Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 1 Issue 3(2009), Pages 16-41.
SPACES OF 𝐷𝐿𝑝 -TYPE AND A CONVOLUTION PRODUCT
ASSOCIATED WITH THE RIEMANN-LIOUVILLE OPERATOR
(DEDICATED IN OCCASION OF THE 65-YEARS OF
PROFESSOR R.K. RAINA)
C.BACCAR AND L.T.RACHDI
Abstract. We define and study the spaces M𝑝 (ℝ2 ), 1 ⩽ 𝑝 ⩽ +∞, that are
of 𝐷𝐿𝑝 -type. Using the harmonic analysis related to the Fourier transform
connected with the Riemann-Liouville operator, we give a new characterization
of the dual space M𝑝′ (ℝ2 ) and we describe its bounded subsets. Next, we define
a convolution product in M𝑝′ (ℝ2 ) × ℳ𝑟 (ℝ2 ), 1 ⩽ 𝑟 ⩽ 𝑝 < +∞, where ℳ𝑟 (ℝ2 )
is the closure of the space S∗ (ℝ2 ) in M𝑟 (ℝ2 ) and we prove some new results.
1. Introduction
𝑛
The space 𝐷𝐿𝑝 (ℝ ); 1 ≤ 𝑝 ≤ +∞; is formed by the measurable functions on ℝ𝑛
such that for all 𝛼 ∈ ℕ𝑛 ; the function 𝐷𝛼 (𝑓 ) belongs to 𝐿𝑝 (ℝ𝑛 , 𝑑𝑥) (the space of
functions with 𝑝𝑡ℎ power integrable on ℝ𝑛 with respect to the Lebesgue measure
𝑑𝑥 on ℝ𝑛 ).
Many aspects of these spaces have been studied [1, 2, 6, 24]. In [8]; the authors
have defined some spaces of functions that are of 𝐷𝐿𝑝 -type but replacing the usual
( 2𝛼+1 𝑑 )
1
𝑑
derivative by the Bessel operator 𝑟2𝛼+1
𝑑𝑟 𝑟
𝑑𝑟 , and they have established
many results for these spaces.
In [3]; we define the so-called Riemann-Liouville operator; defined on C∗ (ℝ2 )
(the space of continuous functions on ℝ2 , even with respect to the first variable) by
⎧
∫ ∫
)
𝛼 1 1 ( √


2 , 𝑥 + 𝑟𝑡

𝑓
𝑟𝑠
1
−
𝑡


⎨ 𝜋 −1 −1
(
) 1(
)𝛼−1
2 𝛼− 2
R𝛼 (𝑓 )(𝑟, 𝑥) =
×
1
−
𝑡
1 − 𝑠2
𝑑𝑡𝑑𝑠; if 𝛼 > 0,

∫ 1 ( √

)

1
𝑑𝑡


𝑓 𝑟 1 − 𝑡2 , 𝑥 + 𝑟𝑡 √
;
if 𝛼 = 0.
⎩
𝜋 −1
1 − 𝑡2
2000 Mathematics Subject Classification. 42B35,46F12.
Key words and phrases. Spaces of 𝐷𝐿𝑝 -type, Riemann-Liouville operator, convolution product,
Fourier transform.
c
⃝2009
Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted September, 2009. Published October, 2009.
16
SPACES OF 𝐷𝐿𝑝 -TYPE
The dual operator 𝑡 R𝛼
⎧






⎨
𝑡
R𝛼 (𝑓 )(𝑟, 𝑥) =






⎩
is defined by
∫ √𝑢2 −𝑟2
∫
2𝛼 +∞
𝑓 (𝑢, 𝑥 + 𝑣)
√
𝜋 𝑟
−( 𝑢2 −𝑟 2
)𝛼−1
× 𝑢2 − 𝑣 2 − 𝑟2
𝑢𝑑𝑢𝑑𝑣;
1
𝜋
∫
𝑓
(√
)
𝑟2 + (𝑥 − 𝑦)2 , 𝑦 𝑑𝑦;
17
if 𝛼 > 0,
if 𝛼 = 0.
ℝ
The operators R𝛼 and 𝑡 R𝛼 generalize the mean operator R0 and its dual 𝑡 R0 ([28]),
defined respectively by
∫ 2𝜋
1
R0 (𝑓 )(𝑟, 𝑥) =
𝑓 (𝑟 sin 𝜃, 𝑥 + 𝑟 cos 𝜃) 𝑑𝜃,
2𝜋 0
and
∫ (√
)
1
𝑡
R0 (𝑓 )(𝑟, 𝑥) =
𝑓
𝑟2 + (𝑥 − 𝑦)2 , 𝑦 𝑑𝑦.
𝜋 ℝ
The mean operator R0 and its dual 𝑡 R0 play an important role and have many
applications, for example, in image processing of so-called synthetic aperture radar
(SAR) data [12, 14], or in the linearized inverse scaterring problem in acoustics [11].
The Fourier transform F𝛼 connected with the Riemann-Liouville operator R𝛼
is defined by
∫ +∞ ∫
F𝛼 (𝑓 )(𝜇, 𝜆) =
𝑓 (𝑟, 𝑥)𝜑𝜇,𝜆 (𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥),
0
ℝ
where
(
)
. 𝜑𝜇,𝜆 (𝑟, 𝑥) = R𝛼 𝑐𝑜𝑠(𝜇.)𝑒−𝑖𝜆. (𝑟, 𝑥).
𝑟 2𝛼+1√
𝑑𝑟 ⊗ 𝑑𝑥.
. 𝑑𝜈𝛼 (𝑟, 𝑥) = 2𝛼 Γ(𝛼+1)
2𝜋
We have constructed the harmonic analysis related to the Fourier transform F𝛼
(inversion formula, Plancherel formula, Paley-Wiener theorem, Plancherel theorem...)
On the other hand the uncertainty principle play an important role in harmonic
analysis, and many aspects of these principle have been studied. In particular,
the Heisenberg-Pauli-Weyl inequality [13] has been established for several Fourier
transforms [21, 22, 23].
In [17, 18, 19, 20], the author gave many generalizations of this inequality for
the usual Fourier transform. In this context, we have established in [4] an 𝐿𝑝 − 𝐿𝑞
version of Hardy theorem’s for the Fourier transform F𝛼 . Also, in [16] the second
author with the other have established the Heisenberg-Pauli-Weyl inequality for
F𝛼 .
Our investigation in the present work consists to define and study some function
spaces denoted by M𝑝 (ℝ2 ), 1 ⩽ 𝑝 ⩽ +∞, similar to 𝐷𝐿𝑝 , but replacing the usual
derivatives by the operator
∂2
2𝛼 + 1 ∂
∂2
+
+
).
∂𝑟2
𝑟 ∂𝑟 ∂𝑥2
For this, let S∗ (ℝ2 ) be the space of infinitely differentiable functions on ℝ2 ,
rapidly decreasing together with all their derivatives and even with respect to the
first variable and S∗′ (ℝ2 ) its topological dual that is the space of tempered distribution on ℝ2 even with resect to the first variable. Then, the singular partial
Δ𝛼 = −(
18
C.BACCAR AND L.T.RACHDI
differential operator Δ𝛼 is continuous from S∗ (ℝ2 ) into itself. Moreover, for all
𝑇 ∈ S∗′ (ℝ2 ) we define Δ𝛼 (𝑇 ) by
∀𝜑 ∈ S∗ (ℝ2 );
⟨Δ𝛼 (𝑇 ), 𝜑⟩ = ⟨𝑇, Δ𝛼 (𝜑)⟩,
then, Δ𝛼 is also continuous from S∗′ (ℝ2 ) into itself.
The space M𝑝 (ℝ2 ) consists off all measurable functions on ℝ2 even with respect
to the first variable such that for all 𝑘 ∈ ℕ; there exists a function denoted by
Δ𝑘𝛼 (𝑓 ) ∈ 𝐿𝑝 (𝑑𝜈𝛼 ) satisfying Δ𝑘𝛼 (𝑇𝑓 ) = 𝑇Δ𝑘𝛼 (𝑓 ) ; where
. 𝑇𝑓 is the tempered distribution, even with respect to the first variable given
by the function 𝑓 .
. 𝐿𝑝 (𝑑𝜈𝛼 ), 1 ⩽ 𝑝 ⩽ +∞, is the space of measurable functions 𝑓 on [0, +∞[×ℝ,
such that
)1/𝑝
( ∫ +∞ ∫
𝑝
∣𝑓 (𝑟, 𝑥)∣ 𝑑𝜈𝛼 (𝑟, 𝑥)
∥𝑓 ∥𝑝,𝜈𝛼 =
< +∞, 1 ⩽ 𝑝 < +∞;
0
∥𝑓 ∥∞,𝜈𝛼 =
ℝ
𝑒𝑠𝑠 𝑠𝑢𝑝
∣𝑓 (𝑟, 𝑥)∣ < ∞,
𝑝 = +∞.
(𝑟,𝑥)∈[0,+∞[×ℝ
Using the convolution product and the Fourier transform F𝛼 associated with the
Riemann-Liouville operator, we establish firstly the following results which give a
nice characterization of the elements of the dual space M ′ 𝑝 (ℝ2 )
∙ Let 𝑇 ∈ S∗′ (ℝ2 ). Then 𝑇 belongs to M𝑝′ (ℝ2 ), 1 ⩽ 𝑝 < +∞, if and only if there
′
exist 𝑚 ∈ ℕ and {𝑓0 , ..., 𝑓𝑚 } ⊂ 𝐿𝑝 (𝑑𝜈𝛼 );
𝑝
, such that
𝑝′ = 𝑝−1
𝑇 =
𝑚
∑
Δ𝑘𝛼 (𝑇𝑓𝑘 ),
𝑘=0
where
∀𝜑 ∈ M𝑝 (ℝ2 );
⟨Δ𝑘𝛼 (𝑇𝑓 ), 𝜑⟩ =
∫
0
+∞
∫
𝑓 (𝑟, 𝑥)Δ𝑘𝛼 (𝜑)(𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥);
ℝ
∙ Let 𝑇 ∈ S∗′ (ℝ2 ), 𝑝 ∈ [1, +∞[, and 𝑝′ =
𝑝
𝑝−1 .
Then T belongs to M𝑝′ (ℝ2 ) if,
′
and only if for every 𝜑 ∈ D∗ (ℝ2 ), the function 𝑇 ∗ 𝜑 belongs to the space 𝐿𝑝 (𝑑𝜈𝛼 ),
where
. 𝑇 ∗ 𝜑 is the function defined by 𝑇 ∗ 𝜑(𝑟, 𝑥) = ⟨𝑇, 𝜏(𝑟,−𝑥) (𝜑)⟩,
ˇ with 𝜑(𝑟,
ˇ 𝑥) =
𝜑(𝑟, −𝑥).
. 𝜏(𝑟,𝑥) is the translation operator associated with the Riemann-Liouville transform, defined on 𝐿𝑝 (𝑑𝜈𝛼 ) by
∫ 𝜋 (√
)
Γ(𝛼 + 1)
)
(
𝜏(𝑟,𝑥) 𝑓 (𝑠, 𝑦) = √
𝑓
𝑟2 + 𝑠2 + 2𝑟𝑠 cos 𝜃, 𝑥 + 𝑦 sin2𝛼 (𝜃)𝑑𝜃.
1
𝜋 Γ 𝛼+ 2 0
. D∗ (ℝ2 ) is the space of infinitely differentiable functions on ℝ2 , even with respect
to the first variable and with compact support.
Next, by the fact that a subset of M𝑝′ (ℝ2 ) is bounded if, and only if it is equicontinuous, we show the coming result that is a good description of the bounded subsets
of the dual space M𝑝′ (ℝ2 )
∙ Let 𝑝 ∈ [1, ∞[ and let 𝐵 ′ be a subset of M𝑝′ (ℝ2 ). The following assertions are
equivalent
(i) 𝐵 ′ is weakly (equivalently strongly) bounded in M𝑝′ (ℝ2 ),
SPACES OF 𝐷𝐿𝑝 -TYPE
19
(ii) there exist 𝐶 > 0 and 𝑚 ∈ ℕ, such that for every 𝑇 ∈ 𝐵 ′ , it is possible to
′
find 𝑓0 , ..., 𝑓𝑚 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ) satisfying
𝑇 =
𝑚
∑
Δ𝑘𝛼 (𝑇𝑓𝑘 ) with max
0⩽𝑘⩽𝑚
𝑘=0
∥𝑓𝑘 ∥𝑝′ ,𝜈𝛼 ⩽ 𝐶,
{
}
′
(iii) for every 𝜑 ∈ D∗ (ℝ2 ), the set 𝑇 ∗ 𝜑, 𝑇 ∈ 𝐵 ′ is bounded in 𝐿𝑝 (𝑑𝜈𝛼 ).
Finally, we define and study a convolution product on the space M𝑝′ (ℝ2 ) ×
ℳ𝑟 (ℝ2 ), 1 ⩽ 𝑟 ⩽ 𝑝 < +∞, where ℳ𝑟 (ℝ2 ) is the closure of the space S∗ (ℝ2 ) in
M𝑟 (ℝ2 ). More precisely
∙ Let 𝑝 ∈ [1, +∞[. For every (𝑟, 𝑥) ∈ [0, +∞[×ℝ, the translation operator 𝜏(𝑟,𝑥)
is continuous from M𝑝 (ℝ2 ) into itself.
∙ Let 1 ⩽ 𝑟 ⩽ 𝑝 < ∞ and 𝑞 ∈ [1, +∞], such that
1 1
1
= − .
𝑞
𝑟
𝑝
Then for every 𝑇 ∈ M ′ 𝑝 (ℝ2 ), the mapping
𝜙 −→ 𝑇 ∗ 𝜙
is continuous from ℳ𝑟 (ℝ2 ) into M𝑞 (ℝ2 ).
2. The Fourier transform associated with the Riemann-Liouville
operator
In this section, we recall some harmonic analysis results related to the convolution
product and the Fourier transform associated with Riemann-Liouville operator. For
more details see [3, 5, 7, 16].
Let 𝐷 and Ξ be the singular partial differential operators defined by
⎧
∂



