International Journal of Analysis and Applications
ISSN 2291-8639
Volume 9, Number 1 (2015), 19-28
http://www.etamaths.com

HARMONIC ANALYSIS ASSOCIATED WITH THE GENERALIZED
WEINSTEIN OPERATOR

AHMED ABOUELAZ, AZZ-EDINE ACHAK , RADOUAN DAHER AND EL MEHDI LOUALID∗

Abstract. In this paper we consider a generalized Weinstein operator ∆d,α,n on Rd−1×]0,∞[,
which generalizes the Weinstein operator ∆d,α, we define the generalized Weinstein intertwining
operator Rα,n which turn out to be transmutation operator between ∆d,α,n and the Laplacian
operator ∆d. We build the dual of the generalized Weinstein intertwining operator tRα,n, another
hand we prove the formula related Rα,n and tRα,n . We exploit these transmutation operators to
develop a new harmonic analysis corresponding to ∆d,α,n.

1. Introduction

In this paper we consider a generalized Weinstein operator ∆d,α,n on Rd−1×]0,∞[, defined by

(1) ∆d,α,n =
d∑
i=1

∂2

∂x2i
+

2α + 1

xd

∂

∂xd
−

4n(α + n)

x2d
, α > −

1

2

where n = 0, 1, ... . For n = 0, we regain the Weinstein operator

(2) ∆d,α =
d∑
i=1

∂2

∂x2i
+

2α + 1

xd

∂

∂xd
, α > −

1

2

Through this paper, we provide a new harmonic analysis on Rd−1×]0,∞[ corresponding to the
generalized Weinstein operator ∆d,α,n.
The outline of the content of this paper is as follows.
Section 2 is dedicated to some properties and results concerning the Weinstein transform.
In section 3, we construct a pair of transmutation operators Rα,n and tRα,n, afterwards we exploit
these transmutation operators to build a new harmonic analysis on Rd−1×]0,∞[ corresponding to
operator ∆d,α,n.

2. Preliminaries

Throughout this paper, we denote by
• Rd+ = Rd−1×]0,∞[.
• x = (x1, ...,xd) = (x

′
,xd) ∈ Rd−1×]0,∞[.

• λ = (λ1, ...,λd) = (λ
′
,λd) ∈ Cd.

• E(Rd) (resp. D(Rd)) the space of C∞ functions on Rd, even with respect to the last variable
(resp. with compact support).

2010 Mathematics Subject Classification. 42A38, 44A35, 34B30.
Key words and phrases. generalized Weinstein operator; generalized Weinstein transform; generalized convolution;

generalized translation operators; harmonic analysis.

c©2015 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

19



• S(Rd) the Schwartz space of rapidly decreasing functions on Rd which are even with respect
to the last variable.

In this section, we recapitulate some facts about harmonic analysis related to the Weinstein operator
∆d,α. We cite here, as briefly as possible, some properties. For more details we refer to [2, 3, 4].
The Weinstein operator ∆d,α defined on Rd+ by

(3) ∆d,α =
d∑
i=1

∂2

∂x2i
+

2α + 1

xd

∂

∂xd
, α > −

1

2

Then
∆d,α = ∆d + Bα

where ∆d is the Laplacian operator in Rd−1 and Bα the Bessel operator with respect to the variable
xd defined by

(4) Bα =
∂2

∂x2d
+

2α + 1

xd

∂

∂xd
, α > −

1

2
.

The Weinstein kernel is given by

(5) Ψλ,α(x) = e−i<x
′,λ′>jα(xdλd), for all (x,λ) ∈ Rd ×Cd.

Here x′ = (x1, ....,xd−1),λ′ = (λ1, ....,λd−1) and jα is the normalized Bessel function of index α
defined by

(6) jα(z) = Γ(α + 1)
∞∑
n=0

(−1)n(z
2
)2n

n! Γ(n + α + 1)
(z ∈ C).

Proposition 1. Ψλ,α satisfies the differential equation

∆d,αΨλ,α = −‖λ‖2Ψλ,α.
Definition 1. The Weinstein intertwining operator is the operator Rα defined on C(Rd) by

(7) Rαf(x) = aαx−2αd
∫ xd
0

(x2d − t
2)α−

1
2 f(x′, t)dt, xd > 0

where

(8) aα =
2Γ(α + 1)
√
πΓ(α + 1

2
)
.

