Volume 10 (2009), Issue 2, Article 55, 24 pp.
GENERALIZED DUNKL-SOBOLEV SPACES OF EXPONENTIAL TYPE AND
APPLICATIONS
HATEM MEJJAOLI
D EPARTMENT OF M ATHEMATICS
FACULTY OF SCIENCES OF T UNIS
C AMPUS - 1060. T UNIS , T UNISIA .
hatem.mejjaoli@ipest.rnu.tn
Received 22 March, 2008; accepted 23 May, 2009
Communicated by S.S. Dragomir
Dedicated to Khalifa Trimèche.
A BSTRACT. We study the Sobolev spaces of exponential type associated with the Dunkl-Bessel
Laplace operator. Some properties including completeness and the imbedding theorem are
proved. We next introduce a class of symbols of exponential type and the associated pseudodifferential-difference operators, which naturally act on the generalized Dunkl-Sobolev spaces
of exponential type. Finally, using the theory of reproducing kernels, some applications are given
for these spaces.
Key words and phrases: Dunkl operators, Dunkl-Bessel-Laplace operator, Generalized Dunkl-Sobolev spaces of exponential
type, Pseudo differential-difference operators, Reproducing kernels.
2000 Mathematics Subject Classification. Primary 46F15. Secondary 46F12.
1. I NTRODUCTION
The Sobolev space W s,p (Rd ) serves as a very useful tool in the theory of partial differential
equations, which is defined as follows
o
n
s,p
d
0
d
2 ps
p
d
W (R ) = u ∈ S (R ), (1 + ||ξ|| ) F(u) ∈ L (R ) .
In this paper we consider the Dunkl-Bessel Laplace operator 4k,β defined by
1
β≥− ,
2
where 4k is the Dunkl Laplacian on Rd , and Lβ is the Bessel operator on ]0, +∞[. We in∀ x = (x0 , xd+1 ) ∈ Rd ×]0, +∞[,
4k,β = 4k,x0 + Lβ,xd+1 ,
troduce the generalized Dunkl-Sobolev space of exponential type WGs,p
(Rd+1
+ ) by replacing
∗ ,k,β
Thanks to the referee for his suggestions and comments.
090-08
2
H ATEM M EJJAOLI
s
(1 + ||ξ||2 ) p by an exponential weight function defined as follows
n
o
s,p
p
d+1
0
s||ξ||
d+1
WG∗ ,k,β (R+ ) = u ∈ G∗ , e FD,B (u) ∈ Lk,β (R+ ) ,
0
where Lpk,β (Rd+1
+ ) it is the Lebesgue space associated with the Dunkl-Bessel transform and G∗
is the topological dual of the Silva space. We investigate their properties such as the imbedding
theorems and the structure theorems. In fact, the imbedding theorems mean that for s > 0,
d+1
u ∈ WGs,p
(Rd+1
/ | Im z| < s}. For
+ ) can be analytically continued to the set {z ∈ C
∗ ,k,β
the structure theorems we prove that for s > 0, u ∈ WG−s,2
(Rd+1
+ ) can be represented as an
∗ ,k,β
infinite sum of fractional Dunkl-Bessel Laplace operators of square integrable functions g, in
other words,
u=
X sm
m
(−4k,β ) 2 g.
m!
m∈N
We prove also that the generalized Dunkl-Sobolev spaces are stable by multiplication of the
functions of the Silva spaces. As applications on these spaces, we study the action for the class
of pseudo differential-difference operators and we apply the theory of reproducing kernels on
these spaces. We note that special cases include: the classical Sobolev spaces of exponential
type, the Sobolev spaces of exponential types associated with the Weinstein operator and the
Sobolev spaces of exponential type associated with the Dunkl operators.
We conclude this introduction with a summary of the contents of this paper. In Section 2
we recall the harmonic analysis associated with the Dunkl-Bessel Laplace operator which we
need in the sequel. In Section 3 we consider the Silva space G∗ and its dual G∗0 . We study the
action of the Dunkl-Bessel transform on these spaces. Next we prove two structure theorems
for the space G∗0 . We define in Section 4 the generalized Dunkl-Sobolev spaces of exponential type WGs,p
(Rd+1
+ ) and we give their properties. In Section 5 we give two applications on
∗ ,k,β
these spaces. More precisely, in the first application we introduce certain classes of symbols
of exponential type and the associated pseudo-differential-difference operators of exponential
type. We show that these pseudo-differential-difference operators naturally act on the generalized Sobolev spaces of exponential type. In the second, using the theory of reproducing kernels,
some applications are given for these spaces.
2. P RELIMINARIES
In order to establish some basic and standard notations we briefly overview the theory of
Dunkl operators and its relation to harmonic analysis. Main references are [3, 4, 5, 8, 16, 17,
19, 20, 21].
2.1. The Dunkl Operators. Let Rd be the Euclidean space equipped with a scalar product h·, ·i
p
and let ||x|| = hx, xi. For α in Rd \{0}, let σα be the reflection in the hyperplane Hα ⊂ Rd
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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D UNKL -S OBOLEV S PACES
3
orthogonal to α, i.e. for x ∈ Rd ,
σα (x) = x − 2
hα, xi
α.
||α||2
A finite set R ⊂ Rd \{0} is called a root system if R ∩ R α = {α, −α} and σα R = R for all
α ∈ R. For a given root system R, reflections σα , α ∈ R, generate a finite group W ⊂ O(d),
S
called the reflection group associated with R. We fix a β ∈ Rd \ α∈R Hα and define a positive
root system R+ = {α ∈ R | hα, βi > 0}. We normalize each α ∈ R+ as hα, αi = 2. A function
k : R −→ C on R is called a multiplicity function if it is invariant under the action of W . We
introduce the index γ as
γ = γ(k) =
X
k(α).
α∈R+
Throughout this paper, we will assume that k(α) ≥ 0 for all α ∈ R. We denote by ωk the
weight function on Rd given by
ωk (x) =
Y
|hα, xi|2k(α) ,
α∈R+
which is invariant and homogeneous of degree 2γ, and by ck the Mehta-type constant defined
by
Z
−1
2
exp(−||x|| )ωk (x) dx
.
ck =
Rd
We note that Etingof (cf. [6]) has given a derivation of the Mehta-type constant valid for all
finite reflection group.
The Dunkl operators Tj , j = 1, 2, . . . , d, on Rd associated with the positive root system R+
and the multiplicity function k are given by
X
∂f
f (x) − f (σα (x))
Tj f (x) =
(x) +
k(α)αj
,
∂xj
hα,
xi
α∈R
f ∈ C 1 (Rd ).
+
We define the Dunkl-Laplace operator 4k on Rd for f ∈ C 2 (Rd ) by
4k f (x) =
d
X
Tj2 f (x)
j=1
= 4f (x) + 2
X
k(α)
α∈R+
h∇f (x), αi f (x) − f (σα (x))
−
hα, xi
hα, xi2
,
where 4 and ∇ are the usual Euclidean Laplacian and nabla operators on Rd respectively. Then
for each y ∈ Rd , the system
(
Tj u(x, y) = yj u(x, y),
j = 1, . . . , d,
u(0, y) = 1
admits a unique analytic solution K(x, y), x ∈ Rd , called the Dunkl kernel. This kernel has a
holomorphic extension to Cd × Cd , (cf. [17] for the basic properties of K).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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H ATEM M EJJAOLI
2.2. Harmonic Analysis Associated with the Dunkl-Bessel Laplace Operator. In this subsection we collect some notations and results on the Dunkl-Bessel kernel, the Dunkl-Bessel
intertwining operator and its dual, the Dunkl-Bessel transform, and the Dunkl-Bessel convolution (cf. [12]).
In the following we denote by
• Rd+1
= Rd × [0, +∞[.