⎨ 𝐷 = ∂𝑥 ;

2
2


⎩ Ξ = ∂ + 2𝛼 + 1 ∂ − ∂ ;
∂𝑟2
𝑟 ∂𝑟 ∂𝑥2
(𝑟, 𝑥) ∈]0, +∞[×ℝ, 𝛼 ⩾ 0.
For all (𝜇, 𝜆) ∈ ℂ2 ; the system
⎧
𝐷𝑢(𝑟, 𝑥) = −𝑖𝜆𝑢(𝑟, 𝑥);



⎨
Ξ𝑢(𝑟, 𝑥) = −𝜇2 𝑢(𝑟, 𝑥);


∂𝑢

⎩ 𝑢(0, 0) = 1,
(0, 𝑥) = 0; ∀𝑥 ∈ ℝ.
∂𝑟
admits a unique solution 𝜑𝜇,𝜆 , given by
∀(𝑟, 𝑥) ∈ [0, +∞[×ℝ;
( √
)
𝜑𝜇,𝜆 (𝑟, 𝑥) = 𝑗𝛼 𝑟 𝜇2 + 𝜆2 exp (−𝑖𝜆𝑥) ,
where 𝑗𝛼 is the modified Bessel function defined by
+∞
𝑗𝛼 (𝑠) = 2𝛼 Γ (𝛼 + 1)
( 𝑠 )2𝑘
∑
𝐽𝛼 (𝑠)
(−1)𝑘
=
Γ
(𝛼
+
1)
,
𝑠𝛼
𝑘!Γ (𝛼 + 𝑘 + 1) 2
𝑘=0
(2.1)
20
C.BACCAR AND L.T.RACHDI
and 𝐽𝛼 is the Bessel function of first kind and index 𝛼 [9, 10, 15, 29]. The modified
Bessel function 𝑗𝛼 has the integral representation
∫ 1
(
) 1
Γ (𝛼 + 1)
2 𝛼− 2
)
(
𝑗𝛼 (𝑠) = √
1
−
𝑡
exp (−𝑖𝑠𝑡) 𝑑𝑡.
(2.2)
𝜋 Γ 𝛼 + 21 −1
Consequently, for all 𝑘 ∈ ℕ and 𝑠 ∈ ℝ; we have
∣𝑗𝛼(𝑘) (𝑠)∣ ⩽ 1.
(2.3)
The eigenfunction function 𝜑𝜇,𝜆 satisfies the following properties
∙
sup ∣𝜑𝜇,𝜆 (𝑟, 𝑥)∣ = 1
if, and only if (𝜇, 𝜆) ∈ Γ,
(2.4)
(𝑟,𝑥)∈ℝ2
where Γ is the set defined by
{
}
Γ = ℝ2 ∪ (𝑖𝜇, 𝜆); (𝜇, 𝜆) ∈ ℝ2 , ∣𝜇∣ ⩽ ∣𝜆∣ .
(2.5)
∙ The function 𝜑𝜇,𝜆
⎧









⎨
𝜑𝜇,𝜆 (𝑟, 𝑥) =









⎩
has the following Mehler integral representation
∫ ∫
(
)
√
𝛼 1 1
cos 𝜇𝑟𝑠 1 − 𝑡2 exp (−𝑖𝜆(𝑥 + 𝑟𝑡))
𝜋 −1 −1
)𝛼−1
(
)𝛼− 21 (
1 − 𝑠2
𝑑𝑡𝑑𝑠;
if 𝛼 > 0,
× 1 − 𝑡2
∫ 1
)
( √
1
cos 𝑟𝜇 1 − 𝑡2 exp(−𝑖𝜆(𝑥 + 𝑟𝑡))
𝜋 −1
𝑑𝑡
×√
;
if 𝛼 = 0.
1 − 𝑡2
The precedent integral representation allows us to define the Riemann-Liouville
transform R𝛼 associated with the operators Δ1 ans Δ2 by
⎧
∫ ∫
)
𝛼 1 1 ( √



𝑓 𝑟𝑠 1 − 𝑡2 , 𝑥 + 𝑟𝑡


⎨ 𝜋 −1 −1
(
)𝛼− 12 (
)𝛼−1
R𝛼 (𝑓 )(𝑟, 𝑥) =
× 1 − 𝑡2
1 − 𝑠2
𝑑𝑡𝑑𝑠; if 𝛼 > 0,

∫ 1 ( √

)

