ Γ+ g ◦ θ(µ,λ)dγn(µ,λ). In the following, we shall define the translation operator and the convolution product associ- ated with the spherical mean operator. For this, we use the product formula for the function ϕµ,λ, for all (r,x), (s,y) ∈ R × Rn, we have ϕµ,λ(r,x)ϕµ,λ(s,y) Γ(n+1 2 ) √ π Γ ( n 2 ) ∫π 0 ϕµ,λ (√ r2 + s2 + 2rs cos θ,x + y ) sinn−1(θ)dθ (2.4) Definition 2. i) For all (r,x) ∈ [0, +∞[×Rn, the translation operator τ(r,x) associated with the spherical mean operator is defined on Lp(dνn), p ∈ [1, +∞] , by τ(r,x)(f)(s,y) = Γ(n+1 2 ) √ π Γ ( n 2 ) ∫π 0 f (√ r2 + s2 + 2rs cos θ,x + y ) sinn−1(θ)dθ. CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 7 ii) The convolution product of f,g ∈ L1(dνn) is defined by ∀(r,x) ∈ [0, +∞[ × Rn, f ∗ g(r,x) = ∫ Rn ∫ +∞ 0 f(s,y)τ(r,−x)(ǧ)(s,y)dνn(s,y), where ǧ(s,y) = g(s, −y). We have the following properties • For all (r,x), (s,y) ∈ [0, +∞[ × Rn, the relation (2.4) can be written τ(r,x)(ϕµ,λ)(s,y)ϕµ,λ(r,x) ϕµ,λ(s,y). (2.5) • If f ∈ Lp(dνn), 1 6 p 6 +∞, then for all (s,y) ∈ [0, +∞[ × Rn, the function τ(s,y)(f) belongs to Lp(dνn) and we have ∥∥τ(s,y)(f) ∥∥ p,νn 6 ‖f‖ p,νn . (2.6) • Let p, q, r ∈ [1, +∞] such that 1 r = 1 p + 1 q − 1. Then for all f ∈ Lp(dνn) and g ∈ Lq(dνn), the function f ∗ g belongs to Lr(dνn) and we have ‖f ∗ g‖ r,νn 6 ‖f‖ p,νn ‖g‖ q,νn . (2.7) Now, we will define the Fourier transform F connected with the spherical mean operator and we recall some properties that we need in the next section. Definition 3. The Fourier transform associated with the spherical mean operator is defined on L1(dνn) by ∀(µ,λ) ∈ Γ, F (f)(µ,λ) = ∫ Rn ∫ +∞ 0 f(r,x) ϕµ,λ(r,x) dνn(r,x), where Γ is the set defined by the relation (2.2). The Fourier transform F satisfies the properties • For every f in L1(dνn) and (r,x) ∈ [0, +∞[ × Rn, we have ∀(µ,λ) ∈ Γ, F ( τ(r,−x)(f) ) (µ,λ)ϕµ,λ(r,x)F (f)(µ,λ). (2.8) • For all f,g ∈ L1(dνn), we have ∀(µ,λ) ∈ Γ, F (f ∗ g) (µ,λ) = F (f)(µ,λ)F (g)(µ,λ). (2.9) • For all f ∈ L1(dνn), we have ∀(µ,λ) ∈ Γ, F (f) (µ,λ)F̃ (f) ◦ θ(µ,λ), (2.10) 8 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) where ∀(µ,λ) ∈ R × Rn, F̃ (f) (µ,λ)= ∫ Rn ∫ +∞ 0 f(r,x)jn−1 2 (rµ)e−i〈λ|x〉 dνn(r,x) (2.11) and θ is the function defined by the relation (2.3). Theorem 4. (Inversion formula for F ) Let f ∈ L1(dνn) such that the function F (f) belongs to L1(dγn), then for almost every (r,x) ∈ [0, +∞[ × Rn, we have f(r,x) = ∫ ∫ Γ+ F (f)(µ,λ) ϕµ,λ(r,x) dγn(µ,λ). We denote by • E∗ (R × Rn) the space of infinitely differentiable functions on R × Rn, even with respect to the first variable. • S∗ (R × Rn) the subspace of E∗ (R × Rn) consisting of functions rapidly decreasing together with all their derivatives. • S∗ (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, i.e ∀k1,k2 ∈ N, ∀α ∈ Nn, sup (µ,λ)∈Γ ( 1 + µ2 + 2|λ|2 )k1 ∣∣∣ ( ∂ ∂µ )k2 Dαλf(µ,λ) ∣∣∣ < +∞, where ∂f ∂µ (µ,λ)    ∂ ∂r (f(r,λ)) , if µ = r ∈ R 1 i ∂ ∂t (f(it,λ)) , if µ = it, |t| 6 |λ| and Dαλ = ( ∂ ∂λ1 )α1 . . . ( ∂ ∂λn )αn . • S′∗ (R × Rn) and S ′ ∗(Γ) are respectively the topological dual spaces of S∗ (R × Rn) and S∗(Γ). Each of these spaces is equipped with its usual topology. Theorem 5. (Schwartz theorem)[2, 18] i) The Fourier transform F is a topological isomorphism from S∗(R × Rn) onto S∗(Γ). The inverse mapping is given by ∀(r,x) ∈ R × Rn, F −1(f)(r,x) = ∫ ∫ Γ+ f(µ,λ) ϕµ,λ(r,x) dγn(µ,λ). (2.12) ii) (Plancherel formula) For all f,g ∈ S∗(R × Rn), we have ∫ +∞ 0 ∫ Rn f(r,x) g(r,x) dνn(r,x) = ∫ ∫ Γ+ F (f)(µ,λ) F (g)(µ,λ) dγn(µ,λ). CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 9 In particular ‖F (f)‖ 2,γn ‖f‖ 2,νn . Theorem 6. (Plancherel theorem) The Fourier transform F can be extended to an isometric iso- morphism from L2(dνn) onto L 2(dγn). For T ∈ S′∗(R × Rn), we put 〈F (T),ϕ〉 = 〈T, F −1(ϕ)〉; ϕ ∈ S∗(Γ). (2.13) Then from Theorem 5, we get the following result Corollary 7. The transform F defined by the relation (2.13) is a topological isomorphism from S ′ ∗(R × Rn) onto S ′ ∗(Γ). Proposition 8. i) Let f ∈ E∗ (R × Rn) , f slowly increasing and let g ∈ S∗(R × Rn). Then the function f ∗ g belongs to the space E∗ (R × Rn) . ii) For all f ∈ S∗(R × Rn) and T ∈ S ′ ∗(R × Rn). The function T ∗ f defined by ∀(r,x) ∈ R × Rn, T ∗ f(r,x) = 〈T,τ(r,−x)(f̌)〉 belongs to the space E∗ (R × Rn) and is slowly increasing. Moreover, we have F ( TT∗f ) = F (f̌)F (T). 