+
• x = (x1 , . . . , xd , xd+1 ) = (x0 , xd+1 ) ∈ Rd+1
+ .
d+1
• C∗ (R ) the space of continuous functions on Rd+1 , 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.
• D∗ (Rd+1 ) the space of C ∞ -functions on Rd+1 which are of compact support, even
with respect to the last variable.
• S 0 ∗ (Rd+1 ) the space of temperate distributions on Rd+1 , even with respect to the last
variable. It is the topological dual of S∗ (Rd+1 ).
We consider the Dunkl-Bessel Laplace operator 4k,β defined by
∀ x = (x0 , xd+1 ) ∈ Rd ×]0, +∞[,
(2.1)
4k,β f (x) = 4k,x0 f (x0 , xd+1 ) + Lβ,xd+1 f (x0 , xd+1 ), f ∈ C∗2 (Rd+1 ),
where 4k is the Dunkl-Laplace operator on Rd , and Lβ the Bessel operator on ]0, +∞[ given
by
d2
2β + 1 d
+
,
2
dxd+1
xd+1 dxd+1
The Dunkl-Bessel kernel Λ is given by
Lβ =
(2.2)
Λ(x, z) = K(ix0 , z 0 )jβ (xd+1 zd+1 ),
1
β>− .
2
(x, z) ∈ Rd+1 × Cd+1 ,
where K(ix0 , z 0 ) is the Dunkl kernel and jβ (xd+1 zd+1 ) is the normalized Bessel function. The
Dunkl-Bessel kernel satisfies the following properties:
i) For all z, t ∈ Cd+1 , we have
(2.3)
Λ(z, t) = Λ(t, z);
Λ(z, 0) = 1
and
for all λ ∈ C.
Λ(λz, t) = Λ(z, λt),
ii) For all ν ∈ Nd+1 , x ∈ Rd+1 and z ∈ Cd+1 , we have
|Dzν Λ(x, z)| ≤ ||x|||ν| exp(||x|| || Im z||),
(2.4)
where Dzν =
(2.5)
∂ν
νd+1
ν1
∂z1 ···∂zd+1
and |ν| = ν1 + · · · + νd+1 . In particular
|Λ(x, y)| ≤ 1,
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
for all x, y ∈ Rd+1 .
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D UNKL -S OBOLEV S PACES
5
The Dunkl-Bessel intertwining operator is the operator Rk,β defined on C∗ (Rd+1 ) by
(2.6) Rk,β f (x0 , xd+1 )

Z xd+1
1

 √2Γ(β + 1) x−2β
(x2d+1 − t2 )β− 2 Vk f (x0 , t)dt,
d+1
1
πΓ(β + 2 )
0
=


f (x0 , 0),
xd+1 > 0,
xd+1 = 0,
where Vk is the Dunkl intertwining operator (cf. [16]).
Rk,β is a topological isomorphism from E∗ (Rd+1 ) onto itself satisfying the following transmutation relation
for all f ∈ E∗ (Rd+1 ),
4k,β (Rk,β f ) = Rk,β (4d+1 f ),
(2.7)
where 4d+1 =
d+1
P
∂j2 is the Laplacian on Rd+1 .
j=1
The dual of the Dunkl-Bessel intertwining operator Rk,β is the operator t Rk,β defined on
D∗ (Rd+1 ) by: ∀y = (y 0 , yd+1 ) ∈ Rd × [0, ∞[,
t
(2.8)
2Γ(β + 1)
Rk,β (f )(y , yd+1 ) = √
πΓ β + 12
0
Z
∞
1
2
(s2 − yd+1
)β− 2 t Vk f (y 0 , s)sds,
yd+1
t
where Vk is the dual Dunkl intertwining operator (cf. [20]).
t
Rk,β is a topological isomorphism from S∗ (Rd+1 ) onto itself satisfying the following transmutation relation
t
(2.9)
Rk,β (4k,β f ) = 4d+1 ( t Rk,β f ),
for all f ∈ S∗ (Rd+1 ).
d+1
We denote by Lpk,β (Rd+1
such that
+ ) the space of measurable functions on R+
! p1
Z
|f (x)|p dµk,β (x) dx
||f ||Lp
=
||f ||L∞
= ess sup |f (x)| < +∞,
d+1
k,β (R+ )
d+1
k,β (R+ )
Rd+1
+
< +∞,
if
1 ≤ p < +∞,
x∈Rd+1
+
where dµk,β is the measure on Rd+1
given by
+
0
dµk,β (x0 , xd+1 ) = ωk (x0 )x2β+1
d+1 dx dxd+1 .
The Dunkl-Bessel transform is given for f in L1k,β (Rd+1
+ ) by
Z
0
(2.10)
FD,B (f )(y , yd+1 ) =
f (x0 , xd+1 )Λ(−x, y)dµk,β (x),
Rd+1
+
for all y = (y 0 , yd+1 ) ∈ Rd+1
+ .
Some basic properties of this transform are the following:
i) For f in L1k,β (Rd+1
+ ),
(2.11)
||FD,B (f )||L∞
d+1
k,β (R+ )
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
≤ ||f ||L1
d+1
k,β (R+ )
.
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6
H ATEM M EJJAOLI
ii) For f in S∗ (Rd+1 ) we have
FD,B (4k,β f )(y) = −||y||2 FD,B (f )(y),
(2.12)
for all y ∈ Rd+1
+ .
iii) For all f ∈ S(Rd+1
∗ ), we have
FD,B (f )(y) = Fo ◦ t Rk,β (f )(y),
(2.13)
for all y ∈ Rd+1
+ ,
where Fo is the transform defined by: ∀ y ∈ Rd+1
+ ,
Z
0 0
(2.14)
Fo (f )(y) =
f (x)e−ihy ,x i cos(xd+1 yd+1 )dx,
Rd+1
+
f ∈ D∗ (Rd+1 ).
d+1
1
iv) For all f in L1k,β (Rd+1
+ ), if FD,B (f ) belongs to Lk,β (R+ ), then
Z
(2.15)
f (y) = mk,β
FD,B (f )(x)Λ(x, y)dµk,β (x), a.e.
Rd+1
+
where
mk,β =
(2.16)
c2k
d
4γ+β+ 2 (Γ(β + 1))2
.
v) For f ∈ S∗ (Rd+1 ), if we define
FD,B (f )(y) = FD,B (f )(−y),
then
FD,B FD,B = FD,B FD = mk,β Id.
(2.17)
Proposition 2.1.
i) The Dunkl-Bessel transform FD,B is a topological isomorphism from S∗ (Rd+1 ) onto
itself and for all f in S∗ (Rd+1 ),
Z
Z
2
(2.18)
|f (x)| dµk,β (x) = mk,β
Rd+1
+
Rd+1
+
|FD,B (f )(ξ)|2 dµk,β (ξ).
1
2
ii) In particular, the renormalized Dunkl-Bessel transform f → mk,β
FD,B (f ) can be
uniquely extended to an isometric isomorphism on L2k,β (Rd+1
+ ).
By using the Dunkl-Bessel kernel, we introduce a generalized translation and a convolution
structure. For a function f ∈ S∗ (Rd+1 ) and y ∈ Rd+1
the Dunkl-Bessel translation τy f is
+
defined by the following relation:
FD,B (τy f )(x) = Λ(x, y)FD,B (f )(x).
If f ∈ E∗ (Rd+1 ) is radial with respect to the d first variables, i.e. f (x) = F (||x0 ||, xd+1 ), then it
follows that
q
h p
i
2
2
0
2
0
2
0
(2.19) τy f (x) = Rk,β F
||x || + ||y || + 2hy , ·i, xd+1 + yd+1 + 2yd+1 (x0 , xd+1 ).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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D UNKL -S OBOLEV S PACES
7
By using the Dunkl-Bessel translation, we define the Dunkl-Bessel convolution product f ∗D,B g
of functions f, g ∈ S∗ (Rd+1 ) as follows:
Z
f ∗D,B g(x) =
(2.20)
Rd+1
+
τx f (−y)g(y)dµk,β (y).
