Sobolev Type Spaces Asociated with the Weinstein Operator 1 ... - Hikari

Int. Journal of Math. Analysis, Vol. 5, 2011, no. 28, 1353 - 1373

Sobolev Type Spaces Asociated with the Weinstein Operator Hassen Ben Mohamed Department of Mathematics, Faculty of Sciences of Gabes, Tunisia [email protected] N´ eji Bettaibi Qassim University, P.O. Box 6640 Buraydah 51452, Saudi Arabia [email protected] Sidi Hamidou Jah Qassim University, P.O. Box 6640 Buraydah 51452, Saudi Arabia [email protected] Abstract In this paper, we introduce and study the Sobolev spaces associated with the Weinstein operator and investigate their properties. Next, we introduce a class of symbols and their associated pseudo-differential operators. Moreover, a generalized Weinstein potential is defined and its properties are investigated.

Mathematics Subject Classification: 42B10, 42B30, 42B35 Keywords: Weinstein operator, Sobolev spaces, pseudo-differential operators.

1

Introduction

The Sobolev spaces have served as a very useful tool in the theory of partial differential equations, mostly those related to continuum mechanics or physics. Their uses and the study of their properties were facilitated by the theory of distributions and Fourier analysis. The Sobolev space H s (Rn ) is defined by the use of the classical Fourier transform as the set of all tempered distribution

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H. Ben Mohamed, N. Bettaibi and S. H. Jah

u such that its classical Fourier transform u  is a function satisfying:  (1 + ξ2)s | u (ξ)| dξ < ∞. n Ê

(1.1)

Generalization of the Sobolev space have been studied by replacing in (1.1), the classical Fourier transform by a generalized one (see [1, 6]) or by using a more general tempered weight function instead of the kernel (1 + ξ2 )s (see [9, 10, 12]). d+1 In this paper, we consider the Weinstein operator Δα,d = W defined on R+ Rd × ]0, +∞[ , by : Δα,d W

d+1  2α + 1 ∂ 1 ∂2 = + = Δ + L , α > − , d α ∂x2i xd+1 ∂xd+1 2 i=1

(1.2)

where Δd is the Laplacian for the d first variables and Lα is the Bessel operator for the last variable defined on ]0, +∞[ by :   ∂ ∂2u 2α + 1 ∂u 1 2α+1 ∂u = 2α+1 x . Lα u = 2 + ∂xd+1 xd+1 ∂xd+1 xd+1 ∂xd+1 d+1 ∂xd+1 The main objective of this paper is to generalize the important subject of H s (Rn ) to the case of the harmonic analysis associated with the Weinstein operator Δα,d W . In fact, we replace in (1.1) the classical Fourier transform α,d : for s ∈ R, we define the Sobolevby the Weinstein-Fourier transform FW s,α d+1 Weinstein space HS∗ (R+ ) as the set of all u ∈ S∗ ( the strong dual of the α,d space S∗ (Rd+1 )) such that FW (u) is a function and  uH s,α = S∗

Êd+1 +

 2  α,d  (1 + ξ2)s FW (u) (ξ) dμα,d (ξ)

12 < ∞,

by : where μα,d is the measure defined on Rd+1 + dμα,d (x) =

x2α+1 d+1 d

dx.

(1.3)

(2π) 2 2α Γ(α + 1)

and dx is the Lebesgue measure on Rd+1 . Next, we investigate the properties of HSs,α (Rd+1 + ). Moreover, we introduce a ∗ class of symbols and their associated pseudo-differential operators. Finally, we define the generalized Weinstein potential and we study its propperties. This paper is organized as follows: in Section 2, we recall some elements of harmonic analysis associated with the Weinstein operator, which will bee needed in the sequel. In Section 3, we define and study the Sobolev-Weinstein

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space HSs,α (Rd+1 + ). Section 4 is devoted to introduce certain classes of symbols ∗ and study their associated pseudo-differential operators. For instance, we give an integral representation for pseudo-differential operators and we show that, in certain cases, they map continuously two Weinstein-Sobolev spaces. Finally, in the last section, we define the generalized Weinstein potential and we study its propperties.

2

Harmonic analysis associated with the Weinstein operator

In this section, we collect some harmonic analysis results related to the Weinstein operator Δα,d W given by (1.2). All these results can be founded in [3]. Notations. In what follows, we need the following notations : • C∗ (Rd+1 ), the space of continuous functions on Rd+1 , even with respect to the last variable. • C∗,c (Rd+1 ), the space of continuous functions on Rd+1 with compact support, even with respect to the last variable. • C∗p (Rd+1 ), the space of functions of class C p on Rd+1 , even with respect to the last variable. • E∗ (Rd+1 ), the space of C ∞ -functions on Rd+1 , even with respect to the last variable. • S∗ (Rd+1 ), the Schwartz space of rapidly decreasing functions on Rd+1 , even with respect to the last variable. 

h,k This space is equipped with the topology defined by the seminorms N1 h,k>0

given by :  N1h,k (ϕ)

=

sup x ∈ Rd+1 μ ∈ Nd+1

 (1 + x2 )k |∂ μ ϕ(x)| < ∞, h|μ| μ!

h, k > 0.