3 Besov spaces This section contains the main result of this paper. Indeed, we define and study a class of Besov spaces B γ,φ p,q ( [0, +∞[×Rn ) , where φ is a smooth function. We show that this space is independant of the choice of φ and is a Banach space for γ < (2n+1)/p. Next, we prove that B γ,φ p,q ( [0, +∞[ × R n ) is an homogeneous space with degree equal to (2n + 1)/p − γ − 2n − 1. Lemma 9. Let a, b, a1, b1 be real numbers such that 0 < a1 < a < b < b1. Then there exists a function ψ ∈ S∗(R × Rn) satisfying the following assumptions i) ∀(µ,λ) ∈ Γ, F (ψ)(µ,λ) > 0. ii) ∀(µ,λ) ∈ Γ ; a2 6 µ2 + 2|λ|2 6 b2, F (ψ)(µ,λ) = C where C is a positive constante. iii) F (ψ)(µ,λ) = 0 if µ2 + 2|λ|2 > b21 or µ 2 + 2|λ|2 < a21. 10 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) iv) For all (µ,λ) ∈ Γ \ {(0,0)}, ∫ +∞ 0 ( F (ψ)(tµ,tλ) )2dt t = 1. Proof. From Uryshon’s lemma, there exists an infinitely differentiable function ω on R such that • ∀t ∈ R; 0 6 ω(t) 6 1. • ∀t ∈ [a,b]; ω(t) = 1. • supp(ω) ⊂]a1,b1[. Let g be the function defined on R × Rn by g(r,x) = ω (√ r2 + |x|2 ) (∫ +∞ 0 (ω(t))2 dt t )1 2 , then the function g belongs to the space S∗(R × Rn). Since, the transform F̃ defined by the relation (2.11) is a topological isomorphism from the space S∗(R × Rn) onto itself [24, 25], then there exists ψ ∈ S∗(R × Rn) such that F̃ (ψ) = g. Thus, by the relation (2.10), we deduce that the function ψ satisfies the hypothesis of the lemma. We denote by • D∗(Γ) the space of real infinitely differentiable functions g on Γ, even with respect to the first variable such that, there exist two positive real numbers 0 < a < b verifying g(µ,λ) = 0 if µ2 + 2|λ|2 < a2 or µ2 + 2|λ|2 > b2. • S∗,0(R × Rn) the subspace of S∗(R × Rn) consisting of functions f such that F (f) belongs to the space D∗(Γ). • S1∗,0(R × Rn) the subspace of S∗,0(R × Rn) formed by the functions f such that ∀(µ,λ) ∈ Γ \ {(0,0)}, ∫ +∞ 0 ( F (f)(tµ,tλ) )2dt t = 1. (3.1) These functions are known as wavelets on [0, +∞[ × Rn [19, 26]. CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 11 • Lp(dt t ); p ∈ [1, +∞], the space of measurable functions on ]0, +∞[ such that ∥∥f ∥∥ Lp( dt t ) =    (∫ +∞ 0 ∣∣f(t) ∣∣pdt t ) 1 p < +∞, 1 6 p < +∞; esssup t>0 ∣∣f(t) ∣∣ < +∞, p = +∞. • ⋆ the convolution product defined on the group ( ]0, +∞[, . ) by f ⋆ g(s) = ∫ +∞ 0 f(t) g( s t ) dt t . (3.2) • For all measurable function φ on [0, +∞[ × Rn, the dilated φt; t > 0 of φ is defined by ∀(r,x) ∈ [0, +∞[ × Rn, φt(r,x) = 1 t2n+1 φ( r t , x t ). Then we have the following properties • Let p, q, r ∈ [1, +∞] such that 1 p + 1 q = 1 + 1 r . Then for all f ∈ Lp(dt t ) and g ∈ Lq(dt t ), the function f ⋆ g belongs to Lr(dt t ) and we have ∥∥f ⋆ g ∥∥ Lr( dt t ) 6 ‖f‖Lp( dt t )‖g‖Lq( dt t ). (3.3) • For every φ ∈ Lp(dνn); p ∈ [1, +∞] , the function φt belongs to Lp(dνn) and we have ∥∥φt ∥∥ p,νn = t − 2n+1 p ′ ∥∥φ ∥∥ p,νn , (3.4) where p ′ = p/(p − 1). • For all φ ∈ L1(dνn) and for every (µ,λ) ∈ Γ, F (φt)(µ,λ) = F (φ)(tµ,tλ). (3.5) Definition 10. Let p, q ∈ [1, +∞] , γ ∈ R and φ ∈ S1∗,0(R × Rn). We define the Besov space B γ,φ p,q ( [0, +∞[ × Rn ) to be the space of tempered distributions f on R × Rn, even with respect to the first variable and satisfying • For all t > 0, the function f ∗ φt belongs to the space Lp(dνn). • The function t 7−→ ‖f ∗ φt‖p,νn tγ belongs to the space Lq(dt t ). • The integral (r,x) 7−→ ∫ +∞ 0 f ∗ φt ∗ φt(r,x) dt t 12 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) is convergent in S ′ ∗(R × Rn) and f = ∫ +∞ 0 f ∗ φt ∗ φt dt t . (3.6) The space B γ,φ p,q ( [0, +∞[ × Rn ) is equipped with the norm Mγ,φp,q (f) =    (∫ +∞ 0 (‖f ∗ φt‖p,νn tγ )qdt t ) 1 q , if 1 6 q < +∞; esssup t>0 ‖f ∗ φt‖p,νn tγ , if q = +∞. Lemma 11. let ψ ∈ S∗(R × Rn) and let φ ∈ S∗,0(R × Rn). Then for all k ∈ N, there exists φk ∈ S∗,0(R × Rn) such that ψ ∗ φt = t2k ( ∆kψ ) ∗ (φk)t, where ∆ is the differential operator defined by ∆ = − ( ∂2 ∂r2 + n r ∂ ∂r + n∑ j=1 ( ∂ ∂xj )2) . Moreover, for all p ∈ [1, +∞] ∥∥ψ ∗ φt ∥∥ p,νn 6 t2k ∥∥∆kψ ∥∥ p,νn ∥∥φk ∥∥ 1,νn (3.7) and ∥∥ψ ∗ φt ∥∥ p,νn 6 t − 2n+1 p ′ ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ p,νn . (3.8) Proof. The operator ∆ is continuous from S∗(R × Rn) into itself and for all f ∈ S∗(R × Rn), we have F ( ∆f ) (µ,λ) = ( µ2 + 2|λ|2 ) F (f)(µ,λ). (3.9) Let ψ ∈ S∗(R × Rn) and let φ ∈ S∗,0(R × Rn). From the relations (2.9) and (3.5), we get F ( ψ ∗ φt ) (µ,λ) = F (ψ)(µ,λ) F (φ)(tµ,tλ) = t2 ( µ2 + 2|λ|2 ) F (ψ)(µ,λ) F (φ)(tµ,tλ) t2 ( µ2 + 2|λ|2 ), and from the equality (3.9), we obtain F ( ψ ∗ φt ) (µ,λ) = t2F (∆ψ)(µ,λ) F (φ)(tµ,tλ) t2 ( µ2 + 2|λ|2 ). (3.10) CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 13 Since, the function φ belongs to the space S∗,0(R × Rn) then the function (µ,λ) 7−→ F (φ)(µ,λ) µ2 + 2|λ|2 belongs to the space S∗(Γ) and from Theorem 5, there exists φ1 ∈ S∗(R × Rn) such that F (φ1)(µ,λ) = F (φ)(µ,λ) µ2 + 2|λ|2 . In particular, φ1 lies in S∗,0(R × Rn) and the relation (3.10) leads to F ( ψ ∗ φt ) (µ,λ) = t2F (∆ψ)(µ,λ) F ( (φ1)t ) (µ,λ), which implies that ψ ∗ φt = t2 (∆ψ) ∗ (φ1)t. By induction, for all k ∈ N∗, there exists φk ∈ S∗,0(R × Rn) verifying ψ ∗ φt = t2k (∆kψ) ∗ (φk)t. (3.11) On the other hand, for every t > 0 and by the relation (3.4), we get ∥∥ψ ∗ φt ∥∥ p,νn 6 ‖ψ‖1,νn ‖φt‖p,νn = t − 2n+1 p ′ ‖ψ‖1,νn ‖φ‖p,νn as the same way and using the relation (3.11), it follows that ∥∥ψ ∗ φt ∥∥ p,νn 6 t2k ‖∆kψ‖p,νn‖φk‖1,νn. Proposition 12. Let φ ∈ S1∗,0(R × Rn). i) For all f ∈ L2(dνn) we have f = ∫ +∞ 0 f ∗ φt ∗ φt dt t ; in L2(dνn). ii) Let γ ∈ R; γ < (2n + 1)/p and f ∈ S′∗(R × Rn) such that for all t > 0, the function f ∗ φt belongs to Lp(dνn) and the function t 7−→ ‖f ∗ φt‖p,νn tγ belongs to the space Lq(dt t ). Then the integral ∫ +∞ 0 f ∗ φt ∗ φt dt t converges in S ′ ∗(R × Rn). 14 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) Proof. i) Let f ∈ L2(dνn) and let Fa,b(f) be the function defined by ∀(r,x) ∈ [0, +∞[ × Rn, Fa,b(f)(r,x) = ∫b a f ∗ φt ∗ φt(r,x) dt t ; 0 < a < b. The function Fa,b(f) is well defined and by the relation (3.4) we have ∣∣Fa,b(f)(r,x) ∣∣ 6 ∫b a ‖f‖2,νn ‖φt ∗ φt‖2,νn dt t 6 ‖f‖2,νn ∫b a ‖φt‖1,νn ‖φt‖2,νn dt t 6 ‖f‖2,νn ‖φ‖1,νn ‖φ‖2,νn ∫b a t− 2n+1 2 −1dt < +∞. Moreover, the function Fa,b(f) belongs to L 2(dνn). Indeed by Minkowski’s inequality [12] and the relation (3.4) we get ∥∥Fa,b(f) ∥∥ 2,νn 6 ∫b a ∥∥f ∗ φt ∗ φt ∥∥ 2,νn dt t 6 ∫b a ‖f‖2,νn ‖φt‖21,νn dt t = ‖f‖2,νn ‖φ‖21,νn log( b a ) < +∞. On the other hand, by Fubini’s theorem and the relation (3.5), we have F ( Fa,b(f) ) (µ,λ) = F (f)(µ,λ) ∫b a ( F (φ)(tµ,tλ) )2dt t . Thus, by the Plancherel theorem ∥∥f − Fa,b(f) ∥∥2 2,νn = ∥∥F (f) − F (Fa,b(f)) ∥∥2 2,γn = ∫ ∫ Γ+ ∣∣F (f)(µ,λ) ∣∣2 ∣∣∣1 − ∫b a ( F (φ)(tµ,tλ) )2dt t ∣∣∣dγn(µ,λ). Using the fact that ∫ +∞ 0 ( F (φ)(tµ,tλ) )2dt t = 1, we have ∀(µ,λ) ∈ Γ\{(0,0)}, ∣∣∣1 − ∫b a ( F (φ)(tµ,tλ) )2dt t ∣∣∣ 6 1 and applying the dominated convergence theorem, we deduce that lim a→0+ b→ +∞ ∥∥f − Fa,b(f) ∥∥ 2,νn = 0. CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 15 ii) Let f be in S ′ ∗(R × Rn) satisfying the hypothesis, then the function Fa,b(f) defined above is bounded on R × Rn. In fact ∣∣Fa,b(f)(r,x) ∣∣ 6 ∫b a ∥∥f ∗ φt ∥∥ p,νn ∥∥φt ∥∥ p ′ ,νn dt t = ∥∥φ ∥∥ p ′ ,νn ∫b a ∥∥f ∗ φt ∥∥ p,νn tγ t − 2n+1 p +γ dt t 6 ∥∥φ ∥∥ p ′ ,νn [∫b a (∥∥f ∗ φt ∥∥ p,νn tγ )qdt t ] 1 q [∫b a t (− 2n+1 p +γ) q ′ dt t ] 1 q ′ < +∞, where q ′ is the conjugate exponent of q. Thus for all a, b ∈ R; b > a > 0, the function Fa,b(f) defines an element of S ′ ∗(R × Rn). Let ψ ∈ S∗(R × Rn), by Fubini’s theorem, we have 〈Fa,b(f),ψ〉 = ∫ +∞ 0 ∫ Rn { ∫b a f ∗ φt ∗ φt(r,x) ψ(r,x) dt t } dνn(r,x) = ∫b a { ∫ +∞ 0 ∫ Rn f ∗ φt ∗ φt(r,x) ψ(r,x) dνn(r,x) } dt t = ∫b a { ∫ +∞ 0 ∫ Rn ψ(r,x) [∫ +∞ 0 ∫ Rn f ∗ φt(s,y) τ(r,−x)(φ̌t)(s,y) dνn(s,y) ] dνn(r,x) } dt t = ∫b a { ∫ +∞ 0 ∫ Rn f ∗ φt(s,y) [∫ +∞ 0 ∫ Rn ψ(r,x) τ(s,−y)(φt)(r,x) dνn(r,x) ] dνn(s,y) } dt t = ∫b a [∫ +∞ 0 ∫ Rn f ∗ φt(s,y) φ̌t ∗ ψ(s,y) dνn(s,y) ] dt t . However, ∫ +∞ 0 [∫ +∞ 0 ∫ Rn ∣∣f ∗ φt(s,y) ∣∣ ∣∣φ̌t ∗ ψ(s,y) ∣∣ dνn(s,y) ] dt t 6 ∫ +∞ 0 ∥∥f ∗ φt ∥∥ p,νn ∥∥φ̌t ∗ ψ ∥∥ p ′ ,νn dt t 6 ∫1 0 ∥∥f ∗ φt ∥∥ p,νn ∥∥φ̌t ∗ ψ ∥∥ p ′ ,νn dt t + ∫ +∞ 1 ∥∥f ∗ φt ∥∥ p,νn ∥∥φ̌t ∗ ψ ∥∥ p ′ ,νn dt t . 16 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) Using the relations (3.7) and (3.8), we get ∫ +∞ 0 { ∫ +∞ 0 ∫ Rn ∣∣f ∗ φt(s,y) ∣∣ ∣∣φ̌t ∗ ψ(s,y) ∣∣ dνn(s,y) } dt t 6 ∥∥∆kψ ∥∥ p ′ ,νn ∥∥φk ∥∥ 1,νn ∫1 0 t2k ∥∥f ∗ φt ∥∥ p,νn dt t + ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ p ′ ,νn ∫ +∞ 1 t − 2n+1 p ∥∥f ∗ φt ∥∥ p,νn dt t ∥∥∆kψ ∥∥ p ′ ,νn ∥∥φk ∥∥ 1,νn ∫ +∞ 0 t2k+γ 1[0,1](t) ∥∥f ∗ φt ∥∥ p,νn tγ dt t + ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ p ′ ,νn ∫ +∞ 0 t − 2n+1 p +γ 1[1,+∞ [(t) ∥∥f ∗ φt ∥∥ p,νn tγ dt t . Let k be sufficiently large. Using the hypothesis γ < (2n + 1)/p and applying Hölder’s inequality, we obtain ∫ +∞ 0 { ∫ +∞ 0 ∫ Rn ∣∣f ∗ φt(s,y) ∣∣ ∣∣φ̌t ∗ ψ(s,y) ∣∣ dνn(s,y) } dt t 6 ∥∥∆kψ ∥∥ p ′ ,νn ∥∥φk ∥∥ 1,νn ∥∥∥t2k+γ 1[0,1] ∥∥∥ Lq ′ ( dt t ) ∥∥∥ ∥∥f ∗ φt ∥∥ p,νn tγ ∥∥∥ Lq( dt t ) + ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ p ′ ,νn ∥∥∥t− 2n+1 p +γ 1[1,+∞ [ ∥∥∥ Lq ′ ( dt t ) ∥∥∥ ∥∥f ∗ φt ∥∥ p,νn tγ ∥∥∥ Lq( dt t ) < +∞. This shows that for all ψ ∈ S∗(R × Rn), lim a→0+ b→ +∞ 〈Fa,b(f),ψ〉 exists and lim a→0+ b→ +∞ 〈Fa,b(f),ψ〉 = ∫ +∞ 0 ∫ +∞ 0 ∫ Rn f ∗ φt(s,y) φ̌t ∗ ψ(s,y) dνn(s,y) dt t . This means that the integral ∫ +∞ 0 f ∗ φt ∗ φt dt t converges in S ′ ∗(R × Rn). Lemma 13. 