1
𝑑𝑡


𝑓 𝑟 1 − 𝑡2 , 𝑥 + 𝑟𝑡 √
;
if 𝛼 = 0.
⎩
𝜋 −1
1 − 𝑡2
where 𝑓 is any continuous functions on ℝ2 , even with respect to the first variable.
∙ From the precedent integral representation of the eigenfunction 𝜑𝜇,𝜆 , we have
∀(𝑟, 𝑥) ∈ [0, +∞[×ℝ,
𝜑𝜇,𝜆 (𝑟, 𝑥) = R𝛼 (cos(𝜇.)𝑒−𝜆. )(𝑟, 𝑥).
In the following, we will define the convolution product and the Fourier transform
associated with the Riemann-Liouville transform. For this, we use the product
formula for the function 𝜑𝜇,𝜆 given by
∀(𝑟, 𝑥), (𝑠, 𝑦) ∈ [0, +∞[×ℝ,
∫ 𝜋
(√
)
Γ(𝛼 + 1)
)
𝜑𝜇,𝜆 (𝑟, 𝑥) 𝜑𝜇,𝜆 (𝑠, 𝑦) = √ (
𝜑𝜇,𝜆
𝑟2 + 𝑠2 + 2𝑟𝑠 cos 𝜃, 𝑥 + 𝑦
1
𝜋Γ 𝛼 + 2 0
× sin2𝛼 (𝜃)𝑑𝜃.
(2.6)
Definition 2.1.
(i) The translation operator associated with Riemann-Liouville transform is defined
on 𝐿1 (𝑑𝜈𝛼 ), by for all (𝑟, 𝑥), (𝑠, 𝑦) ∈ [0, +∞[×ℝ,
∫ 𝜋 (√
)
Γ(𝛼 + 1)
2 + 𝑠2 + 2𝑟𝑠 cos 𝜃, 𝑥 + 𝑦 sin2𝛼 (𝜃)𝑑𝜃.
)
𝜏(𝑟,𝑥) 𝑓 (𝑠, 𝑦) = √ (
𝑓
𝑟
𝜋Γ 𝛼 + 12 0
SPACES OF 𝐷𝐿𝑝 -TYPE
21
(ii) The convolution product of 𝑓, 𝑔 ∈ 𝐿1 (𝑑𝜈𝛼 ) is defined for all (𝑟, 𝑥) ∈ [0, +∞[×ℝ,
by
∫ +∞ ∫
𝜏(𝑟,−𝑥) (𝑓ˇ)(𝑠, 𝑦)𝑔(𝑠, 𝑦)𝑑𝜈𝛼 (𝑠, 𝑦),
𝑓 ∗ 𝑔(𝑟, 𝑥) =
0
ℝ
where 𝑓ˇ(𝑠, 𝑦) = 𝑓 (𝑠, −𝑦).
We have the following properties
∙ The product formula (2.6), can be written
𝜏(𝑟,𝑥) (𝜑𝜇,𝜆 )(𝑠, 𝑦) = 𝜑𝜇,𝜆 (𝑟, 𝑥)𝜑𝜇,𝜆 (𝑠, 𝑦).
∙ For all 𝑓 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ), 1 ⩽ 𝑝 ⩽ +∞, and for all (𝑟, 𝑥) ∈ [0, +∞[×ℝ, the
function 𝜏(𝑟,𝑥) (𝑓 ) belongs to 𝐿𝑝 (𝑑𝜈𝛼 ) and we have
𝜏(𝑟,𝑥) (𝑓 )
⩽ ∥𝑓 ∥
.
(2.7)
𝑝,𝜈𝛼
𝑝,𝜈𝛼
1
∙ For 𝑓, 𝑔 ∈ 𝐿 (𝑑𝜈𝛼 ), the function 𝑓 ∗ 𝑔 belongs to 𝐿1 (𝑑𝜈𝛼 ); the convolution
product is commutative, associative and we have
∥𝑓 ∗ 𝑔∥1,𝜈𝛼 ⩽ ∥𝑓 ∥1,𝜈𝛼 ∥𝑔∥ ∣1,𝜈𝛼 .
Moreover, if 1 ⩽ 𝑝, 𝑞, 𝑟 ⩽ +∞ are such that 1𝑟 = 𝑝1 + 1𝑞 − 1 and if 𝑓 ∈
𝐿𝑝 (𝑑𝜈𝛼 ), 𝑔 ∈ 𝐿𝑞 (𝑑𝜈𝛼 ), then the function 𝑓 ∗ 𝑔 belongs to 𝐿𝑟 (𝑑𝜈𝛼 ), and we
have
∥𝑓 ∗ 𝑔∥𝑟,𝜈𝛼 ⩽ ∥𝑓 ∥𝑝,𝜈𝛼 ∥𝑔∥𝑞,𝜈𝛼 .
(2.8)
In the sequel, we use the following notations
∙ Γ+ is the subset of Γ given by
{
}
Γ+ = ℝ+ × ℝ ∪ (𝑖𝑡, 𝑥); (𝑡, 𝑥) ∈ ℝ2 ; 0 ⩽ 𝑡 ⩽ ∣𝑥∣ .
∙ BΓ+ is the 𝜎-algebra defined on Γ+ by
{
}
BΓ+ = 𝜃−1 (𝐵) , 𝐵 ∈ B𝐵𝑜𝑟 ([0, +∞[×ℝ) ,
where 𝜃 is the bijective function defined on the set Γ+ by
√
𝜃(𝜇, 𝜆) = ( 𝜇2 + 𝜆2 , 𝜆).
(2.9)
∙ 𝑑𝛾𝛼 is the measure defined on BΓ+ by
∀ 𝐴 ∈ BΓ+ ; 𝛾𝛼 (𝐴) = 𝜈𝛼 (𝜃(𝐴)).
𝑝
∙ 𝐿 (𝑑𝛾𝛼 ), 𝑝 ∈ [1, +∞[, is the space of measurable functions 𝑓 on Γ+ , such
that
∫ ∫
1
∥𝑓 ∥𝑝,𝛾𝛼 = (
∣𝑓 (𝜇, 𝜆)∣𝑝 𝑑𝛾𝛼 (𝜇, 𝜆)) 𝑝 < ∞, if 𝑝 ∈ [1, +∞[,
Γ+
∥𝑓 ∥∞,𝛾𝛼 =
∣𝑓 (𝑟, 𝑥)∣ < +∞,
ess sup
if 𝑝 = +∞.
(𝑟,𝑥)∈[0,+∞[×ℝ
Proposition 2.2.
i) For all non negative measurable function 𝑔 on Γ+ , we have
∫ ∫
( ∫ +∞ ∫
1
√
𝑔(𝜇, 𝜆)𝑑𝛾𝛼 (𝜇, 𝜆) =
𝑔(𝜇, 𝜆)(𝜇2 + 𝜆2 )𝛼 𝜇𝑑𝜇𝑑𝜆
2𝛼 Γ(𝛼 + 1) 2𝜋 0
ℝ
Γ+
∫ ∫ ∣𝜆∣
)
+
𝑔(𝑖𝜇, 𝜆)(𝜆2 − 𝜇2 )𝛼 𝜇𝑑𝜇𝑑𝜆 .
ℝ 0
22
C.BACCAR AND L.T.RACHDI
ii) For all nonnegative measurable function 𝑓 on [0, +∞[×ℝ (respectively integrable
on [0, +∞[×ℝ with respect to the measure 𝑑𝜈𝛼 ) 𝑓 𝑜𝜃 is a nonnegative measurable
function on Γ+ (respectively integrable on Γ+ with respect to the measure 𝑑𝛾𝛼 ) and
we have
∫
∫ ∫
∫
+∞
(𝑓 ∘ 𝜃)(𝜇, 𝜆)𝑑𝛾𝛼 (𝜇, 𝜆) =
Γ+
𝑓 (𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥).
0
ℝ
Definition 2.3. The Fourier transform associated with the Riemann-Liouville operator is defined on 𝐿1 (𝑑𝜈𝛼 ), by
∫ +∞ ∫
∀(𝜇, 𝜆) ∈ Γ,
F𝛼 (𝑓 )(𝜇, 𝜆) =
𝑓 (𝑟, 𝑥)𝜑𝜇,𝜆 (𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥),
0
ℝ
where 𝜑𝜇,𝜆 is the eigenfunction given by the relation (2.1) and Γ is the set defined
by the relation (2.5).
We have the following properties
∙ From the relation (2.4), we deduce that for 𝑓 ∈ 𝐿1 (𝑑𝜈𝛼 ) the function F𝛼 (𝑓 )
belongs to the space 𝐿∞ (𝑑𝛾𝛼 ) and we have
F𝛼 (𝑓 )
.
(2.10)
⩽ ∥𝑓 ∥
1,𝜈𝛼
∞,𝛾𝛼
1
∙ For 𝑓 ∈ 𝐿 (𝑑𝜈𝛼 ), we have
˜𝛼 (𝑓 ) ∘ 𝜃(𝜇, 𝜆),
F𝛼 (𝑓 ) (𝜇, 𝜆) = F
∀(𝜇, 𝜆) ∈ Γ,
(2.11)
where
∀(𝜇, 𝜆) ∈ ℝ2 ,
˜𝛼 (𝑓 ) (𝜇, 𝜆) =
F
∫
+∞
∫
𝑓 (𝑟, 𝑥)𝑗𝛼 (𝑟𝜇) exp(−𝑖𝜆𝑥)𝑑𝜈𝛼 (𝑟, 𝑥), (2.12)
0
ℝ
and 𝜃 is the function defined by (2.9).
∙ Let 𝑓 ∈ 𝐿1 (𝑑𝜈𝛼 ) such that the function F𝛼 (𝑓 ) belongs to the space 𝐿1 (𝑑𝛾𝛼 ),
then we have the following inversion formula for F𝛼 , for almost every
(𝑟, 𝑥) ∈ [0, +∞[×ℝ,
∫ ∫
𝑓 (𝑟, 𝑥) =
F𝛼 (𝑓 )(𝜇, 𝜆)𝜑𝜇,𝜆 (𝑟, 𝑥)𝑑𝛾𝛼 (𝜇, 𝜆).
Γ+
1
∙ Let 𝑓 ∈ 𝐿 (𝑑𝜈𝛼 ). For all (𝑠, 𝑦) ∈ [0, +∞[×ℝ, we have
(
)
∀(𝜇, 𝜆) ∈ Γ,
F𝛼 𝜏(𝑠,𝑦) (𝑓 ) (𝜇, 𝜆) = 𝜑𝜇,𝜆 (𝑠, 𝑦)F𝛼 (𝑓 )(𝜇, 𝜆).
∙ For 𝑓, 𝑔 ∈ 𝐿1 (𝑑𝜈𝛼 ), we have
∀(𝜇, 𝜆) ∈ Γ,
F𝛼 (𝑓 ∗ 𝑔) (𝜇, 𝜆) = F𝛼 (𝑓 )(𝜇, 𝜆)F𝛼 (𝑔)(𝜇, 𝜆).
∙ Let 𝑝 ∈ [1, +∞]. the function 𝑓 belongs to 𝐿𝑝 (𝑑𝜈𝛼 ) if, and only if the
function 𝑓 ∘ 𝜃 belongs to the space 𝐿𝑝 (𝑑𝛾𝛼 ) and we have
𝑓 ∘ 𝜃
= ∥𝑓 ∥𝑝,𝜈𝛼 .
(2.13)
𝑝,𝛾𝛼
˜𝛼 is an isometric isomorphism from 𝐿2 (𝑑𝜈𝛼 ) onto itself,
Since the mapping F
then the relations (2.11) and (2.13) show that the Fourier transform F𝛼
is an isometric isomorphism from 𝐿2 (𝑑𝜈𝛼 ) into 𝐿2 (𝑑𝛾𝛼 ), namely, for every
𝑓 ∈ 𝐿2 (𝑑𝜈𝛼 ), the function F𝛼 (𝑓 ) belongs to the space 𝐿2 (𝑑𝛾𝛼 and we have
∥F𝛼 (𝑓 )∥2,𝛾𝛼 = ∥𝑓 ∥2,𝜈𝛼 .
(2.14)
SPACES OF 𝐷𝐿𝑝 -TYPE
23
Proposition 2.4. For all 𝑓 in 𝐿𝑝 (𝑑𝜈𝛼 ), 𝑝 ∈ [1, 2]; the function F𝛼 (𝑓 ) lies in
′
𝑝
𝐿𝑝 (𝑑𝛾𝛼 ), 𝑝′ = 𝑝−1
, and we have
F𝛼 (𝑓 ) ′
⩽ ∥𝑓 ∥𝑝,𝜈𝛼 .
𝑝 ,𝛾𝛼
Proof. The result follows from the relations (2.10), (2.14) and the Riesz-Thorin
theorem’s [25, 26].
□
We denote by
∙ S∗ (Γ) (see [3, 28]) the space of functions 𝑓 : Γ −→ ℂ infinitely differentiable, even with respect to the first variable and rapidly decreasing together
with all their derivatives, that means for all 𝑘1 , 𝑘2 , 𝑘3 ∈ ℕ,
( ) ( )
(
) ∂ 𝑘2 ∂ 𝑘3
2
2 𝑘1 sup
1 + 𝜇 + 2𝜆
𝑓 (𝜇, 𝜆) < +∞,
∂𝜇
∂𝜆
(𝜇,𝜆)∈Γ
where
⎧
∂