Proposition 2. Rα is a topological isomorphism from E(Rd) onto itself satisfying the following
transmutation relation

(9) ∆d,α(Rαf) = Rα(∆df), for all f ∈ E(Rd),
where ∆d is the Laplacian on Rd.
Proposition 3. ∆d,α is self-adjoint, i.e∫

Rd
+

∆d,αf(x)g(x)dµα(x) =

∫
Rd

+

f(x)∆d,αg(x)dµα(x)

for all f ∈ E(Rd) and g ∈ D(Rd).
Definition 2. The dual of the Weinstein intertwining operator Rα is the operator tRα defined on
D(Rd) by

(10) tRα(f)(y) = aα
∫ ∞
yd

(s2 −y2d)
α−1

2 f(y′,s)sds.

20



Proposition 4. tRα is a topological isomorphism from S(Rd) onto itself satisfying the following
transmutation relation

(11) tRα(∆d,αf) = ∆d(tRαf), for all f ∈ S(Rd),
where ∆d is the Laplacian on Rd.
It satisfies for f ∈ D(Rd) and g ∈ E(Rd) the following relation

(12)
∫
Rd

+

tRα(f)(y)g(y)dy =
∫
Rd

+

f(y)Rα(g)(y)dµα(y).

Definition 3. The Weinstein transform FW,α is defined on L1α(Rd+) by

(13) FW,α(f)(λ) =
∫
Rd

+

f(x)Ψλ,α(x)dµα(x), for all λ ∈ Rd.

Proposition 5. (i) For all f ∈ L1(Rd+), the function FW,α(f) is continuous on Rd and we have
(14) ‖FW,α(f)‖α,∞ ≤‖f‖α,1.

(ii) For all f ∈ S(Rd) we have
(15) FW,α(f)(y) = F0 ◦t Rα(f)(y), ∀y ∈ Rd+,

where F0 is the transformation defined by, for all y ∈ Rd+

(16) F0(f)(y) =
∫
Rd

+

f(x)e−i<y
′,x′> cos(xdyd)dx, ∀f ∈ D(Rd).

(iii) For all f ∈ S(Rd) and m ∈ N, we have
(17) FW,α(∆d,αf)(λ) = −‖λ‖2FW,α(f)(λ).

.

Theorem 1. (i) Plancherel formula: For all f ∈ S(Rd) we have

(18)
∫
Rd

+

|f(x)|2dµα(x) = C(α)
∫
Rd

+

|FW,α(f)(λ)|2dµα(λ)

where

(19) C(α) =
1

(2π)d−122α(Γ(α + 1))2
.

(ii) For all f ∈ L1α(Rd+), if FW,α(f) ∈ L1α(Rd+), then

(20) f(y) = C(α)
∫
Rd

+

FW,α(f)(x)Ψλ,α(−x)dµα(x)

where C(α) is given by (19).

Definition 4. The translation operators τxα, x ∈ Rd+, associated with the operator ∆d,α are defined
by

(21) τxαf(y) =
Γ(α + 1)
√
πΓ(α + 1

2

)

∫ π
0

f(x′ + y′,
√
x2d + y

2
d + 2xdyd cos θ)(sin θ)

2αdθ

where f ∈ C(Rd+).

Proposition 6. For all f ∈ Lpα,n(Rd), p ∈ [1,∞], and for all x ∈ Rd+
τxα,n(Φλ,α,n(y)) = Φλ,α,n(x)Φλ,α,n(y).

21



Proposition 7. The translation operator τxα, x ∈ Rd+ satisfies the following properties:
(i) ∀x ∈ Rd+, we have

(22) ∆d,α ◦ τxα = τ
x
α ◦ ∆d,α.

(ii) For all f in E(Rd) and g in S(Rd) we have

(23)
∫
Rd

+

τxαf(y)g(y)dµα(y) =

∫
Rd

+

f(y)τxαg(y)dµα(y).

(iii) For all f in Lpα(R
d
+), p ∈ [1,∞], and x ∈ Rd+ we have

(24) ‖τxα‖p,α ≤‖f‖p,α.