• D∗ (Rd+1 ), the space of C ∞ -functions on Rd+1 which are of compact support, even with respect to the last variable. • Lpα (Rd+1 + ), 1 ≤ p ≤ +∞, the Lebesgue space constituted of measurable such that functions on Rd+1 +  f α,p

=

Ê

1p d+1 +

|f (x)|p dμα,d (x)

f α,∞ = ess sup |f (x)| < +∞, x∈Êd+1 +

< +∞, if 1 ≤ p < +∞,

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where μα,d is the measure given by the relation (1.3). • H∗ (Cd+1 ), the space of entire functions on Cd+1 , even with respect to the last variable, rapidly decreasing and of exponential type. • S∗ the strong dual of the space S∗ (Rd+1 ). Let us begin by the following result, which gives the eigenfunctions of the Weinstein operator Δα,d W . Proposition 1 For all λ = (λ1 , λ2 , ..., λd+1 ) ∈ Cd+1 , the system ⎧ ∂2u ⎪ ⎪ (x) = −λ2j u(x), if 1 ≤ j ≤ d ⎪ ⎪ ⎨ ∂x2j = −λ2d+1 u (x) , Lα u (x) ⎪ ⎪ ∂u ∂u ⎪ ⎪ (0) = 0 and (0) = −iλj , if 1 ≤ j ≤ d ⎩ u (0) = 1, ∂xd+1 ∂xj has a unique solution Ψα,d (λ, .) given by 



∀z∈ Cd+1 , Ψα,d (λ, z) = e−iz ,λ  jα (λd+1 zd+1 ),

(2.1)

where z = (z  , zd+1 ), z  = (z1 , z2 , ..., zd ) and jα is the normalized Bessel function of index α, defined by :

z 2n (−1)n ∀z∈ C, jα (z) = Γ(α + 1) . n!Γ(n + α + 1) 2 n=0 ∞ 

Using the properties of the normalized Bessel function and of the exponential function, it is easy to see that the Weinstein kernel Ψα,d satisfies the following properties. Proposition 2 i) For all λ, z ∈ Cd+1 and t ∈ R, we have Ψα,d (λ, 0) = 1, Ψα,d (λ, z) = Ψα,d (z, λ) and Ψα,d (λ, tz) = Ψα,d (tλ, z) . and z ∈ Cd+1 , we have ii) For all ν ∈ Nd+1 , x ∈ Rd+1 + |Dzν Ψα,d (x, z)| ≤ x|ν| exp(x  Im (z) ), where Dzν = In particular

(2.2)

∂ν νd+1 and |ν| = ν1 + ... + νd+1 . ∂z1ν1 ...∂zd+1 ∀x, y ∈ Rd+1 + , |Ψα,d (x, y)| ≤ 1.

(2.3)

The following useful result characterizes the Shwartz’s functions using the Weinstein operator.

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Lemma 1 Let ϕ ∈ E∗ (Rd+1 ). Then ϕ belongs to S∗ (Rd+1 ) if and only if, m

⎡ ⎤ ϕ(x)| (1 + x2 )k | Δα,d W ⎣ ⎦ < +∞. sup ∀h, k > 0, N2h,k (ϕ) = hm m! d+1 x∈R m∈N Moreover the system of seminorms {N2h,k }h, k>0 generates the topology of S∗ (Rd+1 ). Proof. The result can be obtained by a simple calculation by using the same technique as in Proposition 3.1 of [2].  Definition 1 The Fourier-Weinstein transform is given for f ∈ L1α (Rd+1 + ) by  α,d d+1 ∀λ ∈ R+ , FW (f )(λ) = f (x)Ψα,d (x, λ)dμα,d (x), (2.4) Êd+1 + where μα,d is the measure on Rd+1 given by the relation (1.3). + α,d are summarized in the following Some basic properties of the transform FW results. For the proofs, we refer to [3, 4, 5].

Proposition 3 (see [3, 4]) α,d ∞ d+1 i) For f ∈ L1α (Rd+1 + ), we have FW (f ) ∈ f ∈ Lα (R+ ) and α,d FW (f )α,∞ ≤ f α,1.

(2.5)

ii) For m ∈ N and f ∈ S∗ (Rd+1 ), we have 

m  α,d α,d α,d ∀y ∈ Rd+1 , F f (y) = (−1)m y2mFW (f )(y).  + W W

(2.6)

iii) For f in S∗ (Rd+1 ) and m ∈ N, we have

 m  α,d α,d α,d ∀λ ∈ Rd+1 ,  (f ) (λ) = FW (Pm f )(λ), F + W W

(2.7)

where Pm (λ) = (−1)m λ2m . Theorem 1 (see [3, 4]) α,d is a topological isomorphism from i) The Fourier-Weinstein transform FW d+1 d+1 S∗ (R ) onto itself and from D∗ (R ) onto H∗ (Cd+1 ). α,d 1 d+1 ii) Let f ∈ L1α (Rd+1 + ) such that FW (f ) ∈ Lα (R+ ). Then  α,d f (x) = FW (f ) (y) Ψα,d (−x, y)dμα,d(y), a.e x ∈ Rd+1 (2.8) + . d+1 Ê+

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As a consequence, for f ∈ S∗ (Rd+1 ), we have −1

α,d α,d , F (f )(x) = FW (f ) (−x) . ∀x ∈ Rd+1 + W Theorem 2 (see [3, 4]) i) For all f, g ∈ S∗ (Rd+1 ), we have the following Parseval formula   α,d α,d f (x)g(x)dμα,d (x) = FW (f )(λ)FW (g)(λ)dμα,d(λ). d+1 d+1 Ê+ Ê+ ii) ( Plancherel formula ). For all f ∈ S∗ (Rd+1 ), we have :    2  α,d  2 |f (x)| dμα,d (x) = FW (f )(λ) dμα,d (λ). d+1 d+1 Ê+ Ê+

(2.9)

(2.10)

(2.11)

α,d iii) ( Plancherel Theorem ) : The transform FW extends uniquely to an isometric isomorphism on L2α (Rd+1 ). +

Definition 2 The Fourier-Weinstein transform of a distribution S ∈ S∗ is defined by : α,d α,d ∀φ ∈ S∗ (Rd+1 ), FW (S), φ = S, FW (φ) .