⎨ ∂𝑟 (𝑓 (𝑟, 𝜆)) ,
if 𝜇 = 𝑟 ∈ ℝ;
∂𝑓
(𝜇, 𝜆) =

∂𝜇

1 ∂

⎩
(𝑓 (𝑖𝑡, 𝜆)) , if 𝜇 = 𝑖𝑡, ∣𝑡∣ ⩽ ∣𝜆∣.
𝑖 ∂𝑡
( )
( )
∙ S∗′ ℝ2 and S∗′ (Γ) are respectively the dual spaces of S∗ ℝ2 and S∗ (Γ).
Each of these spaces is equipped with its usual topology.
Remark 2.5. (See [3]) The Fourier transform F𝛼 is a topological isomorphism
from S∗ (ℝ2 ) onto S∗ (Γ). The inverse mapping is given by for all (𝑟, 𝑥) ∈ ℝ2 ,
∫ ∫
−1
F𝛼 (𝑓 )(𝑟, 𝑥) =
𝑓 (𝜇, 𝜆)𝜑𝜇,𝜆 (𝑟, 𝑥)𝑑𝛾𝛼 (𝜇, 𝜆).
Γ+
Definition 2.6. The Fourier transform F𝛼 is defined for all 𝑇 ∈ S∗′ (ℝ2 ) by
⟨F𝛼 (𝑇 ), 𝜑⟩ = ⟨𝑇, F𝛼−1 (𝜑)⟩,
𝜑 ∈ S∗ (Γ).
Since the Fourier transform F𝛼 is an isomorphism from S∗ (ℝ2 ) into S∗ (Γ), we
deduce that F𝛼 is also an isomorphism from S∗′ (ℝ2 ) into S∗′ (Γ).
3. The space M𝑝 (ℝ2 )
We denote by
∙ Δ𝛼 the partial differential operator defined by
( 2
)
∂
2𝛼 + 1 ∂
∂2
Δ𝛼 = −
+
+
.
∂𝑟2
𝑟 ∂𝑟 ∂𝑥2
∙ For all 𝑓 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ), 𝑝 ∈ [1, +∞], 𝑇𝑓 is the element of S∗′ (ℝ2 ) defined by
∫ +∞ ∫
∀𝜑 ∈ S∗ (ℝ2 ),
⟨𝑇𝑓 , 𝜑⟩ =
𝑓 (𝑟, 𝑥)𝜑(𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥).
0
ℝ
∙ For all 𝑔 ∈ 𝐿 (𝑑𝛾𝛼 ), 𝑝 ∈ [1, +∞], 𝑇𝑔 is the element of S∗′ (Γ) defined by
∫ ∫
∀𝜓 ∈ S∗ (Γ),
⟨𝑇𝑔 , 𝜓⟩ =
𝑔(𝜇, 𝜆)𝜓(𝜇, 𝜆)𝑑𝛾𝛼 (𝜇, 𝜆),
𝑝
Γ+
24
C.BACCAR AND L.T.RACHDI
From proposition 2.4 and remark 2.5, we deduce that for all 𝑓 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ), 1 ⩽ 𝑝 ⩽ 2,
′
the function F𝛼 (𝑓 ) belongs to the space 𝐿𝑝 (𝑑𝛾𝛼 ) and we have
F𝛼 (𝑇𝑓 ) = 𝑇F𝛼 (𝑓ˇ) ,
(3.1)
𝑝
.
𝑝−1
On the other hand, since the operator Δ𝛼 is continuous from S∗ (ℝ2 ) into itself; we
define Δ𝛼 on 𝑆∗′ (ℝ2 ) by setting
with 𝑝′ =
∀𝜑 ∈ S∗ (ℝ2 ), ∀𝑇 ∈ 𝑆∗′ (ℝ2 );
⟨Δ𝛼 (𝑇 ), 𝜑⟩ = ⟨𝑇, Δ𝛼 (𝜑)⟩.
Then Δ𝛼 becomes a continuous operator from S∗′ (ℝ2 ) into itself; moreover for all
𝑓 ∈ S∗ (ℝ2 ) and for all integer 𝑘 we have
Δ𝑘𝛼 (𝑇𝑓 ) = 𝑇Δ𝑘𝛼 (𝑓 ) .
Definition 3.1. Let 𝑝 ∈ [1, +∞]. We define M𝑝 (ℝ2 ) to be the space of measurable
functions 𝑓 on ℝ2 , even with respect to the first variable, and such that for all
𝑘 ∈ ℕ there exists a function Δ𝑘𝛼 (𝑓 ) ∈ 𝐿𝑝 (𝑑𝜈𝛼 ), satisfying
in S∗′ (ℝ2 ).
Δ𝑘𝛼 (𝑇𝑓 ) = 𝑇Δ𝑘𝛼 (𝑓 ) ,
The space M𝑝 (ℝ2 ) is equipped with the topology generated by the family of
norms
𝑚 ∈ ℕ,
𝛾𝑚,𝑝 (𝑓 ) = 𝑚𝑎𝑥 Δ𝑘𝛼 (𝑓 )𝑝,𝜈𝛼 ,
0⩽𝑘⩽𝑚
Also, we define a distance 𝑑𝑝 , on M𝑝 (ℝ2 ) by
∞
∑
1 𝛾𝑚,𝑝 (𝑓 − 𝑔)
.
𝑚 1+𝛾
2
𝑚,𝑝 (𝑓 − 𝑔)
𝑚=0
(
)
converges to 0 in M𝑝 (ℝ2 ), 𝑑𝑝 if,and only if
∀(𝑓, 𝑔) ∈ M𝑝 (ℝ2 ),
Then, a sequence (𝑓𝑘 )𝑘∈ℕ
𝑑𝑝 (𝑓, 𝑔) =
∀𝑚 ∈ ℕ;
𝛾𝑚,𝑝 (𝑓𝑘 ) −→ 0 .
𝑘→∞
In the following, we will give some properties of the space M𝑝 (ℝ2 ).
(
)
Proposition 3.2. M𝑝 (ℝ2 ), 𝑑𝑝 is a Frchet space.
Proof. Let (𝑓𝑚 )𝑚∈ℕ be a Cauchy sequence in (M𝑝 (ℝ2 ), 𝑑𝑝 ) and (Δ𝑘𝛼 (𝑓𝑚 ))𝑚∈ℕ ⊂
𝐿𝑝 (𝑑𝜈𝛼 ), such that
Δ𝑘𝛼 (𝑇𝑓𝑚 ) = 𝑇Δ𝑘𝛼 (𝑓𝑚 ) ,
𝑘 ∈ ℕ.
Then, for all 𝑘 ∈ ℕ; (Δ𝑘𝛼 (𝑓𝑚 ))𝑚∈ℕ is a Cauchy sequence in 𝐿𝑝 (𝑑𝜈𝛼 ). We put
𝑔𝑘 =
lim Δ𝑘𝛼 (𝑓𝑚 ),
𝑚→+∞
𝑘 ∈ ℕ,
(3.2)
in 𝐿𝑝 (𝑑𝜈𝛼 ). Thus,
∀ 𝑘 ∈ ℕ;
lim Δ𝑘𝛼 (𝑇𝑓𝑚 ) =
𝑚→+∞
lim 𝑇Δ𝑘𝛼 (𝑓𝑚 ) = 𝑇𝑔𝑘 , in S∗′ (ℝ2 ).
𝑚→+∞
Since the operator Δ𝛼 is continuous from S∗′ (ℝ2 ) into itself and using the relation
(3.2) we deduce that for all 𝑘 ∈ ℕ
Δ𝑘𝛼 (𝑇𝑔0 ) = 𝑇𝑔𝑘
This equality shows that the function 𝑔0 belongs to the space M𝑝 (ℝ2 ) and that for
all 𝑘 ∈ ℕ, Δ𝑘𝛼 (𝑔0 ) = 𝑔𝑘 .
SPACES OF 𝐷𝐿𝑝 -TYPE
25
Now, the relation (3.2) implies that the sequence (𝑓𝑚 )𝑚 converges to 𝑔0 in
(M𝑝 (ℝ2 ), 𝑑𝑝 )
□
Remark 3.3. The operator Δ𝛼 is continuous from M𝑝 (ℝ2 ) into itself. Moreover,
for all 𝑚 ∈ ℕ; we have
∀𝑓 ∈ M𝑝 (ℝ2 );
𝛾𝑚,𝑝 (Δ𝛼 (𝑓 )) ⩽ 𝛾𝑚+1,𝑝 (𝑓 ).
We denote by