This convolution is commutative and associative and satisfies the following:
d+1
i) For all f, g ∈ S∗ (Rd+1
+ ), f ∗D,B g belongs to S∗ (R+ ) and
FD,B (f ∗D,B g)(y) = FD,B (f )(y)FD,B (g)(y).
(2.21)
1
+ 1q − 1r
p
Lrk,β (Rd+1
+ ) and
ii) Let 1 ≤ p, q, r ≤ ∞ such that
radial, then f ∗D,B g ∈
kf ∗D,B gkLr
(2.22)
d+1
k,β (R+ )
q
d+1
= 1. If f ∈ Lpk,β (Rd+1
+ ) and g ∈ Lk,β (R+ ) is
≤ kf kLp
d+1
k,β (R+ )
kgkLq
d+1
k,β (R+ )
3. S TRUCTURE T HEOREMS ON THE S ILVA S PACE
.
AND ITS
D UAL
Definition 3.1. We denote by G∗ or G∗ (Rd+1 ) the set of all functions ϕ in E∗ (Rd+1 ) such that
for any h, p > 0
Np,h (ϕ) = sup
x∈Rd+1
µ∈Nd+1
ep||x|| |∂ µ ϕ(x)|
h|µ| µ!
is finite. The topology in G∗ is defined by the above seminorms.
Lemma 3.1. Let φ be in G∗ . Then for every h, p > 0
Np,h (ϕ) = sup
x∈Rd+1
m∈N
ep||x|| |4m
k,β ϕ(x)|
hm m!
!
.
Proof. We proceed as in Proposition 5.1 of [13], and by a simple calculation we obtain the
result.
Theorem 3.2. The transform FD,B is a topological isomorphism from G∗ onto itself.
Proof. From the relations (2.8), (2.9) and Lemma 3.1 we see that t Rk,β is continuous from
G∗ onto itself. On the other hand, J. Chung et al. [1] have proved that the classical Fourier
transform is an isomorphism from G∗ onto itself. Thus from the relation (2.13) we deduce that
FD,B is continuous from G∗ onto itself. Finally since G∗ is included in S∗ (Rd+1 ), and FD,B is
an isomorphism from S∗ (Rd+1 ) onto itself, by (2.17) we obtain the result.
We denote by G∗0 or G∗0 (Rd+1 ) the strong dual of the space G∗ .
Definition 3.2. The Dunkl-Bessel transform of a distribution S in G∗0 is defined by
hFD,B (S), ψi = hS, FD,B (ψ)i,
ψ ∈ G∗ .
The result below follows immediately from Theorem 3.2.
Corollary 3.3. The transform FD,B is a topological isomorphism from G∗0 onto itself.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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H ATEM M EJJAOLI
Let τ be in G∗0 . We define 4k,β τ , by
h4k,β τ, ψi = hτ, 4k,β ψi,
for all ψ ∈ G∗ .
This functional satisfies the following property
FD,B (4k,β τ ) = −||y||2 FD,B (τ ).
(3.1)
Definition 3.3. The generalized heat kernel Γk,β is given by
Γk,β (t, x, y) :=
2ck
−
d
Γ(β + 1)(4t)γ+β+ 2 +1
e
||x||2 +||y||2
4t
x
y
Λ −i √ , √
2t 2t
;
x, y ∈ Rd+1
+ , t > 0.
The generalized heat kernel Γk,β has the following properties:
Proposition 3.4. Let x, y in Rd+1
and t > 0. Then we have:
+
Z
i) Γk,β (t, x, y) =
exp(−t||ξ||2 )Λ(x, ξ)Λ(−y, ξ)dµk,β (ξ).
Rd+1
+
Z
ii)
Rd+1
+
Γk,β (t, x, y)dµk,β (x) = 1.
iii) For fixed y in Rd+1
+ , the function u(x, t) := Γk,β (t, x, y) solves the generalized heat
equation:
4k,β u(x, t) =
∂
u(x, t)
∂t
on
Rd+1
+ ×]0, +∞[.
Definition 3.4. The generalized heat semigroup (Hk,β (t))t≥0 is the integral operator given for
f in L2k,β (Rd+1
+ ) by
Hk,β (t)f (x) :=
 Z


Rd+1
+


Γk,β (t, x, y)f (y)dµk,β (y) if t > 0,
f (x)
if t = 0.
From the properties of the generalized heat kernel we have
(
f ∗D,B pt (x) if t > 0,
(3.2)
Hk,β (t)f (x) :=
f (x)
if t = 0,
where
pt (y) =
2ck
d
Γ(β + 1)(4t)γ+β+ 2 +1
e−
||y||2
4t
.
Definition 3.5. A function f defined on Rd+1
is said to be of exponential type if there are
+
d+1
constants k, C > 0 such that for every x ∈ R+
|f (x)| ≤ exp k||x||.
The following lemma will be useful later. For the details of the proof we refer to Komatsu
[9]:
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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D UNKL -S OBOLEV S PACES
9
Lemma 3.5. For any L > 0 and ε > 0 there exist a function v ∈ D(R) and a differential
operator P ( dtd ) of infinite order such that
(3.3)
supp v ⊂ [0, ε],
(3.4)
L
|v(t)| ≤ C exp −
t
P
d
dt
=
∞
X
ak
k=0
d
dt
P
(3.5)
k
d
dt
,
|ak | ≤ C1
,
Lk1
,
k!2
0 < t < ∞;
0 < L1 < L;
v(t) = δ + ω(t),
where ω ∈ D(R), supp ω ⊂ [ 2ε , ε].
Here we note that P (4k,β ) is a local operator where 4k,β is a Dunkl-Bessel Laplace operator.
Now we are in a position to state and prove one of the main theorems in this section.
Theorem 3.6. If u ∈ G∗0 then there exists a differential operator P ( dtd ) such that for some C > 0
and L > 0,
P
d
dt
=
∞
X
an
n=0
d
dt
n
,
|an | ≤ C
Ln
,
n!2
and there are a continuous function g of exponential type and an entire function h of exponential
type in Rd+1
such that
+
u = P (4k,β )g(x) + h(x).
(3.6)
Proof. Let U (x, t) = huy , Γk,β (t, x, y)i. Since pt belongs to G∗ for each t > 0, U (x, t) is well
defined and real analytic in (Rd+1
+ )x for each t > 0.
Furthermore, U (x, t) satisfies
(3.7)
(∂t − 4k,β )U (x, t) = 0
for
(x, t) ∈ Rd+1
+ ×]0, ∞[.
Here u ∈ G∗0 means that for some k > 0 and h > 0
(3.8)
|∂ α φ(x)| exp k||x||
|hu, φi| ≤ C sup
,
h|α| α!
x,α
φ ∈ G∗ .
By Cauchy’s inequality and relations (2.4) and (3.8) we obtain, for t > 0
1
0
0
|U (x, t)| ≤ C exp k ||x|| + t +
t
for some C 0 > 0 and k 0 > 0. If we restrict this inequality on the strip 0 < t < ε then it follows
that
1
|U (x, t)| ≤ C exp k ||x|| +
, 0<t<ε
t
for some constants C > 0 and k > 0. Now let
Z
G(y, t) =
Γk,β (t, x, y)φ(x)dµk,β (x), φ ∈ G∗ .
Rd+1
+
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10
H ATEM M EJJAOLI
Moreover, we can easily see that
G(·, t) → φ
(3.9)
in
G∗ as t → 0+
and
Z
(3.10)
Rd+1
+
U (x, t)φ(x)dµk,β (x) = huy , G(y, t)i.