As an immediate consequence of the two previous theorems, we have: α,d Proposition 4 The transform FW is a topological isomorphism from S∗ onto itself.

Definition 3 Let τ be in S∗ . We define Δα,d W τ , by : α,d ∀φ ∈ S∗ (Rd+1 ), Δα,d W τ, φ = τ, ΔW φ .

Remark 1 Let m ∈ N and τ ∈ S∗ , we have   α,d α,d m α,d , F τ ) (τ ) (y) . (Δ (y) = (−1)m y2m FW ∀y ∈ Rd+1 + W W

(2.12)

(2.13)

Definition 4 The translation operator Tx , x ∈ Rd+1 + , associated with the α,d d+1 Weinstein operator ΔW is defined on C∗ (R ), for all y ∈ Rd+1 + , by :    aα π   2 2 f x + y , xd+1 + yd+1 + 2xd+1 yd+1 cos θ (sin θ)2α dθ, Tx f (y) = 2 0 where x + y  = (x1 + y1 , ..., xd + yd ) and 2Γ (α + 1) . aα = √  πΓ α + 12

(2.14)

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Weinstein Sobolev spaces

The following proposition summarizes some properties of the Weinstein translation operator. Proposition 5 (see [3, 4]) i) For f ∈ C∗ (Rd+1 ), we have ∀x, y ∈ Rd+1 + , Tx f (y) = Ty f (x) and T0 f = f. ii) For all f ∈ E∗ (Rd+1 ) and y ∈ Rd+1 + , the function x → Tx f (y) belongs to E∗ (Rd+1 ). iii) We have α,d α,d ∀x ∈ Rd+1 + , ΔW ◦ Tx = Tx ◦ ΔW . d+1 iv) Let f ∈ Lpα (Rd+1 + ), 1 ≤ p ≤ +∞ and x ∈ R+ . Then Tx f belongs to p d+1 Lα (R+ ) and we have

Tx f α,p ≤ f α,p . v) The function Ψα,d (., λ) , λ ∈ Cd+1 , satisfies the following product formula: ∀y ∈ Rd+1 + , Ψα,d (x, λ) Ψα,d (y, λ) = Tx [Ψα,d (., λ)] (y) .

(2.15)

d+1 vi) Let f ∈ Lpα (Rd+1 + ), p = 1 or 2 and x ∈ R+ , we have α,d α,d ∀y ∈ Rd+1 + , FW (Tx f ) (y) = Ψα,d (x, y) FW (f ) (y) .

(2.16)

vii) The space S∗ (Rd+1 ) is invariant under the operators Tx , x ∈ Rd+1 + . Definition 5 The Weinstein convolution product of f, g ∈ C∗ (Rd+1 ) is given by :  d+1 ∀x ∈ R+ , f ∗W g (x) = Tx f (y) g (y) dμα,d (y). (2.17) Êd+1 + Proposition 6 (see [3, 4]) 1 1 1 + − = 1. p q r q d+1 Then for all f ∈ Lpα (Rd+1 + ) and g ∈ Lα (R+ ), the function f ∗W g belongs to r d+1 Lα (R+ ) and we have : i) Let p, q, r ∈ [1, +∞] such that

f ∗W gα,r ≤ f α,p gα,q . (2.18)   resp. S∗ (Rd+1 ) , f ∗W g belongs to L1α (Rd+1 ii) For all f, g ∈ L1α (Rd+1 + ), + )   d+1 resp. S∗ (R ) and we have α,d α,d α,d FW (f ∗W g) = FW (f )FW (g).

(2.19)

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H. Ben Mohamed, N. Bettaibi and S. H. Jah

Sobolev spaces associated with the Weinstein operator

The goal of this section is to introduce and study the Sobolev spaces associated with the Weinstein operator Δα,d W . Definition 6 For s ∈ R, we define the Sobolev-Weinstein space of order s, α,d  that will be denoted HSs,α (Rd+1 + ), as the set of all u ∈ S∗ such that FW (u) is a ∗ function and  2   2 s  α,d (1 + λ ) FW (u) (λ) dμα,d (λ) < ∞. (3.1) d+1 Ê+ (Rd+1 We provide HSs,α + ) with the scalar product ∗  α,d α,d (1 + ξ2)s FW (u)(ξ)FW (v)(ξ)dμα,d (ξ) u, v s,α = d+1 Ê+

(3.2)

and the norm  uH s,α = S∗

Êd+1 +

12  2   α,d (1 + ξ2)s FW (u) (ξ) dμα,d (ξ) .

(3.3)

Proposition 7 i) For all s ∈ R, we have (Rd+1 S∗ (Rd+1 ) ⊂ HSs,α + ). ∗ ii) We have 2 d+1 (Rd+1 HS0,α + ) = Lα (R+ ). ∗

iii) For all s, t ∈ R, t > s, the space HSt,α (Rd+1 + ) is continuously contained in ∗ s,α d+1 HS∗ (R+ ). iv) Let P be a linear partial differential operator with constant coefficients, t,α d+1 (Rd+1 s ∈ R, u ∈ HSs,α + ) and t < s. Then P (u) ∈ HS∗ (R+ ) and the map ∗ v → P (v) is continuous on HSs,α (Rd+1 + ). ∗ Proof. i) and ii) follow from the definition of the Sobolev-Weinstein spaces. iii) Let s, t ∈ R such that t > s and u ∈ HSt,α (Rd+1 + ). Then, since s < t, we ∗ have ∀λ ∈ Rd+1 + ,

(1 + λ2 )s = (1 + λ2 )s−t (1 + λ2)t ≤ (1 + λ2 )t .