1) Let f ∈ Bγ,φp,q ( [0, +∞[ × Rn ) . Then i) For all ψ ∈ S∗(R × Rn), we have f ∗ ψ = ∫ ∞ 0 f ∗ φt ∗ φt ∗ ψ dt t . ii) For all ψ ∈ S1∗,0(R × Rn), f = ∫ +∞ 0 f ∗ ψρ ∗ ψρ dρ ρ . CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 17 2) For all g ∈ S∗(R × Rn) and for all ψ ∈ S1∗,0(R × Rn), we have ∫ +∞ 0 g ∗ ψρ ∗ ψρ dρ ρ = g. Proof. 1) Let f ∈ Bγ,φp,q ( [0, +∞[ × Rn ) . i) For every ψ ∈ S∗(R × Rn), we have f ∗ ψ(r,x) = 〈f,τ(r,−x)ψ̌〉 = lim a→0+ b→ +∞ 〈 ∫b a f ∗ φt ∗ φt dt t ,τ(r,−x)ψ̌〉 = lim a→0+ b→ +∞ ∫ +∞ 0 ∫ Rn (∫b a f ∗ φt ∗ φt(s,y) dt t ) τ(r,−x)ψ̌(s,y) dνn(s,y), and by Fubini’s theorem, we obtain f ∗ ψ(r,x) = lim a→0+ b→ +∞ ∫b a (∫ +∞ 0 ∫ Rn f ∗ φt ∗ φt(s,y) τ(r,−x)ψ̌(s,y) dνn(s,y) ) dt t = lim a→0+ b→ +∞ ∫b a f ∗ φt ∗ φt ∗ ψ(r,x) dt t = ∫ +∞ 0 f ∗ φt ∗ φt ∗ ψ(r,x) dt t . ii) Let ψ ∈ S1∗,0(R × Rn). For all positive real number ρ, we have ψρ ∗ ψρ = (ψ ∗ ψ)ρ. (3.12) Applying i) we get f ∗ ψρ ∗ ψρ = ∫ +∞ 0 f ∗ φt ∗ φt ∗ ψρ ∗ ψρ dt t . Now, let a1, a2, b1, b2 be positive real numbers such that F (φ)(µ,λ) = 0 if µ2 + 2|λ|2 < a21 or µ 2 + 2|λ|2 > b21 and F (ψ)(µ,λ) = 0 if µ2 + 2|λ|2 < a22 or µ 2 + 2|λ|2 > b22 then F (φt)(µ,λ)F (ψρ)(µ,λ) = 0 if t ρ /∈ [ a1 b2 , b1 a2 ] = [α, β] , and consequently, by the relation (2.9) and Theorem 4 φt ∗ ψρ = 0 if t ρ /∈ [α, β] . (3.13) 18 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) Thus, f ∗ ψρ ∗ ψρ = ∫ρβ ρα f ∗ φt ∗ φt ∗ ψρ ∗ ψρ dt t . So for all a, b ∈ R; 0 < a < b, ∫b a f ∗ ψρ ∗ ψρ dρ ρ = ∫b a (∫ρβ ρα f ∗ φt ∗ φt ∗ ψρ ∗ ψρ dt t ) dρ ρ . By Fubini’s theorem, we get ∫b a f ∗ ψρ ∗ ψρ dρ ρ = ∫bβ aα (∫ t α t β f ∗ φt ∗ φt ∗ ψρ ∗ ψρ dρ ρ ) dt t . (3.14) On the other hand, we have ∫ t α t β f ∗ φt ∗ φt ∗ ψρ ∗ ψρ(r,x) dρ ρ = ∫ t α t β (∫ +∞ 0 ∫ Rn φt ∗ ψρ ∗ ψρ(s,y) τ(r,−x) ˇ(f ∗ φt)(s,y)dνn(s,y) ) dρ ρ = ∫ +∞ 0 ∫ Rn τ(r,−x)( ˇf ∗ φt)(s,y) (∫ t α t β φt ∗ ψρ ∗ ψρ(s,y) dρ ρ ) dνn(s,y) = ∫ +∞ 0 ∫ Rn τ(r,−x)( ˇf ∗ φt)(s,y) (∫ +∞ 0 φt ∗ ψρ ∗ ψρ(s,y) dρ ρ ) dνn(s,y). However by i) of Proposition 12, it follows that ∫ t α t β f ∗ φt ∗ φt ∗ ψρ ∗ ψρ(r,x) dρ ρ = ∫ +∞ 0 ∫ Rn φt(s,y) τ(r,−x)( ˇf ∗ φt)(s,y) dνn(s,y) = f ∗ φt ∗ φt(r,x). Replacing in the equality (3.14), we obtain ∫b a f ∗ ψρ ∗ ψρ dρ ρ = ∫bβ aα f ∗ φt ∗ φt dt t . 2) We know that for all g ∈ S∗(R × Rn) and ψ ∈ S1∗,0(R × Rn), the function g ∗ ψρ ∗ ψρ belongs to the space S∗(R × Rn). By Theorem 4 and the relation (3.5), we have g ∗ ψρ ∗ ψρ(r,x) = ∫ ∫ Γ+ F (g)(µ,λ) ( F (ψ)(ρµ,ρλ) )2 ϕµ,λ(r,x) dγn(µ,λ), then ∫ +∞ 0 g ∗ ψρ ∗ ψρ(r,x) dρ ρ ∫ ∫ Γ+ F (g)(µ,λ)ϕµ,λ(r,x) [∫ +∞ 0 ( F (ψ)(ρµ,ρλ) )2 dρ ρ ] dγn(µ,λ), CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 19 and by the relation (3.1) and Theorem 4, we get ∫ +∞ 0 g ∗ ψρ ∗ ψρ(r,x) dρ ρ = ∫ ∫ Γ+ F (g)(µ,λ) ϕµ,λ(r,x) dγn(µ,λ) = g(r,x). Theorem 14. Let p, q ∈ [1, +∞] and γ ∈ R, the space Bγ,φp,q ( [0, +∞[ × Rn ) is independent of the choice of the function φ in S1∗,0 ( R × Rn ) and will be denoted by B γ p,q ( [0, +∞[ × Rn ) . Proof. Let f ∈ Bγ,φp,q ( [0, +∞[ × Rn ) and let ψ ∈ S1∗,0 ( R × Rn ) . From Lemma 13 and the relation (3.13), we have f ∗ ψρ = ∫ +∞ 0 f ∗ φt ∗ φt ∗ ψρ dt t = ∫ρβ ρα f ∗ φt ∗ φt ∗ ψρ dt t = ∫β α f ∗ φρs ∗ φρs ∗ ψρ ds s . Thus, from Minkowski’s inequality and the relations (2.7) and (3.4), we get ∥∥f ∗ ψρ ∥∥ p,νn 6 ∫β α ∥∥f ∗ φρs ∗ ψρ ∗ φρs ∥∥ p,νn ds s 6 ∫β α ∥∥f ∗ φρs ∥∥ p,νn ∥∥ψρ ∗ φρs ∥∥ 1,νn ds s 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn ∫β α ∥∥f ∗ φρs ∥∥ p,νn ds s , (3.15) and by Hölder’s inequality, it follows that ∥∥f ∗ ψρ ∥∥ p,νn 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn (∫β α (∥∥f ∗ φρs ∥∥ p,νn (ρs)γ )qds s ) 1 q (∫β α ( ρs )γ q′ ds s ) 1 q ′ 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn Mγ,φp,q (f)ρ γ (∫β α sγ q ′ ds s ) 1 q ′ < +∞, where q ′ is the conjugate exponent of q. Now, by the relation (3.15), we have ∥∥f ∗ ψρ ∥∥ p,νn ργ 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn ∫β α sγ ∥∥f ∗ φρs ∥∥ p,νn (ρs)γ ds s 