Then it follows from (3.9) and (3.10) that
in G∗0 .
lim+ U (x, t) = u
(3.11)
t→0
Now choose functions u, w and a differential operator of infinite order as in Lemma 3.5. Let
Z ∞
(3.12)
Ũ (x, t) =
U (x, t + s)v(s)ds.
0
Then by taking ε = 2 and L > k in Lemma 3.5 we have
|Ũ (x, t)| ≤ C 0 exp k ||x|| + t ,
(3.13)
t ≥ 0.
Therefore, Ũ (x, t) is a continuous function of exponential type in
n
o
d+1
Rd+1
×
[0,
∞[=
(x,
t)
:
x
∈
R
,
t
≥
0
.
+
+
Moreover, Ũ satisfies
(∂t − 4k,β )Ũ (x, t) = 0
(3.14)
in
Rd+1
+ ×]0, ∞[.
Hence if we set g(x) = Ũ (x, 0) then g is also a continuous function of exponential type, so that
g belongs to G∗0 .
Using (3.5) in Lemma 3.5, we obtain for t > 0
d
Ũ (x, t)
P (−4k,β )Ũ (x, t) = P −
dt
Z ∞
(3.15)
= U (x, t) +
U (x, t + s)w(s)ds.
0
If we set h(x) = −
R∞
0
U (x, s)w(s)ds then h is an entire function of exponential type. As
t → 0+ , (3.15) becomes
u = P (−4k,β )g(x) + h(x)
which completes the proof by replacing the coefficients an of P by (−1)n an .
Theorem 3.7. Let U (x, t) be an infinitely differentiable function in Rd+1
+ ×]0, ∞[ satisfying the
conditions:
i) (∂t − 4k,β )U (x, t) = 0 in Rd+1
+ ×]0, ∞[.
ii) There exist k > 0 and C > 0 such that
1
(3.16)
|U (x, t)| ≤ C exp k ||x|| +
, 0 < t < ε,
t
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
x ∈ Rd+1
+
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D UNKL -S OBOLEV S PACES
11
for some ε > 0. Then there exists a unique element u ∈ G∗0 such that
U (x, t) = huy , Γk,β (t, x, y)i,
t>0
and
lim+ U (x, t) = u
t→0
in G∗0 .
Proof. Consider a function, as in (3.12)
∞
Z
Ũ (x, t) =
U (x, t + s)v(s)ds.
0
Then it follows from (3.13), (3.14) and (3.15) that Ũ (x, t) and H(x, t) are continuous on
Rd+1
+ × [0, ∞[ and
U (x, t) = P (−4k,β )Ũ (x, t) + H(x, t),
(3.17)
where
Z
∞
H(x, t) = −
U (x, t + s)w(s)ds.
0
Furthermore, g(x) = Ũ (x, 0) and h(x) = H(x, 0) are continuous functions of exponential type
on Rd+1
+ . Define u as
u = P (−4k,β )g(x) + h(x).
Then since P (−4k,β ) is a local operator, u belongs to G∗0 and
lim U (x, t) = u
t→0+
in
G∗0 .
Now define the generalized heat kernels for t > 0 as
Z
A(x, t) = (g ∗D,B pt )(x) =
g(y)Γk,β (t, x, y)dµk,β (y)
Rd+1
+
and
Z
B(x, t) = (h ∗D,B pt )(x) =
Rd+1
+
h(y)Γk,β (t, x, y)dµk,β (y).
Then it is easy to show that A(x, t) and B(x, t) converge locally uniformly to g(x) and h(x)
respectively so that they are continuous on Rd+1
+ × [0, ∞[, A(x, 0) = g(x), and B(x, 0) = h(x).
Now let V (x, t) = Ũ (x, t) − A(x, t) and W (x, t) = H(x, t) − B(x, t). Then, since g and h
are of exponential type, V (·, t) and W (·, t) are continuous functions of exponential type and
V (x, 0) = 0, W (x, 0) = 0. Then by the uniqueness theorem of the generalized heat equations
we obtain that
Ũ (x, t) = g ∗D,B pt (x)
and
H(x, t) = h ∗D,B pt (x).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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12
H ATEM M EJJAOLI
It follows from these facts and (3.17) that
h
i
u ∗D,B pt = P (−4k,β )g + h ∗D,B pt
= P (−4k,β )Ũ (·, t) + H(·, t)
= U (·, t).
Now to prove the uniqueness of existence of such u ∈ G∗0 we assume that there exist u, v ∈ G∗0
such that
U (x, t) = u ∗D,B pt (x) = v ∗D,B pt (x).
Then
FD,B (u)FD,B (pt ) = FD,B (v)FD,B (pt )
which implies that FD,B (u) = FD,B (v), since FD,B (pt ) 6= 0. However, since the Dunkl-Bessel
transformation is an isomorphism we have u = v, which completes the proof.
4. T HE G ENERALIZED D UNKL -S OBOLEV S PACES OF E XPONENTIAL T YPE
Definition 4.1. Let s be in R, 1 ≤ p < ∞. We define the space WGs,p
(Rd+1
+ ) by
∗ ,k,β
n
o
u ∈ G∗0 : es||ξ|| FD,B (u) ∈ Lpk,β (Rd+1
+ ) .
The norm on WGs,p
(Rd+1
+ ) is given by
∗ ,k,β
Z
||u||WGs,p,k,β =
∗
mk,β
Rd+1
+
! p1
eps||ξ|| |FD,B (u)(ξ)|p dµk,β (ξ)
.
For p = 2 we provide this space with the scalar product
Z
(4.1)
hu, viW s,2 = mk,β
e2s||ξ|| FD,B (u)(ξ)FD,B (v)(ξ)dµk,β (ξ),
G∗ ,k,β
Rd+1
+
and the norm
||u||2W s,2
(4.2)
G∗ ,k,β
= hu, uiW s,2
G∗ ,k,β
.
Proposition 4.1.
i) Let 1 ≤ p < +∞. The space WGs,p
(Rd+1
+ ) provided with the norm k·kWGs,p,k,β is a
∗ ,k,β
∗
Banach space.
ii) We have
2
d+1
WG0,2
(Rd+1
+ ) = Lk,β (R+ ).
∗ ,k,β
iii) Let 1 ≤ p < +∞ and s1 , s2 in R such that s1 ≥ s2 then
,p
s2 ,p
d+1
WGs∗1,k,β
(Rd+1
+ ) ,→ WG∗ ,k,β (R+ ).
ps||ξ||
Proof. i) It is clear that the space Lp (Rd+1
dµk,β (ξ)) is complete and since FD,B is an
+ ,e
isomorphism from G∗0 onto itself, WGs,p
(Rd+1
+ ) is then a Banach space.
∗ ,k,β
The results ii) and iii) follow immediately from the definition of the generalized DunklSobolev space of exponential type.
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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D UNKL -S OBOLEV S PACES
13
Proposition 4.2. Let 1 ≤ p < +∞, and s1 , s, s2 be three real numbers satisfying s1 < s < s2 .
Then, for all ε > 0, there exists a nonnegative constant Cε such that for all u in WGs,p
(Rd+1
+ )
∗ ,k,β
(4.3)
,p
,p .
||u||WGs,p,k,β ≤ Cε ||u||WGs1 ,k,β
+ ε||u||WGs2 ,k,β
∗
∗
∗
Proof. We consider s = (1 − t)s1 + ts2 , (with t ∈]0, 1[). Moreover it is easy to see
t
s ,p .
||u||WGs,p,k,β ≤ ||u||1−t
s ,p ||u||
W 2
W 1
∗
G∗ ,k,β
G∗ ,k,β
Thus
1−t t
t
,p
,p
||u||WGs,p,k,β ≤ ε− 1−t ||u||WGs1 ,k,β
ε||u||WGs2 ,k,β
∗
∗
≤ε
t
− 1−t
∗
,p
,p .
||u||WGs1 ,k,β
+ ε||u||WGs2 ,k,β
∗
Hence the proof is completed for Cε = ε
t
− 1−t
∗
.