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Weinstein Sobolev spaces

So,



2    α,d (1 + λ2)s FW (u) (λ) dμα,d (λ)

Êd+1 +   2  α,d  (1 + λ2 )t FW (u) (λ) dμα,d (λ) < ∞. ≤ Êd+1 +

(Rd+1 Then u ∈ HSs,α + ) and uH s,α ≤ u ∗ S∗

H

t,α S∗

. 

iv) The result can be proved in the same spirit of iii).

s,α (Rd+1 Proposition 8 The space HSs,α + ) provided with the norm .HS∗ is a Ba∗ nach space.

Proof. Let (fn )n∈Æ be a Cauchy sequence of HSs,α (Rd+1 + ). From the definition of the ∗ α,d norm .H s,α , it is easy to see that FW (fn ) is a Cauchy sequence of S∗ n∈Æ 2 s L2 (Rd+1 + , (1 + x ) dμα,d (x)). 2 s Since L2 (Rd+1 + , (1 + x ) dμα,d (x)) is complete, there exists a function f ∈ 2 d+1 2 s L (R+ , (1 + x ) dμα,d (x)) such that α,d (fn ) − f L2 (Êd+1 , (1+ x 2 )s dμα,d (x)) = 0. lim FW

(3.4)

+

n→+∞

Then f ∈ S∗ and from Proposition 4, we obtain

−1 α,d h = FW (f ) ∈ S∗ . α,d 2 s So, FW (h) = f ∈ L2 (Rd+1 + , (1 + x ) dμα,d (x)), which proves that h ∈ (Rd+1 HSs,α + ). Furthermore, using the relation (2.11) and (3.4), we obtain : ∗ α,d fn − hHSs,α = FW (fn ) − f L2 (Êd+1 , (1+ x 2 )s dμα,d (x)) +

∗

→ 0.

n→+∞

(Rd+1 Hence, HSs,α + ) is complete. ∗



Proposition 9 Let s, t ∈ R. Then, the operator s−t,α (Rd+1 (Rd+1 ∇t : HSs,α + ) → HS ∗ + ) ∗

by : defined for all x ∈ Rd+1 +  α,d (1 + ξ)tΨ(−x, ξ)FW (u) (ξ) dμα,d (ξ), ∇t u (x) = d+1 Ê+ is an isomorphism.

(3.5)

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Proof. (Rd+1 Let s, t ∈ R and u ∈ HSs,α + ). On the one hand, the function ξ → (1 + ∗ s−t α,d t 2 2 ξ) (1 + ξ ) FW (u) (ξ) belongs to L2α (Rd+1 + ) and α,d t α,d ∀ξ ∈ Rd+1 + , FW (∇t u) (ξ) = (1 + ξ) FW (u) (ξ) .

Furthermore, 

2 s−t

2    α,d FW (∇t u) (ξ) dμα,d (ξ)

(1 + ξ ) Êd+1 +   2 (1 + ξ)2t  2 s  α,d (1 + ξ ) FW (u) (ξ) dμα,d (ξ) = 2 t (1 + ξ ) Êd+1 +  2      α,d t ≤ 1+2 (1 + ξ2 )s FW (u) (ξ) dμα,d (ξ) . d+1 Ê+ (Rd+1 Hence, ∇t u ∈ HSs−t,α + ) and we have ∗ √ ∇t us−t ≤ 1 + 2t us . (Rd+1 On the other hand, let v ∈ HSs−t,α + ) and put ∗ −1

  α,d α,d (1 + ξ)−t FW (v) . u = FW From the definition of the operator ∇t , we have ∇t u = v and  2    α,d (1 + ξ2)s FW (u) (ξ) dμα,d (ξ) d+1 Ê+   2 (1 + ξ2 )t  2 s−t  α,d (1 + ξ ) (v) (ξ) = F   dμα,d (ξ) W d+1 (1 + ξ)2t Ê+     2 (1 + ξ2)t  2 s−t  α,d × ≤ sup (1 + ξ ) (v) (ξ) F   dμα,d (ξ) W 2t d+1 (1 + ξ) d+1 Ê+ ξ∈Ê+   2  α,d  |t| ≤2 (1 + ξ2 )s−t FW (v) (ξ) dμα,d (ξ) . d+1 Ê+ So, u ∈ HSs,α (Rd+1 + ) and we have ∗ |t|

|t|

uH s,α ≤ 2 2 vH s−t,α = 2 2 ∇t uH s−t,α . S∗

S∗

S∗

This proves that the operator ∇t is an isomorphism.   s,α d+1  s,α d+1 The following theorem deals with the dual space HS∗ (R+ ) of HS∗ (R+ )   d+1  and gives a relation between HSs,α (R ) and HS−s,α (Rd+1 + + ). ∗ ∗

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−s,α d+1 Theorem 3 The dual of HSs,α (Rd+1 + ) can be identified with HS∗ (R+ ). The ∗ relation of the identification is as follows:  α,d α,d FW (u)(ξ)FW (v)(ξ)dμα,d (ξ), (3.6) u, v 0,α = d+1 Ê+ −s,α d+1 with u ∈ HSs,α (Rd+1 + ) and v ∈ HS∗ (R+ ). ∗