∙ C∗ (ℝ2 ) the space of continuous functions on ℝ2 , even with respect to the first
variable.
∙ E∗ (ℝ2 ) the subspace of C∗ (ℝ2 )consisting of infinitely differentiable functions
on ℝ2 .
Proposition 3.4. Let 𝑝 ∈ [1, 2] and 𝑓 ∈ M𝑝 (ℝ2 ) then
(i) For all 𝑘 ∈ ℕ, the function
(𝜇, 𝜆) 7−→ (1 + 𝜇2 + 2𝜆2 )𝑘 F𝛼 (𝑓 )(𝜇, 𝜆)
′
belongs to the space 𝐿𝑝 (𝑑𝛾𝛼 ), with 𝑝′ =
(ii) M𝑝 (ℝ2 ) ∩ C∗ (ℝ2 ) ⊂ E∗ (ℝ2 ).
𝑝
𝑝−1 .
Proof. (i) Let 𝑓 ∈ M𝑝 (ℝ2 ), 1 ⩽ 𝑝 ⩽ 2. From the relation (3.1), we have
F𝛼 (Δ𝑘𝛼 (𝑇𝑓 )) = F𝛼 (𝑇Δ𝑘𝛼 (𝑓 ) ) = 𝑇F𝛼 (Δ𝑘𝛼 (𝑓ˇ)) .
On the other hand,
F𝛼 (Δ𝑘𝛼 (𝑇𝑓 )) = (𝜇2 + 2𝜆2 )𝑘 F𝛼 (𝑇𝑓 ),
= 𝑇(𝜇2 +2𝜆2 )𝑘 F𝛼 (𝑓ˇ) ,
hence,
(𝜇2 + 2𝜆2 )𝑘 F𝛼 (𝑓 ) = F𝛼 (Δ𝑘𝛼 (𝑓 )),
this equality, together with the fact that the function F𝛼 (Δ𝑘𝛼 (𝑓 )) belongs to the
′
space 𝐿𝑝 (𝑑𝜈𝛼 ) implies (i).
(ii) Let 𝑓 ∈ M𝑝 (ℝ2 ) ∩ C∗ (ℝ2 ). From the assertion (i) and the relations (2.11)
and (2.13), we deduce that for all 𝑘 ∈ ℕ, the function
˜𝛼 (𝑓 )(𝜇, 𝜆),
(𝜇, 𝜆) 7−→ (𝜇2 + 𝜆2 )𝑘 F
′
belongs to the space 𝐿𝑝 (𝑑𝜈𝛼 ). Hence, by using Hölder’s inequality, we deduce that
˜𝛼 (𝑓 ) belongs to the space 𝐿1 (𝑑𝜈𝛼 ) ∩ 𝐿2 (𝑑𝜈𝛼 ).
the function F
˜𝛼 is an isometric isomorphism from 𝐿2 (𝑑𝜈𝛼 )
On the other hand, the transform F
˜𝛼 , and using the continuity of the
onto itself, then from the inversion formula for F
2
function 𝑓 , we have for all (𝑟, 𝑥) ∈ ℝ ,
∫ +∞ ∫
˜𝛼 (𝑓 )(𝜇, 𝜆)𝑗𝛼 (𝑟𝜇) exp (𝑖𝜆𝑥)𝑑𝜈𝛼 (𝜇, 𝜆).
𝑓 (𝑟, 𝑥) =
F
(3.3)
0
ℝ
Then, the result follows from the derivative theorem, the relations (2.3) and (3.3).
□
Proposition 3.5. Let 𝑝 ∈ [1, 2]. Then, for all 𝜂 ∈ [2, +∞],
M𝑝 (ℝ2 ) ∩ C∗ (ℝ2 ) ⊂ M𝜂 (ℝ2 ).
26
C.BACCAR AND L.T.RACHDI
Proof. Let 𝑓 ∈ M𝑝 (ℝ2 ) ∩ C∗ (ℝ2 ), 𝑝 ∈ [1, 2], 𝜂 ⩾ 2 and 𝜂 ′ = 𝜂/(𝜂 − 1). From
proposition 3.4, we deduce that 𝑓 ∈ E∗ (ℝ2 ) and that for all 𝑘 ∈ ℕ, the function
˜𝛼 (𝑓 )(𝜇, 𝜆),
(𝜇, 𝜆) 7−→ (𝜇2 + 𝜆2 )𝑘 F
′
belongs to the space 𝐿𝑝 (𝑑𝜈𝛼 ). By applying Hölder’s inequality it follows that this
′
last function belongs to the space 𝐿𝜂 (𝑑𝜈𝛼 ).
On the other hand, for all (𝑟, 𝑥) ∈ ℝ2 ,
∫ +∞ ∫
( 2
)𝑘
𝑘
˜𝛼 (𝑓 )(𝜇, 𝜆)𝑗𝛼 (𝑟𝜇) exp (𝑖𝜆𝑥)𝑑𝜈𝛼 (𝜇, 𝜆),
𝜇 + 𝜆2 F
Δ𝛼 (𝑓 )(𝑟, 𝑥) =
0
ℝ
(
)
˜𝛼 (𝜇2 + 𝜆2 )𝑘 F
˜𝛼 (𝑓ˇ) (𝑟, 𝑥).
=F
′
From proposition 2.4 and the fact that for all 𝑔 ∈ 𝐿𝜂 (𝑑𝜈𝛼 ),
˜
,
∥F𝛼 (𝑔)∥𝜂,𝛾 = F
𝛼 (𝑔)
𝜂,𝜈𝛼
we deduce that for all 𝑘 ∈ ℕ, the function Δ𝑘𝛼 (𝑓 ) belongs to the space 𝐿𝜂 (𝑑𝜈𝛼 ). □
4. The dual space M𝑝′ (ℝ2 )
In this section, we will give a new characterization of the dual space M𝑝′ (ℝ2 ) of
M𝑝 (ℝ2 ).
It is well known that for every 𝑓 ∈ M𝑝 (ℝ2 ), the family {𝔑𝑚,𝑝,𝜀 (𝑓 ), 𝑚 ∈ ℕ, 𝜀 > 0},
defined by
{
}
𝔑𝑚,𝑝,𝜀 (𝑓 ) = 𝑔 ∈ M𝑝 (ℝ2 ), 𝛾𝑚,𝑝 (𝑓 − 𝑔) < 𝜀
(
)
is a basis of neighborhoods of 𝑓 in M𝑝 (ℝ2 ), 𝑑𝑝 .
Hence, 𝑇 ∈ M𝑝′ (ℝ2 ) if, and only if there exist 𝑚 ∈ ℕ and 𝐶 > 0, such that
∀ 𝑓 ∈ M𝑝 (ℝ2 );
∣⟨𝑇, 𝑓 ⟩∣ ⩽ 𝐶𝛾𝑚,𝑝 (𝑓 ).
(4.1)
′
For 𝑓 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ) and 𝜑 ∈ M𝑝 (ℝ2 ), we put
∫ +∞ ∫
⟨Δ𝑘𝛼 (𝑇𝑓 ), 𝜑⟩ =
𝑓 (𝑟, 𝑥)Δ𝑘𝛼 (𝜑)(𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥);
0
(4.2)
ℝ
with Δ𝑘𝛼 (𝑇𝜑 ) = 𝑇Δ𝑘𝛼 (𝜑) . Then
∣⟨Δ𝑘𝛼 (𝑇𝑓 ), 𝜑⟩∣ ⩽ ∥𝑓 ∥𝑝′ ,𝜈𝛼 ∥Δ𝑘𝛼 (𝜑)∥𝑝,𝜈𝛼
⩽ ∥𝑓 ∥𝑝′ ,𝜈𝛼 𝛾𝑘,𝑝 (𝜑),
′
this proves that for all 𝑓 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ) and 𝑘 (∈ ℕ,) the functional Δ𝑘𝛼 (𝑇𝑓 ) defined by
the relation (4.2), belongs to the space M𝑝′ ℝ2 .
In the following, we will prove that every element of M𝑝′ (ℝ2 ) is also of this type.
Theorem 4.1. Let 𝑇 ∈ S∗′ (ℝ2 ). Then 𝑇 belongs to M𝑝′ (ℝ2 ), 1 ⩽ 𝑝 < +∞, if and
′
only if there exist 𝑚 ∈ ℕ and {𝑓0 , ..., 𝑓𝑚 } ⊂ 𝐿𝑝 (𝑑𝜈𝛼 ), such that
𝑇 =
𝑚
∑
Δ𝑘𝛼 (𝑇𝑓𝑘 ),
𝑘=0