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn ∫ 1 α 1 β t−γ ∥∥f ∗ φρ t ∥∥ p,νn ( ρ t )γ dt t = ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn [ t−γ1[ 1β, 1 α ] ⋆ (∥∥f ∗ φt ∥∥ p,νn tγ )] (ρ), 20 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) where ⋆ is the convolution product defined on ]0, +∞[ by the relation (3.2). By the relation (3.3), we obtain Mγ,ψp,q (f) 6 ∥∥ψ ∥∥ 1,νn ∥∥φ ∥∥ 1,νn ∥∥∥t−γ1[ 1β, 1α ] ∥∥∥ L1( dt t ) Mγ,φp,q (f) < +∞, and the proof is complete if we take into account Lemma 13. Proposition 15. Let p, q ∈ [1, +∞] and γ ∈ R. The Besov space B γ p,q ( [0, +∞[ × Rn ) is homogeneous of degree equal to (2n + 1)/p − γ − 2n − 1, that is for every f ∈ Bγp,q ( [0, +∞[ × Rn ) and t > 0, the distribution ft belongs to the space B γ p,q ( [0, +∞[×Rn ) and we have Mγ,φp,q (ft) = t 2n+1 p −γ−2n−1 Mγ,φp,q (f), where 〈ft,ϕ〉 = 〈f, 1 t2n+1 ϕ1 t 〉; ϕ ∈ S∗(R × Rn). Proof. Let φ ∈ S1∗,0(R × Rn), we have ft ∗ φρ(r,x) = 〈ft,τ(r,−x)(φ̌ρ)〉 = 〈f, 1 t2n+1 ( τ(r,−x)(φ̌ρ) ) 1 t 〉. However, 1 t2n+1 ( τ(r,−x)(φ̌ρ) ) 1 t (s,y) = τ(r,−x)(φ̌ρ)(ts,ty) = 1 t2n+1 τ( r t ,− x t )(φ̌ρ t )(s,y) consequently, ft ∗ φρ(r,x) = 〈f, 1 t2n+1 τ( r t ,− x t )(φ̌ρ t )〉 = ( f ∗ φρ t ) t (r,x). (3.16) Hence, from the relation (3.4), we get ∥∥ft ∗ φρ ∥∥ p,νn = t − 2n+1 p ′ ∥∥f ∗ φρ t ∥∥ p,νn , this shows that for all ρ > 0, the function ft ∗ φρ belongs to Lp(dνn) and we have ∥∥∥ ∥∥ft ∗ φρ ∥∥ p,νn ργ ∥∥∥ q Lq( dρ ρ ) = t − 2n+1 p ′ q ∫ +∞ 0 (∥∥f ∗ φρ t ∥∥ p,νn ργ )q dρ ρ = t − 2n+1 p ′ q t−γ q ∫ +∞ 0 (∥∥f ∗ φs ∥∥ p,νn sγ )q ds s = t −q ( 2n+1 p ′ +γ )[ Mγ,φp,q (f) ]q , CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 21 which proves that the function ρ 7−→ ∥∥ft ∗ φρ ∥∥ p,νn ργ belongs to the space Lq(dρ ρ ) and that Mγ,φp,q (ft) = t 2n+1 p −γ−2n−1 Mγ,φp,q (f). On the other hand, from the relations (3.12) and (3.16), we have ∫ +∞ 0 ft ∗ φρ ∗ φρ(r,x) dρ ρ 1 t2n+1 ∫ +∞ 0 f ∗ ( φ ∗ φ ) s ( r t , x t ) ds s = 1 t2n+1 ∫ +∞ 0 f ∗ φs ∗ φs( r t , x t ) ds s , and from the relation (3.6), it follows ∫ +∞ 0 ft ∗ φρ ∗ φρ(r,x) dρ ρ 1 t2n+1 f( r t , x t ) = ft(r,x). This completes the proof. Proposition 16. Let p, q ∈ [1, +∞] and γ ∈ R. The space B γ p,q ( [0, +∞[ × Rn ) ∩ E∗ ( R × Rn ) is dense in B γ p,q ( [0, +∞[ × Rn ) . Proof. Let f ∈ Bγp,q ( [0, +∞[ × Rn ) and φ ∈ S1∗,0(R × Rn). For all t > 0, the function (r,x) 7−→ f ∗ φt(r,x) = 〈f,τ(r,−x)(φ̌t)〉 belongs to the space E∗ ( R × Rn ) and is slowly increasing. From i) of Proposition 8, we deduce that the function f ∗ φt ∗ φt belongs to the space E∗ ( R × Rn ) . Thus, from derivative’s theorem it follows that for all k ∈ N∗; the function fk(r,x) = ∫k 1 k f ∗ φt ∗ φt(r,x) dt t is infinitely differentiable on R × Rn, even with respect to the first variable. On the other hand, let ψ ∈ S1∗,0(R × Rn), by Fubini’s theorem, we have fk ∗ ψρ = ∫k 1 k f ∗ φt ∗ φt ∗ ψρ dt t . And by the same way as the proof of Theorem 14, we deduce that for all ρ > 0, the function fk ∗ ψρ belongs to Lp(dνn) and that the function ρ 7−→ ∥∥fk ∗ ψρ ∥∥ p,νn ργ 22 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) belongs to Lq(dρ ρ ). Again, by Fubini’s theorem, for all ψ ∈ S1∗,0(R × Rn), ∫ +∞ 0 fk ∗ ψρ ∗ ψρ dρ ρ = ∫ +∞ 0 (∫k 1 k f ∗ φt ∗ φt ∗ ψρ ∗ ψρ dt t ) dρ ρ = ∫k 1 k (∫ +∞ 0 f ∗ ψρ ∗ ψρ ∗ φt ∗ φt dρ ρ ) dt t , and by Lemma 13 and Theorem 14, we obtain ∫ +∞ 0 fk ∗ ψρ ∗ ψρ dρ ρ = ∫k 1 k f ∗ φt ∗ φt dt t = fk. This shows that for all k ∈ N∗, the function fk belongs to the space B γ p,q ( [0, +∞[ × Rn ) ∩ E∗ ( R × Rn ) . Moreover, for every ϕ ∈ S1∗,0(R × Rn), we have fk ∗ ϕρ = ∫k 1 k f ∗ φt ∗ φt ∗ ϕρ dt t , and by i) of Lemma 13, we get f ∗ ϕρ = ∫ +∞ 0 f ∗ φt ∗ φt ∗ ϕρ dt t , Thus, ( f − fk ) ∗ ϕρ = ∫ 1 k 0 f ∗ φt ∗ φt ∗ ϕρ dt t + ∫ +∞ k f ∗ φt ∗ φt ∗ ϕρ dt t = ∫ [0, 1k ]∪[k,+∞ [ f ∗ φt ∗ φt ∗ ϕρ dt t . Now using the relation (3.13), we obtain ( f − fk ) ∗ ϕρ = ∫ ( [0, 1kρ ]∪[ k ρ ,+∞ [ ) ∩[α,β] f ∗ φρs ∗ φρs ∗ ϕρ ds s = ∫ +∞ 0 1( [0, 1kρ ]∪[ k ρ ,+∞ [ ) ∩[α,β] (s) f ∗ φρs ∗ φρs ∗ ϕρ ds s . Now Minkowski’s inequality leads to ∥∥(f − fk ) ∗ ϕρ ∥∥ p,νn 6 ∫β α 1( [0, 1kρ ]∪[ k ρ ,+∞ [ )(s) ∥∥f ∗ φρs ∗ φρs ∗ ϕρ ∥∥ p,νn ds s 6 ∫β α 1( [0, 1kρ ]∪[ k ρ ,+∞ [ )(s) ∥∥f ∗ φρs ∥∥ p,νn ∥∥φρs ∗ ϕρ ∥∥ 1,νn ds s 6 ‖φ‖1,νn ‖ϕ‖1,νn ∫β α 1( [0, 1kρ ]∪[ k ρ ,+∞ [ )(s) ∥∥f ∗ φρs ∥∥ p,νn ds s . CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 23 Consequently; ∥∥(f − fk ) ∗ ϕρ ∥∥ p,νn ργ 6 ‖φ‖1,νn ‖ϕ‖1,νn ∫β α 1( [0, 1k ]∪[k,+∞ [ )(ρs) ∥∥f ∗ φρs ∥∥ p,νn ργ ds