Proof. −s,α d+1 (Rd+1 It is easy to see that for all u ∈ HSs,α + ) and v ∈ HS∗ (R+ ), we have ∗ |u, v 0,α | ≤ uH s,α vH −s,α . S∗

(3.7)

S∗

So, (u, v) → u, v 0,α is a continuous bilinear form on −s,α d+1 (Rd+1 HSs,α + ) × HS∗ (R+ ). ∗ Now, fix v ∈ HS−s,α (Rd+1 + ) and consider the function φ : u → u, v 0,α . ∗ It is clear, from (3.7), that φ is a continuous linear form on HSs,α (Rd+1 + ) and ∗ φ ≤ vH −s,α . S∗

  −1

α,d 2 −s α,d (1 + λ ) FW (v) (λ), we obtain Moreover, if we put u0 (λ) = FW 2 (Rd+1 u0 ∈ HSs,α + ) and u0 , v 0,α = vH −s,α , ∗ S∗

which proves that the norm of φ is equal to vH −s,α .

S∗  s,α d+1  d+1 (R Hence v → ., v 0,α is an isometry from HS−s,α + ) into HS∗ (R+ ) . ∗ It remains, to prove that this isometry is surjective. For this purpose,  then,d+1  let v ∗ ∈ HSs,α (R + ) . By the Riesz representation theorem and the relation ∗ (3.2), one can see that there exists w ∈ HSs,α (Rd+1 + ), such that for all u ∈ ∗ s,α d+1 HS∗ (R+ ), we have  α,d α,d ∗ v (u) = u, w s,α = (1 + λ2 )s FW (w) (λ) FW (u) (λ)dμα,d (λ) . d+1 Ê+

 −1

 α,d α,d (w) (λ) . (1 + λ2 )s FW We put v (λ) = FW (Rd+1 Then, we obtain v ∈ HS−s,α + ) and ∗

∗ ∀u ∈ HSs,α (Rd+1 + ), v (u) = u, v 0,α . ∗

Which completes the proof.



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Proposition 10 ( Convexity ) Let s1 , s2 ∈ R, such that s1 < s2 and s = (1 − t) s1 + ts2 , t ∈ ]0, 1[. Then we have 1−t t s,α ∀u ∈ HSs2∗,α (Rd+1 + ), uHS ≤ uH s1 ,α × uH s2 ,α . ∗

(3.8)

S∗

S∗

Proof. Let s1 , s2 ∈ R, such that s1 < s2 and s = (1 − t) s1 + ts2 , t ∈ ]0, 1[. Let 1 1 u ∈ HSs2∗,α (Rd+1 + ) and put t = , and 1 − t = . p q Then, using the H¨older’s inequality, we get  2    α,d 2 uH s,α = (1 + λ2 )s FW (u) (λ) dμα,d (λ) S∗ d+1 Ê+    2 2 s1 s2  α,d  α,d q p = (1 + λ2 ) q FW (u) (λ) (1 + λ2 ) p FW (u) (λ) dμα,d (λ) Êd+1 +  2q1  2   α,d ≤ (1 + λ2 )s1 FW (u) (λ) dμα,d (λ) × d+1 Ê+ 2p1  2    α,d (1 + λ2 )s2 FW (u) (λ) dμα,d (λ) d+1 Ê+ t ≤ u1−t s ,α × u s2 ,α . H H 1 S∗

S∗

 Proposition 11 Let s1 , s, s2 ∈ R, satisfying s1 < s < s2 . Then, for all ε > 0, there exists a nonnegative constant Cε such that for all u ∈ HSs2∗,α (Rd+1 + ), we have uHSs,α ≤ Cε uHSs1 ,α + εuHSs2 ,α . ∗

∗

(3.9)

∗

Proof. Let s1 , s2 ∈ R, s1 < s2 and s ∈ ]s1 , s2 [ . Then there exists t ∈ ]0, 1[ such that s = (1 − t) s1 + ts2 . From the previous proposition, we have for u ∈ HSs2∗,α (Rd+1 + ), 1−t

t

−t t 1−t u s1 ,α s2 ,α uHSs,α ≤ u1−t ε × εu . s1 ,α × u s2 ,α = HS HS H H ∗

S∗

S∗

∗

But, ∀a, b > 0,

at b1−t ≤ a + b,

∗

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Weinstein Sobolev spaces

then, −t

uHSs,α ≤ ε 1−t uHSs1 ,α + εuHSs2 ,α . ∗

∗

−t

∗

s−s1



Hence, it suffices to take Cε = ε 1−t = ε s−s2 .

Proposition 12 Let s ∈ R and m ∈ N. Then for all ε > 2m, the operator m

α,d s−ε,α is continuous from HSs,α (Rd+1 (Rd+1 ΔW + ) into HS∗ + ). ∗ Proof. Let m ∈ N, ε > 2m, s ∈ R and u ∈ HSs,α (Rd+1 + ). ∗ From (2.6), we have   

m  2  α,d 2 s−ε  α,d (1 + λ ) FW u (λ) dμα,d (λ) ΔW Êd+1 +   2  α,d  λ4m (1 + λ2 )s−ε FW (u) (λ) dμα,d (λ) = d+1 Ê+    2 λ4m  2 s  α,d (1 + λ ) (u) (λ) ≤ sup F   dμα,d (λ) W 2 )ε d+1 (1 + λ d+1 Ê+ λ∈Ê+   2  α,d  ≤ (1 + λ2 )s FW (u) (λ) dμα,d (λ) . d+1 Ê+

m (u) ∈ HSs−ε,α (Rd+1 Then Δα,d + ) and W ∗

m  Δα,d (u) H s−ε,α ≤ uHSs,α . W S∗

∗

 d (Rd+1 Proposition 13 For s > + α + 1, the Hilbert space HSs,α + ) admits the ∗ 2 reproducing kernel:  (1 + ξ2 )−s Ψ(−x, ξ)Ψ(y, ξ)dμα,d(ξ), (3.10) Θα,d (x, y) = d+1 Ê+ that is : i) For every y ∈ Rd+1 + , the distribution given by the function x → Θα,d (x, y) d+1 (R belongs to HSs,α + ). ∗ ii) For every f ∈ HSs,α (Rd+1 + ), we have ∗ ∀y ∈ Rd+1 + , f, Θα,d (., y) s,α = f (y).