Proposition 4.3. For s in R, 1 ≤ p < ∞ and m in N, the operator 4m
k,β is continuous from
s−ε,p
d+1
WGs,p
(Rd+1
+ ) into WG∗ ,k,β (R+ ) for any ε > 0.
∗ ,k,β
Proof. Let u be in WGs,p
(Rd+1
+ ), and m in N. From (3.1) we have
∗ ,k,β
Z
Z
p(s−ε)||ξ||
m
p
e
|FD,B (4k,β u)(ξ)| dµk,β (ξ) =
ep(s−ε)||ξ|| ||ξ||2mp |FD,B (u)(ξ)|p dµk,β (ξ).
Rd+1
+
Rd+1
+
As supξ∈Rd+1 ||ξ||2mp e−ε||ξ|| < ∞, for every m ∈ N and ε > 0
+
Z
Z
p(s−ε)||ξ||
m
p
e
|FD,B (4k,β u)(ξ)| dµk,β (ξ) ≤
eps||ξ|| |FD,B (u)(ξ)|p dµk,β (ξ) < +∞.
Rd+1
+
Rd+1
+
s−ε,p
d+1
Then 4m
k,β u belongs to WG∗ ,k,β (R+ ), and
s,p
||4m
k,β u||W s−ε,p ≤ ||u||WG ,k,β .
∗
G∗ ,k,β
1
Definition 4.2. We define the operator (−4k,β ) 2 by ∀x ∈ Rd+1
+ ,
Z
1
(−4k,β ) 2 u(x) = mk,β
Λ(x, ξ)||ξ||FD,B (u)(ξ)dµk,β (ξ),
Rd+1
+
1
Proposition 4.4. Let P ((−4k,β ) 2 ) =
u ∈ S∗ (Rd+1 ).
m
P
m∈N
am (−4k,β ) 2 be a fractional Dunkl-Bessel Laplace
operators of infinite order such that there exist positive constants C and r such that
(4.4)
|am | ≤ C
rm
,
m!
for all m ∈ N.
1
Let 1 ≤ p < +∞ and s in R. If an element u is in WGs,p
(Rd+1
+ ), then P ((−4k,β ) 2 )u belongs
∗ ,k,β
s−r,p
d+1
to WG∗ ,k,β (R+ ), and there exists a positive constant C such that
1
||P ((−4k,β ) 2 )u||W s−r,p ≤ C||u||WGs,p,k,β .
G∗ ,k,β
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
∗
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14
H ATEM M EJJAOLI
Proof. The condition (4.4) gives that
|P (||ξ||)| ≤ C exp(r ||ξ||).
Thus the result is immediate.
1
Proposition 4.5. Let 1 ≤ p < +∞, t, s in R. The operator exp(t(−4k,β ) 2 ) defined by
Z
1
d+1
2
∀x ∈ R+ , exp(t(−4k,β ) )u(x) = mk,β
Λ(x, ξ)et||ξ|| FD,B (u)(ξ)dµk,β (ξ), u ∈ G∗ .
Rd+1
+
s−t,p
d+1
is an isomorphism from WGs,p
(Rd+1
+ ) onto WG∗ ,k,β (R+ ).
∗ ,k,β
Proof. For u in WGs,p
(Rd+1
+ ) it is easy to see that
∗ ,k,β
1
k exp(t(−4k,β ) 2 )ukW s−t,p = ||u||WGs,p,k,β ,
∗
G∗ ,k,β
and thus we obtain the result.
−s,2
d+1
Proposition 4.6. The dual of WGs,2
(Rd+1
+ ) can be identified as WG∗ ,k,β (R+ ). The relation of
∗ ,k,β
the identification is given by
Z
(4.5)
hu, vik,β = mk,β
Rd+1
+
FD,B (u)(ξ)FD,B (v)(ξ)dµk,β (ξ),
−s,2
d+1
with u in WGs,2
(Rd+1
+ ) and v in WG∗ ,k,β (R+ ).
∗ ,k,β
−s,2
d+1
Proof. Let u be in WGs,2
(Rd+1
+ ) and v in WG∗ ,k,β (R+ ). The Cauchy-Schwartz inequality
∗ ,k,β
gives
|hu, vik,β | ≤ ||u||W s,2
G∗ ,k,β
||v||W −s,2 .
G∗ ,k,β
WG−s,2
(Rd+1
+ )
∗ ,k,β
fixed we see that u 7−→ hu, vik,β is a continuous linear form on
s,2
d+1
−1
WG∗ ,k,β (R+ ) whose norm does not exceed ||v||W −s,2 . Taking the u0 = FD,B
e−2s||ξ|| FD,B (v)
G∗ ,k,β
d+1
element of WGs,2
(R
),
we
obtain
hu
,
vi
=
||v||W −s,2 .
0
k,β
+
∗ ,k,β
G ,k,β
Thus for v in
∗
Thus the norm of u 7−→ hu, vik,β is equal to ||v||W −s,2 , and we have then an isometry:
G∗ ,k,β
WG−s,2
R+
∗ ,k,β
d+1
d+1 0
−→ WGs,2
(R
)
.
+
∗ ,k,β
0
Conversely, let L be in WGs,2
(Rd+1
+ ) . By the Riesz representation theorem and (4.1) there
∗ ,k,β
exists w in WGs,2
(Rd+1
+ ) such that
∗ ,k,β
L(u) = hu, wiW s,2
Z G∗ ,k,β
= mk,β
e2s||ξ|| FD,B (u)(ξ)FD,B (w)(ξ)dµk,β (ξ),
Rd+1
+
for all u ∈ WGs,2
(Rd+1
+ ).
∗ ,k,β
If we set v = FD−1 e2s||ξ|| FD,B (w) then v belongs to WG−s,2
(Rd+1
+ ) and L(u) = hu, vik,β for
∗ ,k,β
all u in WGs,2
(Rd+1
+ ), which completes the proof.
∗ ,k,β
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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D UNKL -S OBOLEV S PACES
15
Proposition 4.7. Let s > 0. Then every u in WG−s,2
(Rd+1
+ ) can be represented as an infinite sum
∗ ,k,β
of fractional Dunkl-Bessel Laplace operators of square integrable function g, in other words,
X sm
m
(−4k,β ) 2 g.
u=
m!
m∈N
Proof. If u in WG−s,2
(Rd+1
+ ) then by definition
∗ ,k,β
e−s||ξ|| FD,B (u)(ξ) ∈ L2k,β (Rd+1
+ ),
which Proposition 2.1 ii) implies that
FD,B (u)(ξ)
FD,B (g)(ξ) = P
∈ L2k,β (Rd+1
+ ).
sm
m
||ξ||
m∈N m!
Hence, we have
X sm
||ξ||m FD,B (g), in G∗0
m!
m∈N
X sm
m =
FD,B (−4k,β ) 2 g , in
m!
m∈N
FD,B (u) =
G∗0
This completes the proof.
Proposition 4.8. Let 1 ≤ p < +∞. Every u in WGs,p
(Rd+1
+ ) is a holomorphic function in the
∗ ,k,β
strip {z ∈ Cd+1 , || Im z|| < s} for s > 0.
Proof. Let
Z
u(z) = mk,β
Rd+1
+
Λ(z, ξ)FD,B (u)(ξ)dµk,β (ξ),
z = x + iy.
From (2.4), for each µ in Nd+1 we have
µ
Dz Λ(z, ξ)FD,B (u)(ξ) ≤ ||ξ|||µ| e||y|| ||ξ|| |FD,B (u)(ξ)|.