s 6 ‖φ‖1,νn ‖ϕ‖1,νn ∫ 1 α 1 β 1( [0, 1k ]∪[k,+∞ [ )( ρ t ) t−γ ∥∥f ∗ φρ t ∥∥ p,νn( ρ t )γ dt t = ‖φ‖1,νn ‖ϕ‖1,νn ( t−γ 1[ 1β, 1 α ] ⋆ ∥∥f ∗ φt ∥∥ p,νn tγ 1( [0, 1k ]∪[k,+∞ [ ) ) (ρ). Thus, by the relation (3.3), we obtain Mγ,ϕp,q (fk − f) 6 ‖φ‖1,νn ‖ϕ‖1,νn ∥∥t−γ 1[ 1 β , 1 α ] ∥∥ L1( dt t ) × [∫ 1 k 0 (∥∥f ∗ φt ∥∥ p,νn tγ )q dt t + ∫ +∞ k (∥∥f ∗ φt ∥∥ p,νn tγ )q dt t ] 1 q . So, lim k→ +∞ Mγ,ϕp,q (fk − f) = 0 because ∫ +∞ 0 (∥∥f ∗ φt ∥∥ p,νn tγ )q dt t < +∞ and the proof is complete. We denote by Lq ( ]0, +∞[ , Lp(dνn), dt t ) the space of measurable functions g on ]0, +∞[ × [0, +∞[ × Rn such that for all t > 0, the function g(t, (., .)) belongs to the space Lp(dνn) and the function t 7−→ ∥∥g(t, (., .)) ∥∥ p,νn belongs to Lq(dt t ). This space is equipped with the norm ∥∥g ∥∥ Lq ( ]0,+∞ [, Lp(dνn), dt t ) = (∫ +∞ 0 ∥∥g(t, (., .)) ∥∥q p,νn dt t ) 1 q . Then we have Lemma 17. Let p,q ∈ [1, +∞] and let γ < (2n + 1)/p. For all φ ∈ S1∗,0(R × Rn), the mapping F defined by F(g)(r,x) = ∫ +∞ 0 tγ g ( t, (., .) ) ∗ φt(r,x) dt t is continuous from Lq ( ]0, +∞[ , Lp(dνn), dt t ) into B γ p,q ( [0, +∞[ × Rn ) . 24 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) Proof. Let φ ∈ S1∗,0(R × Rn) and g ∈ Lq ( ]0, +∞[ , Lp(dνn), dt t ) . • Let a, b be real numbers such that b > a > 0 and F (φ)(µ,λ) = 0 if µ2 + 2|λ|2 < a2 or µ2 + 2|λ|2 > b2. Let ψ ∈ S∗(R × Rn) such that F (ψ)(µ,λ) = 1 if a2 6 µ2 + 2|λ|2 6 b2, then from the relation (2.9), we deduce that for every t > 0 ψt ∗ φt = φt. (3.17) For every k ∈ N∗, the function F(g)k defined by F(g)k(r,x) = ∫k 1 k tγ g(t, (., .)) ∗ φt(r,x) dt t . is bounded on R × Rn. In fact, from the relations (2.7) and (3.4), we deduce that for all (r,x) ∈ R × Rn, ∣∣F(g)k(r,x) ∣∣ 6 ∫k 1 k tγ ∥∥g(t, (., .)) ∥∥ p,νn ∥∥φt ∥∥ p ′ ,νn dt t 6 ∥∥φ ∥∥ p ′ ,νn ∫k 1 k t γ − 2n+1 p ∥∥g ( t, (., .) )∥∥ p,νn dt t 6 ∥∥φ ∥∥ p ′ ,νn ∥∥g ∥∥ Lq ( ]0,+∞ [, Lp(dνn), dt t ) [∫k 1 k t (γ − 2n+1 p ) q ′ dt t ] 1 q ′ < +∞. Thus, for all k ∈ N∗ the function F(g)k defines a tempered distribution on R × Rn, even with respect to the first variable. Moreover, for all h ∈ S∗(R × Rn), we have 〈F(g)k,h〉 = ∫k 1 k tγ [∫ +∞ 0 ∫ Rn h(r,x) g(t, (., .)) ∗ φt(r,x) dνn(r,x) ] dt t = ∫k 1 k tγ 〈g(t, (., .)) ∗ φt,h〉 dt t , and by the relation (3.17), it follows that 〈F(g)k,h〉 = ∫k 1 k tγ 〈g(t, (., .)) ∗ φt ∗ ψt,h〉 dt t = ∫k 1 k tγ 〈g(t, (., .)) ∗ φt,h ∗ ψ̌t〉 dt t . (3.18) CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 25 However, ∫ +∞ 0 tγ ∣∣〈g(t, (., .)) ∗ φt,h ∗ ψ̌t〉 ∣∣ dt t 6 ∫ +∞ 0 tγ ∥∥g(t, (., .)) ∗ φt ∥∥ ∞,νn ∥∥h ∗ ψ̌t ∥∥ 1,νn dt t 6 ∫ +∞ 0 tγ ∥∥g(t, (., .)) ∥∥ p,νn ∥∥φt ∥∥ p ′ ,νn ∥∥h ∗ ψ̌t ∥∥ 1,νn dt t = ∥∥φ ∥∥ p ′ ,νn ∫ +∞ 0 t γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn ∥∥h ∗ ψ̌t ∥∥ 1,νn dt t = ∥∥φ ∥∥ p ′ ,νn { ∫1 0 t γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn ∥∥h ∗ ψ̌t ∥∥ 1,νn dt t + ∫ +∞ 1 t γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn ∥∥h ∗ ψ̌t ∥∥ 1,νn dt t } Applying the relations (3.7) and (3.8), we get ∫ +∞ 0 tγ ∣∣〈g(t, (., .)) ∗ φt,h ∗ ψ̌t〉 ∣∣ dt t 6 ∥∥φ ∥∥ p ′ ,νn ‖∆kh‖1,νn ‖ψ̌k‖1,νn ∫1 0 t 2k+γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn dt t + ∥∥φ ∥∥ p ′ ,νn ‖h‖1,νn ‖ψ̌‖1,νn ∫ +∞ 1 t γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn dt t ; and by Hölder’s inequality, we have ∫ +∞ 0 tγ ∣∣〈g(t, (., .)) ∗ φt,h ∗ ψ̌t〉 ∣∣ dt t 6 ∥∥φ ∥∥ p ′ ,νn ∥∥g ∥∥ Lq ( ]0,+∞ [, Lp(dνn), dt t ) { ‖∆kh‖1,νn ‖ψ̌k‖1,νn (∫1 0 t (2k+γ − 2n+1 p ) q ′ dt t ) 1 q ′ + ‖h‖1,νn ‖ψ̌‖1,νn (∫ +∞ 1 t (γ − 2n+1 p ) q ′ dt t ) 1 q ′ } < +∞. The last inequality together with the relation (3.18) show that for all h ∈ S∗(R × Rn), lim k→ +∞ 〈F(g)k,h〉 exists and lim k→ +∞ 〈F(g)k,h〉 = ∫ +∞ 0 tγ 〈g(t, (., .)) ∗ φt,h〉 dt t . Consequently, the function F(g)(r,x) = ∫ +∞ 0 tγ g ( t, (., .) ) ∗ φt(r,x) dt t defines an element of S ′ ∗(R × Rn). 