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Proof. d + α + 1, the function i) Let y ∈ Rd+1 + . From (2.3), we have for all s > 2 2 d+1 ξ → (1 + ξ2)−s Ψ(y, ξ) belongs to L1α (Rd+1 + ) ∩ Lα (R+ ). Then, from the relation (2.11), the function x → Θα,d (x, y) belongs to L2α (Rd+1 + ) and we have α,d 2 −s ∀ξ ∈ Rd+1 + , FW [Θα,d (., y)] (ξ) = (1 + ξ ) Ψ(y, ξ).

(3.11)

Then Θα,d (., y) ∈ HSs,α (Rd+1 + ). ∗ s,α d+1 ii) Let f ∈ HS∗ (R+ ) and y ∈ Rd+1 + . Using the relations (3.2), (3.11) and ( 2.8), we obtain  α,d f, Θα,d (., y) s,α = FW (f ) (ξ) Ψ(−y, ξ)dμα,d (ξ) = f (y). d+1 Ê+ 

4

Pseudo-differential operators

Notations. • For r ≥ 0, we designate by S r , the space of C ∞ −function a on Rd+1 × Rd+1 such that for each compact set K ⊂ Rd+1 and each β, γ ∈ N, there exists a constant C = C (K, β, γ) satisfying : 

  α,d β α,d γ  r d+1  ∀ (x, ξ) ∈ K × R ,  ΔW,ξ ΔW,x a (x, ξ) ≤ C(1 + ξ2) 2 . (4.1) • For r ≥ 0 and l > 0, we denote by S r,l , the space consits of all C ∞ −function a on Rd+1 × Rd+1 such that for each β, γ ∈ N, there exist a positive constant C = C (r, l, β, γ) satisfying the relation 

  α,d β α,d γ  r l d+1 d+1  × R ,  ΔW,ξ ∀ (x, t) ∈ R ΔW,x a (x, t) ≤ C(1 + ξ2) 2 (1 + x2 )− 2 . (4.2)

Definition 7 The pseudo-differential operator A a, Δα,d W

 associated with

a (x, ξ) ∈ S is defined for u ∈ S∗ (R ), by :    

α,d α,d Ψα,d (−x, ξ)a (x, ξ) FW (u) (ξ) dμα,d (ξ) . (4.3) A a, ΔW (u) (x) = d+1 Ê+ r

d+1

Theorem 4 If a (x, ξ) ∈ S r , then its associated pseudo-differential operator 

  is a well-defined mapping from S∗ (Rd+1 ) into C ∞ Rd+1 . A a, Δα,d W

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Proof. d Let a (x, ξ) ∈ S r and s > r + + α + 1. From the relation (4.1), we have for 2 any compact set K ⊂ Rd+1 and any γ ∈ N, 

 γ r   a (x, ξ) ∀ (x, ξ) ∈ K × Rd+1 ,  Δα,d  ≤ C(1 + ξ2 ) 2 . W,x Then, by using the relation (2.3) and the Cauchy-Schwartz inequality, we obtain for any compact set K ⊂ Rd+1 , for any γ ∈ N and any x ∈ K,  

   α,d γ  α,d d+1 ∀u ∈ S∗ (R ),  ΔW,x a (x, ξ) Ψα,d (−x, ξ)FW (u) (ξ) dμα,d (ξ) d+1 Ê+    r   α,d (1 + ξ2 ) 2 FW (u) (ξ) dμα,d (ξ) ≤ Êd+1 +  ≤ C uH s,α , S∗

where C = C



 12 Ê

d+1 +

(1 + ξ2)r−s dμα,d (λ)

.



  α,d In the case γ = 0, this relation proves that A a, ΔW (u) (x) is well-defined and continuous on Rd+1 + . Together with the Leibniz formula, this relation in its general form completes the proof.  The next lemma plays an important role in this section. Lemma 2 Let t ≥ 0. then, for all symbols a (x, ξ) ∈ S r,l , we have :   r t  α,d  ∀ (y, ξ) ∈ Rd+1 × Rd+1 , FW (a (., y)) (ξ) ≤ C(1 + y2) 2 (1 + ξ2)− 2 , (4.4) where C is a constant depending on r, t, α, d and l. Proof. Let k ∈ N. From the relation (2.6) and (4.2), we obtain 

     k    α,d α,d 2k  α,d ξ FW (a (., y)) (ξ) = FW ΔW,ξ a (., y) (ξ)    

k   a (z, y) Ψ (z, ξ)dμ (z) Δα,d =  α,d α,d W,z   Êd+1 +  r l ≤ C1 (1 + y2) 2 (1 + z2 )− 2 dμα,d (z) , Êd+1 +

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H. Ben Mohamed, N. Bettaibi and S. H. Jah

where l > d +2α+ 2.   t t t Let now m = + 1, t ≥ 0, where is the integer part of . 2 2 2 Using the relation m       α,d  α,d   k Cm ξ2k FW (a (., y)) (ξ) , (1 + ξ ) FW (a (., y)) (ξ) = 2 m

k=0

we get   r t  α,d  FW (a (., y)) (ξ) ≤ C(1 + y2 ) 2 (1 + ξ2)− 2 .