where Δ𝑘𝛼 (𝑇𝑓𝑘 ) is given by the relation (4.2).
(4.3)
SPACES OF 𝐷𝐿𝑝 -TYPE
27
Proof. It is clear that if
𝑇 =
𝑚
∑
′
Δ𝑘𝛼 (𝑇𝑓𝑘 ),
{𝑓0 , ..., 𝑓𝑚 } ⊂ 𝐿𝑝 (𝑑𝜈𝛼 ),
𝑘=0
then 𝑇 belongs to the space M𝑝′ (ℝ2 ). Conversely, suppose that 𝑇 ∈ M𝑝′ (ℝ2 ). From
the relation (4.1), there exist 𝑚 ∈ ℕ and 𝐶 > 0, such that
∀ 𝜑 ∈ M𝑝 (ℝ2 ),
∣⟨𝑇, 𝜑⟩∣ ⩽ 𝐶𝛾𝑚,𝑝 (𝜑).
{
}
= (𝑓0 , ..., 𝑓𝑚 ), 𝑓𝑘 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ), 0 ⩽ 𝑘 ⩽ 𝑚 , equipped with the
(
)𝑚+1
Let 𝐿𝑝 (𝑑𝜈𝛼 )
norm
(𝑓0 , ..., 𝑓𝑚 )
(𝐿𝑝 (𝑑𝜈𝛼 ))𝑚+1
=
𝑚𝑎𝑥
0⩽𝑘⩽𝑚
∥𝑓𝑘 ∥𝑝,𝜈𝛼 .
We consider the mappings
(
)𝑚+1
𝒜 : M𝑝 (ℝ2 ) −→ 𝐿𝑝 (𝑑𝜈𝛼 )
𝜑 7−→ (𝜑, Δ𝛼 (𝜑), ..., Δ𝑚
𝛼 (𝜑)),
and
𝔅 : 𝒜(M𝑝 (ℝ2 )) −→ ℂ; 𝔅(𝒜𝜑) = ⟨𝑇, 𝜑⟩.
From the relation (4.1), we deduce that
(
)
𝔅 𝒜(𝜑) = ∣⟨𝑇, 𝜑⟩∣
⩽ 𝐶 𝒜(𝜑)(𝐿𝑝 (𝑑𝜈
𝛼 ))
𝑚+1
,
this means that 𝔅 is a continuous functional on the subspace 𝒜(M𝑝 (ℝ2 )) of the
(
)𝑚+1
space 𝐿𝑝 (𝑑𝜈𝛼 )
. From Hahn Banach theorem, there exists a continuous extension of 𝔅 to (𝐿𝑝 (𝑑𝜈𝛼 ))𝑚+1 , denoted again by 𝔅.
( ′
)𝑚+1
(
)
By Riesz theorem, there exist 𝑓0 , ..., 𝑓𝑚 ∈ 𝐿𝑝 (𝑑𝜈𝛼 )
, such that for all
(
)𝑚+1
(𝜑0 , ..., 𝜑𝑚 ) ∈ 𝐿𝑝 (𝑑𝜈𝛼 )
,
𝔅(𝜑0 , ..., 𝜑𝑚 ) =
𝑚 ∫
∑
𝑘=0
0
+∞
∫
𝑓𝑘 (𝑟, 𝑥)𝜑𝑘 (𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥).
ℝ
By means of the relation (4.2), we deduce that for 𝜑 ∈ M𝑝 (ℝ2 ), we have
𝑚 ∫ +∞ ∫
𝑚
∑
∑
⟨𝑇, 𝜑⟩ =
𝑓𝑘 (𝑟, 𝑥)Δ𝑘𝛼 (𝜑)(𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥) =
⟨Δ𝑘𝛼 (𝑇𝑓𝑘 ), 𝜑⟩.
𝑘=0
0
ℝ
𝑘=0
□
Proposition 4.2. Let 𝑝 ⩾ 2. Then for all 𝑇 ∈ M𝑝′ (ℝ2 ), there exist 𝑚 ∈ ℕ and
𝐹 ∈ 𝐿𝑝 (𝑑𝛾𝛼 ); such that
F𝛼 (𝑇 ) = 𝑇(1+𝜇2 +2𝜆2 )𝑚 𝐹 .
28
C.BACCAR AND L.T.RACHDI
Proof. Let 𝑇 ∈ M𝑝′ (ℝ2 ). From Theorem 4.1, there exist 𝑚 ∈ ℕ and (𝑓0 , ..., 𝑓𝑚 ) ⊂
( ′
)𝑚+1
𝑝
𝐿𝑝 (𝑑𝜈𝛼 )
, such that
, 𝑝′ = 𝑝−1
𝑇 =
𝑚
∑
Δ𝑘𝛼 (𝑇𝑓𝑘 ).
𝑘=0
Consequently,
F𝛼 (𝑇 ) =
𝑚
∑
F𝛼 (Δ𝑘𝛼 (𝑇𝑓𝑘 )) =
𝑘=0
𝑚
∑
(𝜇2 + 2𝜆2 )𝑘 F𝛼 (𝑇𝑓𝑘 ).
𝑘=0
From the relation (3.1), we get
F𝛼 (𝑇 ) = 𝑇(1+𝜇2 +2𝜆2 )𝑚 𝐹 ,
where
𝐹 =
𝑚
∑
𝑘=0
(𝜇2 + 2𝜆2 )𝑘
F𝛼 (𝑓ˇ𝑘 ).
(1 + 𝜇2 + 2𝜆2 )𝑚
□
Proposition 4.3. Let 𝑇 ∈ S∗′ (ℝ2 ), then 𝑇 ∈ M2′ (ℝ2 ) if, and only if there exist
𝑚 ∈ ℕ and 𝐹 ∈ 𝐿2 (𝑑𝛾𝛼 ), such that
F𝛼 (𝑇 ) = 𝑇(1+𝜇2 +2𝜆2 )𝑚 𝐹 .
(4.4)
Proof. From Proposition 4.2, we deduce that if 𝑇 ∈ M2′ (ℝ2 ), then there exist 𝑚 ∈ ℕ
and 𝐹 ∈ 𝐿2 (𝑑𝛾𝛼 ) verifying (4.4).
Conversely, suppose that (4.4) holds with 𝐹 ∈ 𝐿2 (𝑑𝛾𝛼 ). Since F𝛼 is an isometric
isomorphism from 𝐿2 (𝑑𝜈𝛼 ) into 𝐿2 (𝑑𝛾𝛼 ), then there exists 𝐺 ∈ 𝐿2 (𝑑𝜈𝛼 ), such that
F𝛼 (𝐺) = 𝐹 and from the relation (3.1), we have
F𝛼 (𝑇𝐺ˇ ) = 𝑇𝐹 .
Consequently,
F𝛼 (𝑇 ) = F𝛼 ((𝐼 + Δ𝛼 )𝑚 (𝑇𝐺ˇ )),
thus,
𝑇 =
𝑚
∑
𝑘
𝐶𝑚
Δ𝑘𝛼 (𝑇𝐺ˇ ),
𝑘=0
and Theorem 4.1 implies that 𝑇 belongs to M2′ (ℝ2 ).
□
We denote by
∙ D∗ (ℝ2 ) the space of infinitely differentiable functions on ℝ2 , even with
respect to the first variable and with compact support, equipped with its
usual topology.
∙ For 𝑎 > 0, D∗,𝑎 (ℝ2 ) the {subspace of D∗ (ℝ2 ), consisting
of function 𝑓 , such
}
that 𝑠𝑢𝑝𝑝𝑓 ⊂ ℬ(0, 𝑎) = (𝑟, 𝑥) ∈ ℝ2 , 𝑟2 + 𝑥2 ⩽ 𝑎2 .
∙ For 𝑎 > 0, D ′ ∗,𝑎 (ℝ2 ) the dual space of D∗,𝑎 (ℝ2 ).
∙ For 𝑎 > 0 and 𝑚 ∈ ℕ, 𝒲𝑎𝑚 (ℝ2 ) the space of function 𝑓 : ℝ2 7−→ ℂ; of
class 𝐶 2𝑚 on ℝ2 , even with respect to the first variable and with support
in ℬ(0, 𝑎), normed by
𝒩∞,𝑚 (𝑓 ) = 𝑚𝑎𝑥 Δ𝑘𝛼 (𝑓 )
.
0≤𝑘≤𝑚
∞,𝜈𝛼
SPACES OF 𝐷𝐿𝑝 -TYPE
29
Lemma 4.4. For all 𝑚 ∈ ℕ, there exists 𝛽 ∈ ℕ sufficiently large, such that the
function
)
(
1
˜𝛼
(𝜇, 𝜆) 7−→ 𝔤𝛽 (𝜇, 𝜆) = F
(𝜇, 𝜆),
(4.5)
𝛽
(1 + 𝑟2 + 𝑥2 )
is of class 𝐶 2𝑚 on ℝ2 , even with respect to the first variable, and infinitely differentiable on ℝ2 ∖ {(0, 0)}.