26 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) • Let ϕ ∈ S1∗,0(R × Rn), we have F(g) ∗ ϕρ(r,x) = 〈F(g),τ(r,−x)ϕ̌ρ〉 = lim k→ +∞ 〈F(g)k,τ(r,−x)ϕ̌ρ〉 = lim k→ +∞ F(g)k ∗ ϕρ(r,x) = lim k→ +∞ ∫k 1 k tγ g(t, (., .)) ∗ φt ∗ ϕρ(r,x) dt t . However, the relation (3.13) implies ∫ +∞ 0 tγ ∣∣g(t, (., .)) ∗ φt ∗ ϕρ(r,x) ∣∣ dt t ∫ρβ ρα tγ ∣∣g(t, (., .)) ∗ φt ∗ ϕρ(r,x) ∣∣ dt t 6 ∫ρβ ρα tγ ∥∥g(t, (., .)) ∥∥ p,νn ∥∥φt ∗ ϕρ ∥∥ p ′ ,νn dt t 6 ‖φ‖p′,νn‖ϕ‖1,νn ∫ρβ ρα t γ − 2n+1 p ∥∥g(t, (., .)) ∥∥ p,νn dt t , 6 ‖φ‖p′,νn ‖ϕ‖1,νn (∫ρβ ρα t (γ − 2n+1 p ) q ′ dt t ) 1 q ′ ∥∥g ∥∥ Lq ( ]0,+∞ [, Lp(dνn), dt t ) < +∞. Thus, F(g) ∗ ϕρ(r,x) = ∫ +∞ 0 tγ g(t, (., .)) ∗ φt ∗ ϕρ(r,x) dt t = ∫β α (ρs)γ g(ρs, (., .)) ∗ φρs ∗ ϕρ(r,x) ds s . (3.19) By Minkowski’s inequality, we obtain ∥∥F(g) ∗ ϕρ ∥∥ p,νn 6 ∫β α (ρs)γ ∥∥g(ρs, (., .)) ∗ φρs ∗ ϕρ ∥∥ p,νn ds s 6 ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ∫β α (ρs)γ ∥∥g(ρs, (., .)) ∥∥ p,νn ds s < +∞ and ∥∥F(g) ∗ ϕρ ∥∥ p,νn ργ 6 ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ∫β α sγ ∥∥g(ρs, (., .)) ∥∥ p,νn ds s = ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ∫ 1 α 1 β t−γ ∥∥g( ρ t , (., .)) ∥∥ p,νn dt t = ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ( t−γ1[ 1β, 1 α ] ⋆ ∥∥g(t, (., .)) ∥∥ p,νn ) (ρ), CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 27 and by the relation (3.3) it follows that ∥∥∥ ∥∥F(g) ∗ ϕρ ∥∥ p,νn ργ ∥∥∥ Lq( dρ ρ ) 6 ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ∥∥∥t−γ1[ 1β, 1α ] ∥∥∥ L1( dt t ) ∥∥g ∥∥ Lq ( ]0,+∞ [,Lp(dνn), dt t ) < +∞. (3.20) • Let ϕ ∈ S1∗,0(R × Rn), from the relation (3.19), we have F(g) ∗ ϕρ(r,x) = ∫β α (ρs)γ g(ρs, (., .)) ∗ φρs ∗ ϕρ(r,x) ds s , and by Fubini’s theorem, we get F(g) ∗ ϕρ ∗ ϕρ(r,x) = ∫β α (ρs)γ g(ρs, (., .)) ∗ φρs ∗ ϕρ ∗ ϕρ(r,x) ds s = ∫ρβ ρα tγ g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dt t . Thus, ∫k 1 k F(g) ∗ ϕρ ∗ ϕρ(r,x) dρ ρ ∫k 1 k [∫ρβ ρα tγ g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dt t ] dρ ρ = ∫βk α k tγ [∫ t α t β g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dρ ρ ] dt t . (3.21) However, ∫ t α t β g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dρ ρ = ∫ t α t β [∫ +∞ 0 ∫ Rn τ(r,−x)ǧ(t, (., .))(s,y) φt ∗ ϕρ ∗ ϕρ(s,y) dνn(s,y) ] dρ ρ . Again, by Fubini’s theorem, we have ∫ t α t β g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dρ ρ = ∫ +∞ 0 ∫ Rn τ(r,−x)ǧ(t, (., .))(s,y) [∫ +∞ 0 φt ∗ ϕρ ∗ ϕρ(s,y) dρ ρ ] dνn(s,y). applying 2) of Lemma 13, we obtain ∫ t α t β g(t, (., .)) ∗ φt ∗ ϕρ ∗ ϕρ(r,x) dρ ρ = ∫ +∞ 0 ∫ Rn τ(r,−x)ǧ(t, (., .))(s,y) φt(s,y) dνn(s,y) = g(t, (., .)) ∗ φt(r,x). 28 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) Replacing in the equality (3.21), it follows that ∫k 1 k F(g) ∗ ϕρ ∗ ϕρ(r,x) dρ ρ = ∫βk α k tγ g(t, (., .)) ∗ φt(r,x) dt t . Hence, ∫ +∞ 0 F(g) ∗ ϕρ ∗ ϕρ dρ ρ = F(g). This shows that the function F(g) belongs to the space B γ p,q ( [0, +∞[×Rn ) and from the inequality (3.20), we have Mγ,ϕp,q ( F(g) ) 6 ∥∥φ ∥∥ 1,νn ∥∥ϕ ∥∥ 1,νn ∥∥∥t−γ1[ 1β, 1α ] ∥∥∥ L1( dt t ) ∥∥g ∥∥ Lq ( ]0,+∞ [, Lp(dνn), dt t ) which means that the mapping F is continuous from Lq ( ]0, +∞[ , Lp(dνn), dt t ) into B γ p,q ( [0, +∞[ × Rn ) . Theorem 18. Let p,q ∈ [1, +∞] and let γ ∈ R, γ < (2n+1)/p. Then the Besov space Bγp,q ( [0, +∞[× R n ) is a Banach one. Proof. Let φ ∈ S1∗,0(R × Rn). We define the mapping G on the space B γ p,q ( [0, +∞[ × Rn ) by setting G(f)(t, (r,x)) = f ∗ φt(r,x) tγ . The mapping G is continuous from B γ p,q ( [0, +∞[ × Rn ) into Lq ( ]0, +∞[ , Lp(dνn), dt t ) and we have ∥∥G(f) ∥∥ Lq ( ]0,+∞ [, Lp(dνn), dt t ) = Mγ,φp,q (f). (3.22) Moreover, for all f ∈ Bγp,q ( [0, +∞[ × Rn ) , we have F ◦ G(f)(r,x) = ∫ +∞ 0 tγ G(f)(t, (., .)) ∗ φt(r,x) dt t = ∫ +∞ 0 tγ f ∗ φt ∗ φt(r,x) tγ dt t = ∫ +∞ 0 f ∗ φt ∗ φt(r,x) dt t and by ii) of Lemma 13, we get F ◦ G(f) = f. This equality shows that G ( B γ p,q ( [0, +∞[ × Rn )) ker ( G ◦ F − Id (Lq ( ]0,+∞ [, Lp(dνn), dt t ) ) ) . CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 29 In particular, G ( B γ p,q ( [0, +∞[ × Rn )) is a closed subspace of Lq ( ]0, +∞[ , Lp(dνn), dt t ) . Let (fk)k∈N be a Cauchy sequence in B γ p,q ( [0, +∞[ × Rn ) . From the relation (3.22), the sequence ( G(fk) ) k is a Cauchy’s one in Lq ( ]0, +∞[ , Lp(dνn), dt t ) . Since G ( B γ p,q ( [0, +∞[×Rn )) is a closed subspace of Lq ( ]0, +∞[ , Lp(dνn), dt t ) , then there exists a function f in B γ p,q ( [0, +∞[ × Rn ) such that lim k→ +∞ G(fk) = G(f) in L q ( ]0, +∞[ , Lp(dνn), dt t ) . Again by the relation (3.22), lim k→ +∞ fk = f in B γ p,q ( [0, +∞[ × Rn ) . Proposition 19. i) Let q ∈ [1, +∞] , p1, p2 ∈ [1, +∞] ; p1 < p2 and let γ1, γ2 ∈ R such that 2n + 1 p1 − γ1 = 2n + 1 p2 − γ2. (3.23) Then B γ1 p1,q ( [0, +∞[ × Rn ) →֒ Bγ2p2,q ( [0, +∞[ × Rn ) . ii) For all p ∈ [1, +∞] , B 0 p,1 ( [0, +∞[ × Rn ) →֒ Lp(dνn). Proof. i) Let p1, p2, γ1, γ2, q be real numbers satisfying the hypothesis. Let p3 be an exponent such that 1 p1 + 1 p3 = 1 + 1 p2 . (3.24) Finally, let f ∈ Bγ1p1,q ( [0, +∞[ × Rn ) and φ ∈ S1∗,0 ( R × Rn ) such that F (φ)(µ,λ) = 0 if µ2 + 2|λ|2 > b2 or µ2 + 2|λ|2 < a2. Let us take ψ ∈ S∗ ( R × Rn ) satisfying ∀(µ,λ) ∈ Γ ; a2 6 µ2 + 2|λ|2 6 b2, F (ψ)(µ,λ) = 1. Then for all t > 0, we have φt ∗ ψt = φt and Mγ2,φp2,q (f) = (∫ +∞ 0 (‖f ∗ φt‖p2,νn tγ2 )qdt t ) 1 q = (∫ +∞ 0 (‖f ∗ φt ∗ ψt‖p2,νn tγ2 )qdt t ) 1 q . 