The following theorem gives an alternative form of A a, useful in the sequel.

Δα,d W





which will be

 Theorem 5 Let a (x, ξ) ∈ S r,l . Then, the pseudo-differential operator A a, Δα,d W admits the following representation :    

α,d Ψα,d (−x, ξ) × A a, ΔW (u) (x) = Êd+1  + α,d α,d FW (Ty a (., y)) (ξ) FW (u) (y) dμα,d (y) dμα,d (ξ) , d+1 Ê+ (4.5) where all involved integrals are absolutely convergent. Proof. Let

 g (ξ) =

Ê

d+1 +

α,d α,d FW (Ty a (., y)) (ξ) FW (u) (y) dμα,d (y) .

  . We shall prove that g belongs to L1α Rd+1 + r d Let t > d + 2α + 2 and γ > + + α + 1. Using the relations (2.3), (2.16) 2 2 and (4.4), we obtain   r t   α,d α,d (4.6) FW (Ty a (., y)) (ξ) FW (u) (y) ≤ C(1 + y2) 2 −γ (1 + ξ2)− 2 . Hence, we get |g (ξ)| ≤ C  (1 + ξ2 )− 2 , t

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Weinstein Sobolev spaces

where 



C =C

(1 + y2) 2 −γ dμα,d (y) . r

Êd+1 +

  t . Since t > d + 2α + 2, the function ξ → (1 + ξ2 )− 2 belongs to L1α Rd+1 +  d+1  ∞  d+1  1 Then g ∈ Lα R+ ∩Lα R+ . So, the result follows by applying the inverse theorem.  Now, we are in a situation to establish the fundamental result of this section given by the following result.

 d + α + 1, a (x, ξ) ∈ S r,l and A a, Δα,d be the asW 2 

maps continuously sociated pseudo-differential operator. Then A a, Δα,d W Theorem 6 Let s >

s,α d+1 (Rd+1 HSs+r,α + ) to HS∗ (R+ ). Moreover, we have ∗

∀u ∈ S∗ (R

d+1



    α,d ), A x, ΔW (u)

HSs,α ∗

≤ ks uH r+s,α . S∗

(4.7)

Proof. d Let s > + α + 1. We put 2  α,d α,d 2 2s FW (Ty a (., y)) (ξ) FW (u) (y) dμα,d (y) . ϕs (ξ) = (1 + ξ ) Êd+1 + The relation (2.16) together with (4.4) gives     2 2s − 2t 2 r+s 2 − 2s  α,d 2 |ϕs (ξ)| ≤ C1 (1 + ξ ) (1 + y ) (1 + y ) FW (u) (y) dμα,d (y) . d+1 Ê+ Applying now the Cauchy-Schwartz inequality, we get |ϕs (ξ)| ≤ C2 (1 + ξ2 ) 2 − 2 uH r+s,α , s

t

S∗

where  C2 = C1

Ê

d+1 +

(1 + y2)−s dμα,d (y) .

Then 

    α,d A a, ΔW (u)

HSs,α ∗

= ϕs α,2 ≤ ks uH r+s,α , S∗

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H. Ben Mohamed, N. Bettaibi and S. H. Jah

where t > s +

d + α + 1 and 2  k s = C2

 12 Ê

d+1 +

(1 + ξ2)s−t

. 

Corollary 1 Let n ∈ N, l > 0 and ak (x) be a sequence of symbols satisfying : ∀k ∈ N, |ak (x)| ≤ C(1 + x2 )−l .

(4.8)

 given by : Then the pseudo-differential operator Pn x, Δα,d W

Pn x,

Δα,d W

 =

n 

k ak (x) Δα,d , W

k=0 s,α d+1 (Rd+1 maps continuously HSs+2n,α + ) to HS∗ (R+ ). Moreover, we have ∗ 

    α,d d+1 ∀u ∈ S∗ (R ), Pn x, ΔW (u) s,α ≤ Cs uH 2n+s,α . S∗

HS ∗

(4.9)

Proof. From the relation (2.7) and (4.8), we get    

    α,d   α,d  α,d FW Pn x, ΔW u (ξ) ≤ C(1 + ξ2 )n (1 + x2 )−l FW (u) (ξ) . Then, we obtain for all u ∈ S∗ (Rd+1 ), 

    α,d Pn x, ΔW (u) s,α ≤ Cs (1 + x2 )−l uH 2n+s,α ≤ Cs uH 2n+s,α . S∗

HS ∗

S∗



5

Generalized Weinstein potential

Let a (x) be a multiplier in S∗ (Rd+1 ) such that ∀x ∈ R

d+1

, a (x) = 0 and



α,d FW

−1

  (a−m ) ∈ L1 Rd+1 ,

(5.1)

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Weinstein Sobolev spaces

−1

α,d is the Fourier-Weinstein inverse transform. where m ∈ N and FW

For f ∈ S∗ , we can define the products f am ∈ S∗ , m ∈ Z, by means of the following two relations : ∀φ ∈ S∗ (Rd+1 ), f an , φ = f, an φ

and ∀φ ∈ S∗ (Rd+1 ), f a−n , an φ = f, φ , where n ∈ N. Therefore, for all m ∈ Z, f am ∈ S∗ . Let m ∈ Z. We define the generalized Weinstein potential Jm by :

−1 α,d α,d  a−m FW (u) . ∀u ∈ S∗ , Jm u = FW

(5.2)

α,d is a continuous linear Remark 2 Since the Fourier-Weinstein transform FW −1

α,d   map of S∗ onto S∗ , the same being true for FW . So, Jm is a continuous

linear map of S∗ onto S∗ . Clearly Jm is a pseudo-differential operator with symbol a−m . The following lemma gives an easily verified property of Jm . Lemma 3 Let u ∈ S∗ . Then we have J0 u = u and ∀m, l ∈ Z, Jm Jl u = Jm+l u.