Proof. From the relation (2.12), we have for all (𝜇, 𝜆) ∈ ℝ2 ,
∫ +∞ ∫
𝑗𝛼 (𝑟𝜇) exp(−𝑖𝜆𝑥)
𝔤𝛽 (𝜇, 𝜆) =
𝑑𝜈𝛼 (𝑟, 𝑥).
𝛽
0
ℝ (1 + 𝑟 2 + 𝑥2 )
Using the relation (2.3) and the derivative theorem, we can choose 𝛽 sufficiently
large, such that the function 𝔤𝛽 is of class 𝐶 2𝑚 on ℝ2 .
On the other hand, from the integral representation (2.2) of the function 𝑗𝛼 and
applying the Fubini’s theorem, we get
1
𝔤𝛽 (𝜇, 𝜆) =
1
𝜋2𝛼− 2 Γ(𝛼 + 12 )
)
∫
∫ (∫ +∞
( 𝑟 2
)
𝑟
2 𝛼− 12
(𝑟 − 𝑡 )
cos(𝜇𝑡)𝑑𝑡 𝑑𝑟 exp(−𝑖𝜆𝑥)𝑑𝑥
×
(1 + 𝑟2 + 𝑥2 )𝛽 0
ℝ
0
(
)
∫
∫ +∞
∫
)
( +∞ (𝑟2 − 𝑡2 )𝛼− 21
1
=
𝑟𝑑𝑟 𝑑𝑡 exp(−𝑖𝜆𝑥)𝑑𝑥,
cos(𝜇𝑡)
1
(1 + 𝑟2 + 𝑥2 )𝛽
𝜋2𝛼− 2 Γ(𝛼 + 12 ) ℝ
0
𝑡
using the change of variables 𝑠 =
𝔤𝛽 (𝜇, 𝜆) =
𝑟 2 −𝑡2
1+𝑟 2 +𝑥2 ,
Γ(𝛽 − 𝛼 − 12 )
𝜋2𝛼+1/2 Γ(𝛽)
∫ ∫
ℝ
Γ(𝛽 − 𝛼 − 1/2)
=
2𝛼+1/2 Γ(𝛽)
∫
0
we obtain
+∞
0
+∞
cos(𝜇𝑡) exp(−𝑖𝜆𝑥)
𝑑𝑡𝑑𝑥,
(1 + 𝑡2 + 𝑥2 )𝛽−𝛼−1/2
√
𝑗0 (𝑡 𝜇2 + 𝜆2 )
1 𝑡𝑑𝑡.
(1 + 𝑡2 )𝛽−𝛼− 2
again, from the relation (2.2), we deduce that for all 𝑥 ∈ ℝ,
∫ +∞
∫ +∞
∫
)
( +∞ (𝑡2 − 𝑠2 )−1/2
𝑗0 (𝑡𝑥)
𝑡𝑑𝑡
=
cos(𝑠𝑥)
1
1 𝑡𝑑𝑡 𝑑𝑠,
𝛽−𝛼−
𝛽−𝛼−
2
2
2
2
(1 + 𝑡 )
(1 + 𝑡 )
0
0
𝑠
√
∫ +∞
𝜋 Γ(𝛽 − 𝛼 − 1)
cos(𝑠𝑥)
=
𝑑𝑠,
2 Γ(𝛽 − 𝛼 − 21 ) 0
(1 + 𝑠2 )𝛽−𝛼−1
and therefore for all (𝜇, 𝜆) ∈ ℝ2 ,
√
√
∫
𝜋 Γ(𝛽 − 𝛼 − 1) +∞ cos(𝑠 𝜇2 + 𝜆2 )
𝔤𝛽 (𝜇, 𝜆) = 𝛼+ 3
𝑑𝑠.
Γ(𝛽)
(1 + 𝑠2 )𝛽−𝛼−1
2 2
0
Now, from [10, 29] it follows that for all (𝜇, 𝜆) ∈ ℝ2 ∖ {(0, 0)},
√
√
3
1
( 𝜇2 + 𝜆2 )𝛽−𝛼− 2 K𝛽−𝛼− 23 ( 𝜇2 + 𝜆2 )
𝔤𝛽 (𝜇, 𝜆) = 𝛽+1
2
Γ(𝛽)
where K𝛽−𝛼− 23 is the Bessel function of second kind and index 𝛽 − 𝛼 − 32 , called
also the Mac-Donald function.
This shows that the function 𝔤𝛽 is infinitely differentiable on ℝ2 ∖ {(0, 0)}, even
with respect to the first variable.
□
30
C.BACCAR AND L.T.RACHDI
Proposition 4.5. Let 𝑎 > 0 and 𝑚 ∈ ℕ. Then there exists 𝑛𝑜 ∈ ℕ, such that
for every 𝑛 ∈ ℕ, 𝑛 ⩾ 𝑛𝑜 , it is possible to find 𝜑𝑛 ∈ D∗,𝑎 (ℝ2 ) and 𝜓𝑛 ∈ 𝒲𝑎𝑚 (ℝ2 )
satisfying
𝛿 = (𝐼 + Δ𝛼 )𝑛 𝑇𝜑𝑛 + 𝑇𝜓𝑛
in S∗′ (ℝ2 ). Where 𝛿 is the Dirac distribution.
Proof. Let 𝜅 ∈ D∗,𝑎 (ℝ2 ), such that
𝑎2
,
𝜅(𝑟, 𝑥) = 1.
4
From Lemma 4.4; there exists 𝑛0 ∈ ℕ, such that for all 𝑛 ⩾ 𝑛0 , the function
𝔤𝑛 is of class 𝐶 2𝑚 on ℝ2 , even with respect to the first variable and infinitely
˜𝛼 defined by relation (2.12), is
differentiable on ℝ2 ∖{(0, 0)}. Since the transform F
2
an isomorphism from S∗ (ℝ ) onto itself, and for all 𝜑 ∈ S∗ (ℝ2 ), (𝑟, 𝑥) ∈ ℝ2 , we
have
∫ +∞ ∫
˜𝛼 (𝜑)(𝑠, 𝑦)𝑗𝛼 (𝑟𝑠) exp(𝑖𝑥𝑦)𝑑𝜈𝛼 (𝑠, 𝑦).
𝜑(𝑟, 𝑥) =
F
(4.6)
∀ (𝑟, 𝑥) ∈ ℝ2 ,
0
𝑟 2 + 𝑥2 ⩽
ℝ
Then, from the relations (4.5) and (4.6), we deduce that for all 𝜑 ∈ S∗ (ℝ2 ), we
have
⟨(𝐼 + Δ𝛼 )𝑛 𝑇𝔤𝑛 , 𝜑⟩ = ⟨𝑇𝔤𝑛 , (𝐼 + Δ𝛼 )𝑛 𝜑⟩,
∫ +∞ ∫
=
𝔤𝑛 (𝑟, 𝑥)(𝐼 + Δ𝛼 )𝑛 𝜑(𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥)
0
ℝ
∫ +∞ ∫
(
)
1
˜𝛼 (𝐼 + Δ𝛼 )𝑛 (𝜑)(𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥),
F
=
2 + 𝑥2 ) 𝑛
(1
+
𝑟
0
ℝ
∫ +∞ ∫
˜𝛼 (𝜑)(𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥),
=
F
0
ℝ
= 𝜑(0, 0).
This means that for all 𝑛 ⩾ 𝑛0 ; (𝐼 + Δ𝛼 )𝑛 𝑇𝔤𝑛 = 𝛿. Then
𝜅(𝐼 + Δ𝛼 )𝑛 𝑇𝔤𝑛 = (𝐼 + Δ𝛼 )𝑛 𝑇𝔤𝑛 = 𝛿.
(4.7)
2
Using the fact that the function 𝔤𝑛 is infinitely differentiable on ℝ ∖{(0, 0)}, even
with respect to the first variable, we deduce that the function
𝜑𝑛 (𝑟, 𝑥) = (𝜅 − 1)(𝐼 + Δ𝛼 )𝑛 𝔤𝑛 + (𝐼 + Δ𝛼 )𝑛 ((1 − 𝜅)𝔤𝑛 ),
(4.8)
belongs to the space D∗,𝑎 (ℝ2 ). From the relation (4.7), we have
𝑇(𝜅−1)(𝐼+Δ𝛼 )𝑛 𝔤𝑛 = (𝜅 − 1)(𝐼 + Δ𝛼 )𝑛 𝑇𝔤𝑛 = 0,
and this implies, by using the relation (4.8) that
𝑇𝜑𝑛 = 𝑇(𝐼+Δ𝛼 )𝑛 ((1−𝜅)𝔤𝑛 ) = (𝐼 + Δ𝛼 )𝑛 (𝑇(1−𝜅)𝔤𝑛 ).
Hence,
𝑇𝜑𝑛 + (𝐼 + Δ𝛼 )𝑛 𝑇𝜅𝔤𝑛 = (𝐼 + Δ𝛼 )𝑛 𝑇𝔤𝑛 = 𝛿,
and this completes the proof of the proposition if we pick 𝜓𝑛 = 𝜅𝔤𝑛 .