30 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) By the relations (2.7), (3.4), (3.23) and (3.24) we get Mγ2,φp2,q (f) 6 ‖ψ‖p3,νn [∫ +∞ 0 (‖f ∗ φt‖p1,νn tγ1 )qdt t ] 1 q 6 ‖ψ‖p3,νn Mγ1,φp1,q (f). This shows that the space B γ1 p1,q ( [0, +∞[×Rn ) is contained in B γ2 p2,q ( [0, +∞[×Rn ) and that the canonical injection is continuous from B γ1 p1,q ( [0, +∞[ × Rn ) into the space B γ2 p2,q ( [0, +∞[ × Rn ) . ii) Let f ∈ B0p,1 ( [0, +∞[ × Rn ) ; p ∈ [1, +∞] . From ii) of Lemma 13, we have f = ∫ +∞ 0 f ∗ φt ∗ φt dt t ; φ ∈ S1∗,0 ( R × Rn ) thus, ‖f‖p,νn 6 ∫ +∞ 0 ∥∥f ∗ φt ∗ φt ∥∥ p,νn dt t 6 ‖φ‖1,νn M 0,φ p,1 (f). This completes the proof. In the following, we shall define a discrete norm on the Besov space B γ p,q ( [0, +∞[ × Rn ) and we will prove that it is equivalent to the norm M γ,φ p,q ; φ ∈ S1∗,0 ( R × Rn ) . More precisely, we have Theorem 20. Let p, q ∈ [1, +∞], γ ∈ R. Let a, b be real numbers such that 0 < a < b and φ ∈ S∗,0 ( R × Rn ) verifying F (φ)(µ,λ) = 1 if a2 6 µ2 + 2|λ|2 6 b2. Then the mapping N γ,φ p,q defined by Nγ,φp,q (f) =    ( ∑ k∈Z (‖f ∗ φ2k‖p,νn 2kγ )q ) 1 q , if 1 6 q < +∞; esssup k∈Z ‖f ∗ φ2k‖p,νn 2kγ , if q = +∞ is a norm on the Besov space B γ p,q ( [0, +∞[ × Rn ) which defines the same topology as the norm M γ,ψ p,q ; ψ ∈ S1∗,0 ( R × Rn ) . Proof. • From Lemma 9, there exists ψ ∈ S1∗,0 ( R × Rn ) such that F (ψ)(µ,λ) = 0 if µ2 + 2|λ|2 < a2 or µ2 + 2|λ|2 > b2. CUBO 13, 2 (2011) Homogeneous Besov Spaces associated with the spherical . . . 31 Then for all s ∈ [1,2] and k ∈ Z, we have F (ψ)(2ksµ,2ksλ) = F (ψ)(2ksµ,2ksλ) F (φ)(2kµ,2kλ) which leads to ψ2ks = ψ2ks ∗ φ2k and therefore, for all f ∈ Bγp,q ( [0, +∞[ × Rn ) f ∗ ψ2ks = f ∗ φ2k ∗ ψ2ks. (3.25) Then for all q ∈ [1, +∞[ Mγ,ψp,q (f) = (∫ +∞ 0 (‖f ∗ ψt‖p,νn tγ )q dt t ) 1 q = ( ∑ k∈Z ∫2k+1 2k (‖f ∗ ψt‖p,νn tγ )q dt t ) 1 q = ( ∑ k∈Z ∫2 1 (‖f ∗ ψ2ks‖p,νn (2ks)γ )q ds s ) 1 q . Using the relations (2.7), (3.4) and (3.25), we obtain Mγ,ψp,q (f) 6 ‖ψ‖1,νn [ ∑ k∈Z (‖f ∗ φ2k‖p,νn 2kγ )q ∫2 1 ds sγq+1 ] 1 q = ‖ψ‖1,νn (1 − 2−qγ qγ ) 1 q Nγ,φp,q (f). On the other hand, for q = +∞ and again by the relation (3.25), we deduce that for all k ∈ Z and s ∈ [1,2] ‖f ∗ ψ2ks‖p,νn (2ks)γ 6 (1 + 2−γ) ‖ψ‖1,νn ‖f ∗ φ2k‖p,νn 2kγ . Consequently, for all k ∈ Z and t ∈ [2k,2k+1] ‖f ∗ ψt‖p,νn tγ 6 (1 + 2−γ) ‖ψ‖1,νn Nγ,φp,∞ (f), which shows that Mγ,ψp,∞ (f) 6 (1 + 2 −γ) ‖ψ‖1,νn Nγ,φp,∞ (f). • Let a1, b1 be two real numbers; 0 < a1 < a < b < b1 such that F (φ)(µ,λ) = 0 if µ2 + 2|λ|2 < a21 or µ 2 + 2|λ|2 > b21. From Lemma 9, there exists ψ ∈ S1∗,0 ( R × Rn ) such that F (ψ)(µ,λ) = C, for all (µ,λ) ∈ Γ ; a21 6 µ2 + 2|λ|2 6 4b21 32 L.T.Rachdi and A.Rouz CUBO 13, 2 (2011) where C is a positive constant. Then for all k ∈ Z and s ∈ [1,2], CF (φ)(2kµ,2kλ) = F (φ)(2kµ,2kλ) F (ψ)(2kµs,2kλs) so, C.φ2k = φ2k ∗ ψ2ks. Hence, for all f ∈ Bγp,q ( [0, +∞[×Rn ) C f ∗ φ2k = f ∗ ψ2ks ∗ φ2k (3.26) and C ‖f ∗ φ2k‖p,νn 2kγ 6 (1 + 2γ) ‖φ‖1,νn ‖f ∗ ψ2ks‖p,νn (2ks)γ . Integrating over [1,2] with respect to the measure ds s , we get for all q ∈ [1, +∞[, (‖f ∗ φ2k‖p,νn 2kγ )q 6 ( (1 + 2γ) ‖φ‖1,νn )q Cq log 2 ∫2k+1 2k (‖f ∗ ψt‖p,νn tγ )qdt t which leads to Nγ,φp,q (f) 6 1 C (log 2)− 1 q (1 + 2γ) ‖φ‖1,νn Mγ,ψp,q (f). On the other hand, for q = +∞ and using the relation (3.26), we deduce that for all k ∈ Z ‖f ∗ φ2k‖p,νn 2kγ 6 (1 + 2γ) C ‖φ‖1,νn Mγ,ψp,∞ (f), which implies that Nγ,φp,∞ (f) 6 (1 + 2γ) C ‖φ‖1,νn Mγ,ψp,∞ (f). This completes the proof of theorem. Remark 21. 1) From Theorem 14 and Theorem 20, we deduce that the Besov space B γ p,q ( [0, +∞[× R n ) is independent of the choice of the function φ ∈ S∗,0 ( R × Rn ) , when it is endowed with the norm N γ,φ p,q . From Proposition 15 and Theorem 20, we deduce the following proposition Proposition 22. The Besov space B γ p,q ( [0, +∞[ × Rn ) is homogeneous in a weaker sense when equipped with the norm N γ,φ p,q , that is there exist C1, C2 > 0 such that for all f ∈ Bγp,q ( [0, +∞[ × R n ) and t > 0 C1 t 2n+1 p −2n−1−γ Nγ,φp,q (f) 6 N γ,φ p,q (ft) 6 C2 t 2n+1 p −2n−1−γ Nγ,φp,q (f). Proposition 23. Let p ∈ [1, +∞] and γ ∈ R. 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