(iv) For f ∈ S(Rd) and y ∈ Rd+ we have

(25) FW,α (τyαf) (x) = Ψy,α(x)FW,α(f)(x).

Definition 5. The generalized convolution product f ∗W,α g of functions f,g ∈ L1α(Rd+) is defined by

(26) f ∗W,α g(x) =
∫
Rd

+

τxαf(−y
′,y)g(y)dµα(y).

Proposition 8. For all f,g ∈ L1α(Rd+), f ∗W,α g belongs to L1α(Rd+) and

(27) FW,α (f ∗W,α g) = FW,α(f)FW,α(g).

3. Harmonic analysis associated with the generalized Weinstein operator

Transmutation operators.
• Mn the map defined by Mnf(x

′
,xd) = x

2n
d f(x

′
,xd).

• Lpα,n(Rd+) the class of measurable functions f on Rd+ for which

‖f‖α,n,p = ‖M−1n f‖α+2n,p < ∞.

• En(Rd) (resp. Dn(Rd) and Sn(Rd)) stand for the subspace of E(Rd) (resp. D(Rd) and S(Rd))
consisting of functions f such that

f(x
′
, 0) =

(
dkf

dxkd

)
(x

′
, 0) = 0, ∀k ∈{1, ...2n− 1}.

Lemma 1. (i) The map

(28) Mn(f)(x) = x2nd f(x)

is an isomorphism
– from E(Rd) onto En(Rd).
– from S(Rd) onto Sn(Rd).

(ii) For all f ∈ E(Rd) we have

(29) Bα,n ◦Mn(f) = Mn ◦Bα+2n(f),

where Bα,n is the generalized Bessel operator given by (4).
(iii) For all f ∈ E(Rd)

(30) ∆d,α,n ◦Mn(f)(x) = Mn ◦ ∆d,α+2n
where ∆d,α+2n is the Weinstein operator of order α + 2n given by (3).

22



(iv) ∆d,α,n is self-adjoint, i.e

(31)
∫
Rd

+

∆d,α,nf(x)g(x)dµα(x) =

∫
Rd

+

f(x)∆d,α,ng(x)dµα(x)

for all f ∈ E(Rd) and g ∈ Dn(Rd).
Proof. Assertion (i) and (ii) (see [1]).
For assertion (iii) using (1) and (29) we obtain

∆d,α,n ◦Mn(f)(x
′
,xd) = (∆d + Bα,n) ◦Mn(f)(x

′
,xd),

= ∆d(Mnf)(x
′
,xd) + Bα,n(Mnf)(x

′
,xd),

= Mn(∆df)(x
′
,xd) + Mn(Bα+2nf)(x

′
,xd),

= Mn ◦ ∆d,α+2nf(x
′
,xd).

which give (iii).
If f ∈ E(Rd) and g ∈ Dn(Rd), then by Proposition 3 we get∫

Rd
+

∆d,α,nf(x)g(x)dµα(x) =

∫
Rd

+

(
∆d,αf(x) −

4n(α + n)

x2d
f(x)

)
g(x)dµα(x),

=

∫
Rd

+

∆d,αf(x)g(x)dµα(x) −
∫
Rd

+

4n(α + n)

x2d
f(x)g(x)dµα(x),

=

∫
Rd

+

f(x)∆d,αg(x)dµα(x) −
∫
Rd

+

4n(α + n)

x2d
f(x)g(x)dµα(x),

=

∫
Rd

+

f(x)

(
∆d,αg(x) −

4n(α + n)

x2d
g(x)

)
dµα(x),

=

∫
Rd

+

f(x)∆d,α,ng(x)dµα(x).

Definition 6. The generalized Weirstein intertwining operator is the operator Rα,n defined on E(Rd+1)
by

(32) Rα,nf(x) = aα+2nx
−2(α+n)
d

∫ xd
0

(x2d − t
2)α+2n−

1
2 f(x′, t)dt, xd > 0

where aα+2n is given by 8.