On the other hand from the Cauchy-Schwartz inequality we have
Z
Rd+1
+
||ξ µ ||e||y|| ||ξ|| |FD,B (u)(ξ)|dµk,β (ξ) ≤
Z
Rd+1
+
! 1q
||ξ||q|µ| eq||ξ||(||y||−s) dµk,β (ξ)
||u||WGs,p,k,β .
∗
Since the integral in the last part of the above inequality is integrable if ||y|| < s, the result
follows by the theorem of holomorphy under the integral sign.
Notations. Let m be in N. We denote by:
d+1
0
• Em
(Rd+1
with compact support and order less than
+ ) the space of distributions on R+
or equal to m.
d+1
0
0
• Eexp,m
(Rd+1
+ ) the space of distributions u in Em (R+ ) such that there exists a positive
constant C such that
|FD,B (u)(ξ)| ≤ Cem||ξ|| .
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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16
H ATEM M EJJAOLI
Proposition 4.9.
i) Let 1 ≤ p < +∞. For s in R such that s < −m, we have
s,p
0
d+1
Eexp,m
(Rd+1
+ ) ⊂ WG∗ ,k,β (R+ ).
ii) Let 1 ≤ p < +∞. We have, for s < 0 and m an integer,
s,p
0
d+1
Em
(Rd+1
+ ) ⊂ WG∗ ,k,β (R+ ).
Proof. The proof uses the same idea as Theorem 3.12 of [13].
Theorem 4.10. Let ψ be in G∗ . For all s in R, the mapping u 7→ ψu from WGs,2
(Rd+1
+ ) into
∗ ,k,β
itself is continuous.
Proof. Firstly we assume that s > 0.
It is easy to see that
||v||2W s,2
G∗ ,k,β
where k·kḢ s
d+1
k,β (R+ )
≤
∞
X
(2s)j
j!
j=0
||v||2
j
2 (Rd+1 )
Ḣk,β
+
,
designates the norm associated to the homogeneous Dunkl-Bessel-Sobolev
space defined by
||v||2Ḣ s (Rd+1 )
+
k,β
Z
=
Rd+1
+
||ξ||2s |FD,B (v)(ξ)|2 dµk,β (ξ).
On the other hand, proceeding as in Proposition 4.1 of [14] we prove that if u, v belong to
T ∞ d+1
s
s
Ḣk,β
(Rd+1
Lk,β (R+ ), s > 0 then uv ∈ Ḣk,β
(Rd+1
+ )
+ ) and
h
i
||uv||Ḣ s (Rd+1 ) ≤ C ||u||L∞ (Rd+1 ) ||v||Ḣ s (Rd+1 ) + ||v||L∞ (Rd+1 ) ||u||Ḣ s (Rd+1 ) .
k,β
+
+
k,β
k,β
Thus from this we deduce that for s > 0
∞
X
(2s)j
2
||ϕu||W s,2 ≤
||ϕu||2 j
2 (Rd+1 )
G∗ ,k,β
j!
Ḣk,β
+
j=0
h
≤ C ||ϕ||2L∞ (Rd+1 ) ||u||2W s,2
k,β
+
G∗ ,k,β
+
k,β
+ ||u||2L∞
k,β
+
k,β
||ϕ||2W s,2
(Rd+1 )
+
i
+
< +∞.
G∗ ,k,β
For s = 0 the result is immediate.
For s < 0 the result is obtained by duality.
5. A PPLICATIONS
5.1. Pseudo-differential-difference operators of exponential type. The pseudo-differentialdifference operator A(x, 4k,β ) associated with the symbol a(x, ξ) := A(x, −||ξ||2 ) is defined
by
(5.1)
A(x, 4k,β )u (x) = mk,β
Z
Rd+1
+
Λ(x, ξ)a(x, ξ)FD,B (u)(ξ)dµk,β (ξ), u ∈ G∗ ,
r
where a(x, ξ) belongs to the class Sexp
, r ≥ 0, defined below:
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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D UNKL -S OBOLEV S PACES
17
r
Definition 5.1. The function a(x, ξ) is said to be in Sexp
if and only if a(x, ξ) belongs to
C ∞ (Rd+1 × Rd+1 ) and for each compact set K ⊂ Rd+1
and each µ, ν in Nd+1 , there exists
+
a constant CK = Cµ,ν,K such that the estimate
|Dξµ Dxν a(x, ξ)| ≤ CK exp(r ||ξ||),
for all
(x, ξ) ∈ K × Rd+1
+
hold true.
Proposition 5.1. Let A(x, 4k,β ) be the pseudo-differential-difference operator associated with
r
the symbol a(x, ξ) := A(x, −||ξ||2 ). If a(x, ξ) ∈ Sexp
then A(x, 4k,β ) in (5.1) is a well-defined
mapping of G∗ into E∗ (Rd+1 ).
Proof. For any compact set K ⊂ Rd+1
+ , we have
|a(x, ξ)| ≤ CK exp(r ||ξ||), for all (x, ξ) ∈ K × Rd+1
+ .
On the other hand since u is in G∗ the Cauchy-Schwartz inequality gives that
! 12
Z
Z
|Λ(x, ξ)a(x, ξ)FD,B (u)(ξ)|dµk,β (ξ) ≤ CK ||u||W s,2
e−2(s−r)||ξ|| dµk,β (ξ)
G,k,β
Rd+1
+
Rd+1
+
is integrable for s > r. This prove the existence and the continuity of (A(x, 4k,β )u)(x) for all
x in Rd+1
+ . Finally the result follows by using Leibniz formula.
r,l
Now we consider the symbol which belongs to the class Sexp,rad
defined below:
Definition 5.2. Let r, l in R be real numbers with l > 0. The function a(x, ξ) is said to be
r,l
in Sexp,rad
if and only if a(x, ξ) is in C ∞ (Rd+1 × Rd+1 ), radial with respect to the first d + 1
variables and for each L > 0, and for each µ, ν in Nd+1 , there exists a constant C = Cr,µ,ν such
that the estimate
|Dξµ Dxν a(x, ξ)| ≤ CL|µ| µ! exp(r ||ξ|| − l ||x||)
hold true.
To obtain some deep and interesting results we need the following alternative form of A(x, 4k,β ).
Lemma 5.2. Let A(x, 4k,β ) be the pseudo-differential-difference operator associated with the
r,l
symbol a(x, ξ) := A(x, −||ξ||2 ). If a(x, ξ) is in Sexp,rad
then A(x, 4k,β ) in (5.1) is given by:
A(x, 4k,β )u (x)
"Z
#
Z
= mk,β
Λ(x, ξ)
τ−η FD,B (a)(·, η) (ξ)FD,B (u)(η)dµk,β (η) dµk,β (ξ)
Rd+1
+
Rd+1
+
for all u ∈ G∗ where all the involved integrals are absolutely convergent.
Proof. When we proceed as [10] we see that for all L > 0 there exist C > 0 and 0 < τ <
(Ld)−1 such that
(5.2)
|FD,B (a)(ξ, η)| ≤ C exp(r ||η|| − τ ||ξ||).
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
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18
H ATEM M EJJAOLI
On the other hand since u in G∗ , we have
|FD,B (u)(η)| ≤ C exp(−t ||η||), ∀ t > 0.
Thus from the positivity for the Dunkl-Bessel translation operator for the radial functions we
obtain
τ−η FD,B (a)(·, η) (ξ)FD,B (u)(η) ≤ C exp(−(t − r) ||η||)τ−η (exp(−τ k·k))(ξ).