(5.3)

the set of Notation. Let m ∈ Z and 1 ≤ p ≤ +∞, we denote by HSm,p,α  d+1  ∗ p all Fourier ultrahyperfunctions u for which J−m u ∈ Lα R+ . We equip this space with the norm ∀u ∈ HSm,p,α , um,p,α = J−m uα,p . ∗

(5.4)

is a Banach space with respect to the norm Theorem 7 The space HSm,p,α ∗ given by the relation (5.4). Proof. Let (un )n∈Æ be a Cauchy sequence of HSm,p,α . From the definition of the norm ∗   . .m,p,α , the sequence (J−m un )n∈Æ is a Cauchy sequence of the space Lpα Rd+1 +  d+1  p Thus, there exists a function v in Lα R+ such that lim J−m un − vα,p = 0.

n→+∞

Then (Jm v) ∈ HSm,p,α and we have ∗ lim un − Jm vm,p,α = 0.

n→+∞

This completes the proof.



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H. Ben Mohamed, N. Bettaibi and S. H. Jah

Theorem 8 Let m, l ∈ Z. Then Jl is an isometry from HSm,p,α onto HSm+l,p,α ∗ ∗ and we have , Jl um+l,p,α = um,p,α . ∀u ∈ HSm,p,α ∗ Proof. The result follows immediately from the relations (5.3) and (5.4).

(5.5) 

We finish this section by giving an analog of the Sobolev imbedding theorem. Theorem 9 Let 1 < p < +∞ and m, l ∈ Z, m ≤ l. Then HSl,p,α ⊂ HSm,p,α and there exists a constant Cm,l , such that ∗ ∗ , um,p,α ≤ Cm,l ul,p,α . ∀u ∈ HSl,p,α ∗

(5.6)

Proof. Let m, l ∈ Z, m ≤ l and u ∈ HSl,p,α . Using the relations (5.2), (5.3) and (5.4), ∗ we obtain   m−l um,p,α = J−m up,α = Jl−m J−l up,α ≤ sup |a (x)| ul,p,α x∈Êd+1 which achieves the proof.



Acknowledgement The second and the third authors acknowledge research support from the Qassim university, grant SR-D-010-316.

References [1] M. Assal, M. M. Nissibi, Bessel-Sobolev Type Spaces, Mathematica Bolkanica Vol.18, 227-234 (2003). [2] H. Ben Mohamed and K. Trim`eche, Dunkl transform on R and convolution product on new spaces of distributions, Integ. Transf. and Special Func. Vol.14, Nr 5, (2003) p. 437-458. [3] Z. Ben Nahia, Fonctions harmoniques et propriet´es de la moyenne associ´ees a` l’op´erateur de Weinstein, Th`ese 3e`me cycle Maths. (1995) Department of Mathematics Faculty of Sciences of Tunis. Tunisia. [4] Z. Ben Nahia and N. Ben Salem, Spherical harmonics and applications associated with the Weinstein operator. “ Proceedings ” de la Conf´erence Internationale de Th´eorie de Potentiel, I.C.P.T. 94, tenue a` Kouty ( en R´epublique Tch`eque ) du 13-20 Aoˆ ut 1994.

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[5] Z. Ben Nahia and N. Ben Salem, On a mean value property associated with the Weinstein operator. “ Proceedings ” de la Conf´erence Internationale de Th´eorie de Potentiel, I.C.P.T. 94, tenue a` Kouty ( en R´epublique Tch`eque) du 13-20 Aoˆ ut 1994. [6] N. Ben Salem, A. Ould Ahmed Salem, Sobolev type spaces associated with the Jacobi-Dunkl operator, Frac. Calc. Appl. Analy., V 7, Nr 1 (2004). [7] M. Brelot, Equation de Weinstein et potentiels de Marcel Riez, Lecture Notes in Math. 681, S´eminaires de Th´eorie du Potentiel, Paris, Nr 3, p. 18-38. [8] J. Chung, S.Y. Chung and D. Kim, A characterization for the Fourier hyperfunctions, Publ. RIMS, Kyoto Univ. 300, (1994) p. 203-208. [9] L. H¨ormander, The analysis of linear partiel differentiel operator, Springer-Verlag, Berlin-New work (1983). [10] S. Lee, Generalized Sobolev spaces of exponential type, Kangweon-Kyungki Math. J. Vol.8, Nr 1 (2000) p. 73-86. [11] H. Leutwiler, Best constants in the Harnack inequality for the Weinstein equation, Aequationes Mathematicae 34, (1987) p. 304-315. [12] D. H. Pahk and B. H. Kang, Sobolev spaces in the generlized distribution spaces of Beurling type, Tsukuba J. Math. Vol. 15, (1991) p.325-334. [13] R. S. Pathak, Generalized Sobolev spaces and pseudo-differential operators on spaces of ultradistributions, Structure of solutions of diffential equations, Edited y M.Morimoto and T. Kawai, World Scientific, Singafore (1996) p. 343-368. Received: February, 2011