Remark 1. by (7) and (32) we have

(33) Rα,n = Mn ◦Rα+2n.
Proposition 9. Rα,n is a topological isomorphism from E(R) onto En(R) satisfying the following
transmutation relation

(34) ∆d,α,n(Rα,nf) = Rα,n(∆df), forallf ∈ E(Rd+1)
where ∆d is the Laplacian on Rd.
Proof. Using (9), (30) and (33) we obtain

∆d,α,n(Rα,nf) = ∆d,α,n (Mn ◦Rα+2n) (f),
= Mn ◦ ∆d,α+2n(Rα+2nf)
= Mn (Rα+2n ◦ ∆d) (f)
= Rα,n(∆df).

23



Definition 7. The dual of the generalized Weinstein intertwining operator Rα,n is the operator tRα,n
defined on Dn(Rd) by

(35) tRα,n(f)(y) = aα+2n
∫ ∞
yd

(s2 −y2d)
α+2n−1

2 f(y′,s)s1−2nds.

Remark 2. From (10) and (35) we have

(36) tRα,n =t Rα+2n ◦M−1n .

Proposition 10. tRα,n is a topological isomorphism from Sn(Rd+1) onto S(Rd+1) satisfying the
following transmutation relation

(37) tRα,n(∆d,α,nf) = ∆d(tRα,nf), for all f ∈ Sn(Rd+1)
where ∆d is the Laplacian on Rd.

Proof. An easily combination of (11), (30) and (37) shows that
tRα,n(∆d,α,nf) = tRα+2n ◦M−1n

(
Mn ◦ ∆d,α+2n ◦M−1n

)
(f),

= tRα+2n
(
∆d,α+2n ◦M−1n

)
(f),

= ∆d
(
Rα+2n ◦M−1n

)
(f),

= ∆d(
tRα,nf).

Proposition 11. For all f ∈ Dn(Rd) and g ∈ E(Rd)

(38)
∫
Rd

+

tRα,n(f)(y)g(y)dy =
∫
Rd

+

f(y)Rα,n(g)(y)dµα(y).

Proof. Using (12), (33) and (37)∫
Rd

+

tRα,n(f)(x)g(x)dx =
∫
Rd

+

tRα+2n ◦M−1n f(x)g(x)dx

=

∫
Rd

+

M−1n f(x)
tRα+2n(g)(x)dµα+2n(x)

=

∫
Rd

+

f(x)Mn(Rα+2n(g))(x)dµα(x)

=

∫
Rd

+

f(y)Rα,n(g)(y)dµα(y).

Generalized Weinstein transform.
Throughout this section assume α > −1

2
and n a non-negative integer.

For all λ = (λ1, ....,λd) ∈ Cd and x = (x1, ....,xd) ∈ Rd, put

(39) Φλ,α,n(x) = x2nd Ψλ,α+2n(x)

where Ψλ,α+2n(x) is the Weinstein kernel of index α + 2n is given by (5).

Proposition 12. Φλ,α,n satisfies the differential equation

(40) ∆d,α,nΦλ,α,n = −‖λ‖2Φλ,α,n.
24



Proof. From Proposition 1 and (39) we obtain

∆d,α,nΦλ,α,n = Mn ◦ ∆d,α+2nM−1n Φλ,α,n,
= Mn ◦ ∆d,α+2nΨλ,α+2n,
= −‖λ‖2MnΨλ,α+2n,
= −‖λ‖2Φλ,α,n.

Definition 8. The generalized Weinstein transform is defined on L1α,n(R
d
+) by, for all λ ∈ Rd

(41) FW,α,n(f)(λ) =
∫
Rd

+

f(x)Φλ,α,n(x)dµα(x).

Remark 3. By (5), (13) and (41), we have

(42) FW,α,n = FW,α+2n ◦M−1n .

Theorem 2. (i) Inverse formula: Let f ∈ L1α,n(Rd+), if FW,α,n ∈ L1α(Rd+) then

(43) f(x) = C(α + 2n)
∫
Rd

+

FW,α,nf(λ)Φλ,α,n(x)dµα+2n(λ).

(ii) Plancherel formula:

(44)
∫
Rd

+

|f(x)|2dµα(x) = C(α + 2n)
∫
Rd

+

|FW,α,nf(λ)|2dµα+2n(λ)

where C(α + 2n) is given by (19).