Moreover it is clear that the function η 7→ exp(−(t − r) ||η||)τ−η (exp(−τ k·k))(ξ) belongs to
L1k,β (Rd+1
+ ) for t > r, τ > 0. So that
Z
τ−η FD,B (a)(·, η) (ξ)FD,B (u)(η) dµk,β (η)
Rd+1
+
Z
≤C
Rd+1
+
exp(−(t − r) ||η||)τ−η (exp(−τ || · ||))(ξ)dµk,β (η).
The right-hand side is a Dunkl-Bessel convolution product of two integrable functions and hence
is an integrable function on Rd+1
+ . Therefore the function
Z
ξ 7→
τ−η (FD,B (a)(·, η))(ξ)FD,B (u)(η)dµk,β (η)
Rd+1
+
is in L1k,β (Rd+1
+ ). Applying the inverse Dunkl-Bessel transform we get the result.
Now we prove the fundamental result
Theorem 5.3. Let A(x, 4k,β ) be the pseudo-differential-difference operator where the symbol
r,l
a(x, ξ) := A(x, −||ξ||2 ) belongs to Sexp,rad
. Then for all u in G∗ and all s in R
||A(x, 4k,β )u||WGs,p,k,β ≤ Cs ||u||W r+s,p .
(5.3)
∗
G∗ ,k,β
Proof. We consider the function
Z
s||ξ||
Us (ξ) = e
τ−η FD,B (a)(·, η) (ξ)FD,B (u)(η)dµk,β (η),
Rd+1
+
s ∈ R.
Then invoking (5.2) and (2.19) we deduce that
Z
exp((r + s) ||η||)|FD,B (u)(η)|
(5.4) |Us (ξ)| ≤
Rd+1
+
× τ−η exp(−(τ − |s|) ||y||) (ξ)dµk,β (η),
s ∈ R.
The integral of (5.4) can be considered as a Dunkl-Bessel convolution product between
f (ξ) = exp(−(τ − |s|) ||ξ||) and g(ξ) = exp((r + s) ||ξ||)|FD,B (u)(ξ)|. It is clear that f is
radial and belongs to L1k,β (Rd+1
+ ) for τ > |s|, on the other hand since FD,B (u) in G∗ then g in
p
d+1
Lpk,β (Rd+1
+ ). Hence f ∗D,B g is in Lk,β (R+ ) and we have
||f ∗D,B g||Lp
d+1
k,β (R+ )
≤ ||f ||L1
d+1
k,β (R+ )
||g||Lp
d+1
k,β (R+ )
= Cs ||u||W r+s,p .
G∗ ,k,β
Thus
||A(x, 4k,β )u||WGs,p,k,β = ||Us ||Lp
∗
d+1
k,β (R+ )
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
≤ ||f ∗D,B g||Lp
d+1
k,β (R+ )
≤ Cs ||u||W r+s,p .
G∗ ,k,β
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D UNKL -S OBOLEV S PACES
19
This proves (5.3).
5.2. The Reproducing Kernels.
Proposition 5.4. For s > 0, the Hilbert space WGs,2
(Rd+1
+ ) admits the reproducing kernel:
∗ ,k,β
Z
Θk,β (x, y) = mk,β
Λ(x, ξ)Λ(−y, ξ)e−2s||ξ|| dµk,β (ξ),
Rd+1
+
that is:
s,2
d+1
i) For every y in Rd+1
+ , the function x 7→ Θk,β (x, y) belongs to WG∗ ,k,β (R+ ).
d+1
ii) For every f in WGs,2
(Rd+1
+ ) and y in R+ , we have
∗ ,k,β
hf, Θk,β (·, y)iW s,2
G∗ ,k,β
= f (y).
Proof. i) Let y be in Rd+1
+ . From (2.5), the function
ξ 7→ Λ(−y, ξ)e−2s||ξ||
belongs to L1k,β (Rd+1
+ )
T
L2k,β (Rd+1
+ ) for s > 0, then from Proposition 2.1 ii) there exists a
function in L2k,β (Rd+1
+ ), which we denote by Θk,β (·, y), such that
(5.5)
FD,B Θk,β (·, y) (ξ) = Λ(−y, ξ)e−2s||ξ|| .
Thus Θk,β (x, y) is given by
Z
Θk,β (x, y) = mk,β
Rd+1
+
Λ(x, ξ)Λ(−y, ξ)e−2s||ξ|| dµk,β (ξ).
d+1
ii) Let f be in WGs,2
(Rd+1
+ ) and y in R+ . From (4.1),(5.5) and (2.15) we have
∗ ,k,β
Z
hf, Θk,β (·, y)iW s,2 = mk,β
FD,B (f )(ξ)Λ(y, ξ)dµk,β (ξ) = f (y).
G∗ ,k,β
Rd+1
+
Proposition 5.5.
i) Let f be in G∗ . Then u(x, t) = Hk,β (t)f (x) solves the problem
(
∂
4k,β − ∂t
u(x, t) = 0 on Rd+1
+ ×]0, ∞[
u(·, 0) = f.
ii) The integral transform Hk,β (t), t > 0, is a bounded linear operator from WGs,2
(Rd+1
+ ),
∗ ,k,β
s in R, into L2k,β (Rd+1
+ ), and we have
kHk,β (t)f kL2
d+1
k,β (R+ )
s2
≤ e 4t kf kW s,2
G∗ ,k,β
.
Proof. i) This assertion follows from Definition 3.4 and Proposition 3.4 iii).
ii) Let f be in WGs,2
(Rd+1
+ ). Using Proposition 2.1 ii) we have
∗ ,k,β
kHk,β (t)f k2L2
d+1
k,β (R+ )
= mk,β kFD,B (Hk,β (t)f )k2L2
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
d+1
k,β (R+ )
.
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20
H ATEM M EJJAOLI
Invoking the relation ships (2.21) and (3.2) we can write
Z
2
2
e−2t||ξ|| |FD,B (f )(ξ)|2 dµk,β (ξ).
kHk,β (t)f kL2 (Rd+1 ) = mk,β
k,β
+
Rd+1
+
Therefore
s2
kHk,β (t)f kL2
d+1
k,β (R+ )
≤ e 4t kf kW s,2
G∗ ,k,β
.
Definition 5.3. Let r > 0, t ≥ 0 and s in R. We define the Hilbert space HGr,s∗ ,k,β (Rd+1
+ ) as the
subspace of WGs,2
(Rd+1
+ ) with the inner product:
∗ ,k,β
hf, giHGr,s,k,β = rhf, giW s,2
∗
G∗ ,k,β
+ hHk,β (t)f, Hk,β (t)giL2
d+1
k,β (R+ )
,
f, g ∈ WGs,2
(Rd+1
+ ).
∗ ,k,β
The norm associated to the inner product is defined by:
kf k2H r,s
G∗ ,k,β
:= rkf k2W s,2
G∗ ,k,β
+ kHk,β (t)f k2L2
d+1
k,β (R+ )
.
Proposition 5.6. For s ∈ R, the Hilbert space HGr,s∗ ,k,β (Rd+1
+ ) admits the following reproducing
kernel:
Z
Pr (x, y) = mk,β
Rd+1
+
Λ(x, ξ)Λ(−y, ξ)dµk,β (ξ)
.
re2s||ξ|| + e−2t||ξ||2
Proof. i) Let y be in Rd+1
+ . In the same way as in the proof of Proposition 5.4 i), we can prove
that the function x 7→ Pr (x, y) belongs to L2k,β (Rd+1
+ ) and we have
FD,B Pr (·, y) (ξ) =
(5.6)
Λ(−y, ξ)
.
re2s||ξ|| + e−2t||ξ||2
On the other hand we have
(5.7)
FD,B Hk,β (t)(Pr (·, y) (ξ) = exp(−t||ξ||2 )FD,B Pr (·, y) (ξ),
for all ξ ∈ Rd+1
+ .