Proof. By (20), (39) and (42) we obtain

C(α + 2n)

∫
Rd

+

FW,α,nf(λ)Φλ,α,n(x)dµα+2n(λ) = C(α + 2n)
∫
Rd

+

FW,α,nf(λ)x2nd Ψλ,α+2n(x)dµα+2n(λ),

= x2nd C(α + 2n)

∫
Rd

+

FW,α+2n
(
M−1n f

)
(λ)Ψλ,α+2n(x)dµα+2n(λ),

= x2nd M
−1
n f(x),

= f(x).

which proves (i).
For (ii) an easily combination of (18), (39) and (42) shows that∫

Rd
+

|f(x)|2dµα(x) =
∫
Rd

+

|M−1n f(x)|
2dµα+2n(x),

= C(α + 2n)

∫
Rd

+

∣∣FW,α+2n (M−1n f(λ))∣∣2 dµα+2n(λ),
= C(α + 2n)

∫
Rd

+

|FW,α,nf(λ)|2dµα+2n(λ).

Proposition 13. (i) For all f ∈ L1α,n(Rd+), we have

‖FW,α,n(f)‖α,∞ ≤‖f‖α,n,1.
25



(ii) For all f ∈ Sn(Rd) we have

FW,α,n(f)(y) = F0 ◦t Rα,n(f)(y), ∀y ∈ Rd+,

where F0 is the transformation defined by ().
(iii) For all f ∈ Sn(Rd) and m ∈ N, we have

FW,α,n(∆d,αf)(λ) = −‖λ‖2FW,α,n(f)(λ).

Proof. From (14) and (42) we have

‖FW,α,n(f)‖α,n,∞ = ‖FW,α+2n ◦M−1n (f)‖α,n,∞
≤ ‖M−1n f‖α+2n,1
≤ ‖f‖α,n,1

which proves assertion (i).
By (15), (36) and (42) we obtain

FW,α,n(f) = FW,α+2n ◦M−1n (f)
= F0 ◦ tRα+2n ◦M−1n (f)
= F0 ◦ tRα,n(f),

which proves assertion (ii).
Due to (16), (33) and (42) we have

FW,α,n(∆d,α,nf)(λ) = Fd,α+2n ◦M−1n (∆d,α,nf)(λ)
= FW,α+2n ◦M−1n (∆d,α,nf)(λ)
= FW,α+2n(∆d,α+2nM−1n f)(λ)
= −‖λ‖2FW,α+2n ◦M−1n (f)(λ)
= −‖λ‖2FW,α,n(f)(λ).

Generalized convolution product.

Definition 9. The generalized translation operators τxα,n, x ∈ Rd associated with ∆d,α,n are defined
on Rd+ by

(45) τxα,nf = x
2n
d Mnτ

x
α+2nM

−1
n f

where τxα+2n are the Weinstein translation operators of order α + 2n given by (21).

Definition 10. The generalized convolution product of two functions f ∈ E(Rd) and g ∈ D(Rd) is
defined by:

(46) f ∗W,α,n g(x) =
∫
Rd

+

τxα,nf(y)g(y)dµα(y); ∀x ∈ R
d
+.

Proposition 14. Let f and g in Dn(Rd), we have

(47) f ∗W,α,n g = Mn
(
M−1n f ∗W,α+2n M

−1
n g

)
.

26



Proof. Using (23) and (45) we get

f ∗W,α,n g(x) =
∫
Rd

+

τxα,nf(y)g(y)dµα(y)

=

∫
Rd

+

x2nd Mnτ
x
α+2nM

−1
n f(y)g(y)dµα(y)

= x2nd

∫
Rd

+

τxα+2nM
−1
n f(y)M

−1
n g(y)dµα+2n(y)

= Mn
(
M−1n f ∗W,α+2n M

−1
n g

)
(x).

Proposition 15. (i) For all f ∈ Lpα,n(Rd), p ∈ [1,∞], and for all x ∈ Rd+

(48) ‖τxα,nf‖p,α,n ≤ x
2n
d ‖f‖p,α,n.

(ii)

(49) τxα,n(Φλ,α,n(y)) = Φλ,α,n(x)Φλ,α,n(y).