Hence from Proposition 2.1 ii), we obtain
kHk,β (t)(Pr (·, y)k2L2 (Rd+1 )
+
k,β
Z
= mk,β
≤
C
r2
Rd+1
+
2
Z
Therefore we conclude that kPr (·, y)k2H r,s
2 e−2t||ξ|| FD,B Pr (·, y) (ξ) 2 dµk,β (ξ)
Rd+1
+
e−2t||ξ||
dµk,β (ξ) < ∞.
e4s||ξ||
< ∞.
G∗ ,k,β
d+1
ii) Let f be in HGr,s∗ ,k,β (Rd+1
+ ) and y in R+ . Then
(5.8)
hf, Pr (·, y)iHGr,s,k,β = rI1 + I2 ,
∗
where
I1 = hf, Pr (·, y)iW s,2
G∗ ,k,β
and
I2 = hHk,β (t)f, Hk,β (t)(Pr (·, y))iL2
d+1
k,β (R+ )
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
.
http://jipam.vu.edu.au/
D UNKL -S OBOLEV S PACES
21
From (5.6) and (4.1), we have
Z
I1 = mk,β
Rd+1
+
e2s||ξ|| FD,B (f )(ξ)Λ(y, ξ)dµk,β (ξ)
.
re2s||ξ|| + e−2t||ξ||2
From (5.7) and (5.6) and Proposition 2.1 ii) we have
Z
2
e−2t||ξ|| FD,B (f )(ξ)Λ(y, ξ)dµk,β (ξ)
I2 = mk,β
.
2s||ξ|| + e−2t||ξ||2
re
Rd+1
+
The relations (5.8) and (2.15) imply that
hf, Pr (·, y)iHGr,s,k,β = f (y).
∗
5.3. Extremal Function for the Generalized Heat Semigroup Transform. In this subsection, we prove for a given function g in L2k,β (Rd+1
+ ) that the infimum of
n
o
s,2
2
2
d+1
rkf kW s,2 + kg − Hk,β (t)f kL2 (Rd+1 ) , f ∈ WG∗ ,k,β (R+ )
G∗ ,k,β
k,β
+
∗
is attained at some unique function denoted by fr,g
, called the extremal function. We start with
the following fundamental theorem (cf. [11, 18]).
Theorem 5.7. Let HK be a Hilbert space admitting the reproducing kernel K(p, q) on a set E
and H a Hilbert space. Let L : HK → H be a bounded linear operator on HK into H. For
r > 0, introduce the inner product in HK and call it HKr as
hf1 , f2 iHKr = rhf1 , f2 iHK + hLf1 , Lf2 iH .
Then
i) HKr is the Hilbert space with the reproducing kernel Kr (p, q) on E which satisfies the
equation
K(·, q) = (rI + L∗ L)Kr (·, q),
where L∗ is the adjoint operator of L : HK → H.
ii) For any r > 0 and for any g in H, the infimum
n
o
inf rkf k2HK + kLf − gk2H
f ∈HK
∗
is attained by a unique function fr,g
in HK and this extremal function is given by
∗
fr,g
(p) = hg, LKr (·, p)iH .
(5.9)
We can now state the main result of this subsection.
Theorem 5.8. Let s ∈ R. For any g in L2k,β (Rd+1
+ ) and for any r > 0, the infimum
n
o
2
2
(5.10)
inf
rkf
k
+
kg
−
H
(t)f
k
s,2
d+1
k,β
2
L (R
)
W
s,2
f ∈WG∗ ,k,β (Rd+1
+ )
G∗ ,k,β
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
k,β
+
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22
H ATEM M EJJAOLI
∗
is attained by a unique function fr,g
given by
Z
∗
(5.11)
fr,g (x) =
g(y)Qr (x, y)dµk,β (y),
Rd+1
+
where
Qr (x, y) = Qr,s (x, y)
Z
2
e−t||ξ|| Λ(x, ξ)Λ(−y, ξ)
= mk,β
dµk,β (ξ).
re2s||ξ|| + e−2t||ξ||2
Rd+1
+
(5.12)
Proof. By Proposition 5.6 and Theorem 5.7 ii), the infimum given by (5.10) is attained by a
∗
∗
unique function fr,g
, and from (5.9) the extremal function fr,g
is represented by
∗
fr,g
(y) = hg, Hk,β (t)(Pr (·, y))iL2
d+1
k,β (R+ )
,
y ∈ Rd+1
+ ,
where Pr is the kernel given by Proposition 5.6. On the other hand we have
Z
Hk,β (t)f (x) = mk,β
exp(−t||ξ||2 )FD,B (f )(ξ)Λ(x, ξ)dµk,β (ξ), for all x ∈ Rd+1
+ .
Rd+1
+
Hence by (5.6), we obtain
Hk,β (t) Pr (·, y) (x) = mk,β
Z
Rd+1
+
exp(−t||ξ||2 )Λ(x, ξ)Λ(−y, ξ)
dµk,β (ξ)
re2s||ξ|| + e−2t||ξ||2
= Qr (x, y).
This gives (5.12).
Corollary 5.9. Let s ∈ R, δ > 0 and g, gδ in L2k,β (Rd+1
+ ) such that
kg − gδ kL2
d+1
k,β (R+ )
Then
∗
∗
kfr,g
− fr,g
k s,2
δ W
G∗ ,k,β
≤ δ.
δ
≤ √ .
2 r
Proof. From (5.12) and Fubini’s theorem we have
2
(5.13)
∗
FD,B (fr,g
)(ξ) =
e−t||ξ|| FD,B (g)(ξ)
.
re2s||ξ|| + e−2t||ξ||2
Hence
2
e−t||ξ|| FD,B (g − gδ )(ξ)
.
re2s||ξ|| + e−2t||ξ||2
Using the inequality (x + y)2 ≥ 4xy, we obtain
2
1
∗
∗
e2s||ξ|| FD,B (fr,g
− fr,g
)(ξ) ≤ |FD,B (g − gδ )(ξ)|2 .
δ
4r
This and Proposition 2.1 ii) give
mk,β
∗
∗
kfr,g
− fr,g
k2 s,2 ≤
kFD,B (g − gδ )k2L2 (Rd+1 )
δ W
+
k,β
G∗ ,k,β
4r
1
≤ kg − gδ k2L2 (Rd+1 ) ,
+
k,β
4r
from which we obtain the desired result.
∗
∗
FD,B (fr,g
− fr,g
)(ξ) =
δ
J. Inequal. Pure and Appl. Math., 10(2) (2009), Art. 55, 24 pp.
http://jipam.vu.edu.au/
D UNKL -S OBOLEV S PACES
23
Corollary 5.10. Let s ∈ R. If f is in WGs,2
(Rd+1
+ ) and g = Hk,β (t)f . Then
∗ ,k,β
∗
kfr,g
− f k2W s,2
→0
as r → 0.
G∗ ,k,β
Proof. From (3.2), (5.13) we have
FD,B (f )(ξ) = exp(t||ξ||2 )FD,B (g)(ξ)
and
2
∗
FD,B (fr,g
)(ξ)
e−t||ξ|| FD,B (g)(g)(ξ)
.
=
re2s||ξ|| + e−2t||ξ||2
Hence
∗
FD,B (fr,g
− f )(ξ) =
−re2s||ξ|| FD,B (f )(ξ)
.
re2s||ξ|| + e−2t||ξ||2
Then we obtain
∗
kfr,g
−
f k2W s,2
G∗ ,k,β
Z
=
Rd+1
+
with
hr,t,s (ξ) =
hr,t,s (ξ)|FD,B (f )(ξ)|2 dµk,β (ξ),
r2 e6s||ξ||
.
(re2s||ξ|| + e−2t||ξ||2 )2
Since
lim hr,t,s (ξ) = 0
r→0
and
|hr,t,s (ξ)| ≤ e2s||ξ|| ,
we obtain the result from the dominated convergence theorem.
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