Proof. From (24) and (45) we have

‖τxα,nf‖p,α,n = x
2
d‖Mnτ

x
α+2nM

−1
n f‖p,α,n

= x2d‖τ
x
α+2nM

−1
n f‖p,α+2n

≤ x2d‖τ
x
α+2nM

−1
n f‖p,α+2n

≤ x2d‖M
−1
n f‖p,α+2n

= x2nd ‖f‖p,α,n.

which give (i).
From (39), (45) and Proposition 6 we get

τxα,nΦλ,α,n(y) = x
2n
d Mn ◦ τ

x
α+2n ◦M

−1
n Φλ,α,n(y)

= x2nd Mn ◦ τ
x
α+2nΨλ,α,n(y)

= x2nd y
2n
d τ

x
α+2nΨλ,α,n(y)

= x2nd y
2n
d Ψλ,α,n(x)Ψλ,α,n(y)

= Φλ,α,n(x)Φλ,α,n(y).

which prove (ii).

Theorem 3. (i) For f ∈ S(Rd) and y ∈ Rd+

(50) FW,α,n
(
τxα,nf

)
(λ) = Φλ,α,n(x)FW,α,n(f(λ)), λ ∈ Rd+.

(ii) For all f ∈ E(Rd) and g ∈ S(Rd)

(51)
∫
Rd

+

τxα,nf(y)g(y)dµα(y) =

∫
Rd

+

f(y)τxα,ng(y)dµα(y).

(iii) For all f,g ∈ L1α(Rd+), f ∗W,α,n g ∈ L1α(Rd+), and

(52) FW,α,n (f ∗W,α,n g) = FW,α,n(f)FW,α,n(g).
27



Proof. An easily combination of (25), (39), (42) and (45) shows that

FW,α,n
(
τxα,nf

)
(λ) = x2nd FW,α+2n

(
τxα+2nM

−1
n f

)
(λ),

= x2nd Ψλ,α+2n(x)FW,α+2nM
−1
n (f)(λ),

= Φλ,α,n(x)FW,α,n(f(λ)).
which prove (i).
For assertion (ii) using (23) and (45) we obtain∫

Rd
+

τxα,nf(y)g(y)dµα(y) = x
2
d

∫
Rd

+

τxα+2n
(
M−1n f(y)

)(
M−1n g(y)

)
dµα+2n(y),

= x2d

∫
Rd

+

(
M−1n f(y)

)
τxα+2n

(
M−1n g(y)

)
dµα+2n(y),

=

∫
Rd

+

f(y)τxα,ng(y)dµα(y).

which prove (ii).
For the last assertion using (47) we get

f ∗W,α,n g = Mn
[
(M−1n f) ∗W,α+2n (M

−1
n g)

]
using (27) and (42) we get

FW,α,n(f ∗W,α,n g) = FW,α,n ◦Mn
[
(M−1n f) ∗W,α+2n (M

−1
n g)

]
= FW,α+2n ◦M−1n ◦Mn

[
(M−1n f) ∗W,α+2n (M

−1
n g)

]
= FW,α+2n

[
(M−1n f) ∗W,α+2n (M

−1
n g)

]
= FW,α+2n(M−1n f)FW,α+2n(M

−1
n g)

= FW,α,n(f)FW,α,n(g).

References

[1] R. F. Al Subaie and M. A. Mourou, Transmutation Operators Associated with a Bessel Type Operator on The
Half Line and Certain of Their Applications, Tamsui Oxford Journal of Information and Mathematical Sciences,
29(2013), 329-349.

[2] Hassen Ben Mohamed, Nèji Bettaibi, Sidi Hamidou Jah, Sobolev Type Spaces Asociated with the Weinstein
Operator, Int. Journal of Math. Analysis, 5(2011), , 1353-1373.

[3] Hatem Mejjaoli, Ahsaa, Makren Salhi, Uncertainty Principles for the Weinstein transform, Czechoslovak Mathe-
matical Journal, 61(2011), 941-974.

[4] Youssef Othmani and Khalifa Trimèche, Real Paley-Wiener Theorems Associated with the Weinstein Operator,
Mediterr. J. Math. 3(2006), 105-118.

Department of Mathematics, Faculty of Sciences Aïn Chock, University of Hassan II, Casablanca
20100, Morocco

∗Corresponding author

28