□
In the following, we will prove that the elements of all bounded subset ℬ ′ ⊂
can be continuously extended to the space 𝒲𝑎𝑚 (ℝ2 ). For this we define
some new families of norms on the space D∗,𝑎 (ℝ2 ).
For 𝑓 ∈ D∗,𝑎 (ℝ2 ), 𝑎 > 0, we denote by
′
D∗,𝑎
(ℝ2 ),
SPACES OF 𝐷𝐿𝑝 -TYPE
31
∂ 𝑘 ∂ 𝑘 ( ) 1 ( ) 2 𝑓 ,
𝑘1 +𝑘2 ⩽𝑚 ∂𝑟
∂𝑥
∞,𝜈𝛼
∂
˜𝑚 (𝑓 ) =
∙ 𝒫
𝑚𝑎𝑥 ℓ𝑘𝛼1 ( )𝑘2 𝑓 ,
𝑘1 +𝑘2 ⩽𝑚 ∂𝑥
∞,𝜈𝛼
∙ 𝒩𝑝,𝑚 (𝑓 ) = 𝑚𝑎𝑥 ∥Δ𝑘𝛼 (𝑓 )∥𝑝,𝜈𝛼 , 𝑝 ∈ [1, +∞].
∙ 𝒫𝑚 (𝑓 ) =
𝑚𝑎𝑥
0≤𝑘≤𝑚
where ℓ𝛼 is the Bessel operator defined by
ℓ𝛼 =
∂2
2𝛼 + 1 ∂
+
.
∂𝑟2
𝑟 ∂𝑟
Lemma 4.6. (i)For all 𝑚 ∈ ℕ, there exists 𝐶1 > 0, such that
∀ 𝜑 ∈ D∗,𝑎 (ℝ2 ),
˜𝑚 (𝜑).
𝒫𝑚 (𝜑) ⩽ 𝐶1 𝒫
(ii) For all 𝑚 ∈ ℕ, there exist 𝐶2 > 0 and 𝑚′ ∈ ℕ, such that
∀ 𝜑 ∈ D∗,𝑎 (ℝ2 ),
˜𝑚 (𝜑) ⩽ 𝐶2 𝒩𝑝,𝑚′ (𝜑).
𝒫
Proof. (i) Let 𝜑 ∈ D∗,𝑎 (ℝ2 ). By induction on 𝑘1 , we have
(
where
get
∂
∂𝑟 2
(
=
𝑘1
∑
∂
∂ 𝑘1 ∂ 𝑘2
∂
𝑃𝑛 (𝑟)( 2 )𝑛 ( )𝑘2 𝜑(𝑟, 𝑥),
) ( ) 𝜑(𝑟, 𝑥) =
∂𝑟
∂𝑥
∂𝑟
∂𝑥
𝑛=0
1 ∂
𝑟 ∂𝑟
(4.9)
and 𝑃𝑛 is a real polynomial. Also, by induction, for all 𝑛 ⩾ 1, we
∂ 𝑛 ∂ 𝑘2
) ( ) 𝜑(𝑟, 𝑥) =
(4.10)
∂𝑟2
∂𝑥
∫ 1
∫ 1
∂
2𝛼+1+2(𝑛−1)
...
ℓ𝑛𝛼 ( )𝑘2 𝜑(𝑟𝑡1 . . . 𝑡𝑛 , 𝑥)𝑡1
. . . 𝑡𝑛2𝛼+1 𝑑𝑡1 . . . 𝑑𝑡𝑛 ,
∂𝑥
0
0
from the relations (4.9) and (4.10), it follows that for all 𝑚 ∈ ℕ,
˜𝑚 (𝜑).
𝒫𝑚 (𝜑) ⩽ 𝐶1 𝒫
(ii) Let 𝑝 ∈ [1, +∞], 𝑚 ∈ ℕ and 𝑚1 ∈ ℕ, such that
1
)𝑚1 (
1 + 𝑟 2 + 𝑥2
1,𝜈𝛼
< +∞,
32
C.BACCAR AND L.T.RACHDI
then, for all 𝜑 ∈ D∗,𝑎 and 𝑘1 , 𝑘2 ∈ ℕ such that 𝑘1 + 𝑘2 ⩽ 𝑚, we have
−1 ( ( 𝑘 ∂ 𝑘
𝑘 ( ∂ )𝑘 2
) )
˜𝛼 ℓ 1 ( ) 2 (𝜑) ˜ F
F
ℓ𝛼1
=
(𝜑)
𝛼
𝛼
∂𝑥
∂𝑥
∞,𝜈𝛼
∞,𝜈𝛼
(
)
∂
𝑘
2
˜ 𝑘1
⩽
F𝛼 (ℓ𝛼 ∂𝑥 𝜑)
1,𝜈𝛼
2𝑘1 𝑘2 ˜
= 𝜇 𝜆 F𝛼 (𝜑)
1,𝜈𝛼
2
2 𝑚 ˜
⩽ (1 + 𝜇 + 𝜆 ) F𝛼 (𝜑)
1,𝜈𝛼
)
(
1
𝑚+𝑚1
˜
=
𝜑 F𝛼 (𝐼 + Δ𝛼 )
2
2
𝑚
1
(1 + 𝜇 + 𝜆 )
1,𝜈𝛼
(
)
1
˜
⩽
F𝛼 (𝐼 + Δ𝛼 )𝑚+𝑚1 𝜑 (1 + 𝜇2 + 𝜆2 )𝑚1 ∞,𝜈𝛼
1,𝜈
𝛼
1
(𝐼 + Δ𝛼 )𝑚+𝑚1 𝜑
,
⩽
(1 + 𝜇2 + 𝜆2 )𝑚1 1,𝜈𝛼
1,𝜈𝛼
and by Hölder’s inequality, we get
(
) 1′ 𝑘 ( ∂ )𝑘2 1
𝑝
ℓ𝛼1
(𝐼 + Δ𝛼 )𝑚+𝑚1 𝜑
⩽
𝜈(ℬ(0,
𝑎))
,
𝜑
2
2
𝑚
𝑝,𝜈𝛼
1
∂𝑥
(1 + 𝜇 + 𝜆 )
∞,𝜈𝛼
1,𝜈𝛼
(
) 1′
1
𝑝
⩽
𝜈(ℬ(0,
𝑎))
2𝑚+𝑚1 𝒩𝑝,𝑚+𝑚1 (𝜑).
(1 + 𝜇2 + 𝜆2 )𝑚1 1,𝜈𝛼
which implies that
(
)1
˜𝑚 (𝜑) ⩽ 2𝑚+𝑚1 𝜈(ℬ(0, 𝑎)) 𝑝′ ∥
𝒫
(1 +
𝜇2
1
∥1,𝜈𝛼 𝒩𝑝,𝑚+𝑚1 (𝜑).
+ 𝜆2 )𝑚1
□
′
Theorem 4.7. Let 𝑎 > 0 and let 𝐵 ′ be a weakly bounded set of D∗,𝑎
(ℝ2 ). Then,
′
there exists 𝑚 ∈ ℕ, such that the elements of 𝐵 can be continuously extended to
𝒲𝑎𝑚 (ℝ2 ). Moreover, the family of these extensions is equicontinuous.
′
Proof. Let 𝑝 ∈ [1, +∞[. Since 𝐵 ′ is weakly bounded in D∗,𝑎
(ℝ2 ), then from [27]
and Lemma 4.6 there exist a positive constant 𝐶 and 𝑚 ∈ ℕ, such that
∀𝑇 ∈ 𝐵 ′ , ∀𝜑 ∈ D∗,𝑎 (ℝ2 ),
∣⟨𝑇, 𝜑⟩∣ ⩽ 𝐶𝒩𝑝,𝑚 (𝜑).
We consider the mappings
(
)𝑚+1
𝒜 : 𝒲𝑎𝑚 (ℝ2 ) −→ 𝐿𝑝 (𝑑𝜈𝛼 )
,
𝜑 7−→ (Δ𝑘𝛼 (𝜑))0⩽𝑘⩽𝑚 ,
and for all 𝑇 ∈ 𝐵 ′ ,
𝔏𝑇 : 𝒜(D∗,𝑎 (ℝ2 )) −→ ℂ;
⟨𝔏𝑇 , 𝒜𝜑⟩ = ⟨𝑇, 𝜑⟩.
From the relation (4.11), we deduce that
∀ 𝜑 ∈ D∗,𝑎 (ℝ2 );
∣⟨𝔏𝑇 , 𝒜𝜑⟩∣ ⩽ 𝐶 ∥𝒜𝜑∥(𝐿𝑝 (𝑑𝜈𝛼 ))𝑚+1 ,
(4.11)
SPACES OF 𝐷𝐿𝑝 -TYPE
33
(
)
this means that 𝔏𝑇 is a continuous functional on the subspace 𝒜 D∗,𝑎 (ℝ2 ) of the
(
)𝑚+1
space 𝐿𝑝 (𝑑𝜈𝛼 )
, and that for all 𝑇 ∈ 𝐵 ′ ,
∥𝔏𝑇 ∥𝒜(D∗,𝑎 (ℝ2 )) =
sup
∥𝒜𝜑∥(𝐿𝑝 (𝑑𝜈𝛼 ))𝑚+1 ⩽1
∣⟨𝔏𝑇 , 𝒜𝜑⟩∣ ⩽ 𝐶
From the Hahn Banach theorem, 𝔏𝑇 can be continuously extended to the space
(
)𝑚+1
𝐿𝑝 (𝑑𝜈𝛼 )
, denoted again by 𝔏𝑇 . Furthermore, for all 𝑇 ∈ 𝐵 ′
⟨𝔏𝑇 , 𝜓⟩
∥𝔏𝑇 ∥ 𝑝
sup
𝑚+1 =
(𝐿 (𝑑𝜈𝛼 ))
∥𝜓∥(𝐿𝑝 (𝑑𝜈𝛼 ))𝑚+1 ⩽1
= ∥𝔏𝑇 ∥𝒜(D∗,𝑎 (ℝ2 )) ⩽ 𝐶.
(4.12)
′
Now, from the Riesz theorem, for all 𝑇 ∈ 𝐵 ′ , there exists (𝑓𝑘,𝑇 )0⩽𝑘⩽𝑚 ⊂ 𝐿𝑝 (𝑑𝜈𝛼 ),
(
)𝑚+1
such that for all 𝜓 = (𝜓0 , ..., 𝜓𝑚 ) ∈ 𝐿𝑝 (𝑑𝜈𝛼 )
,
⟨𝔏𝑇 , 𝜓⟩ =
𝑚 ∫
∑
0
𝑘=0
+∞
∫
𝑓𝑘,𝑇 (𝑟, 𝑥)𝜓𝑘 (𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥);
ℝ
with
∥𝔏𝑇 ∥(𝐿𝑝 (𝑑𝜈𝛼 ))𝑚+1 =
max ∥𝑓𝑘,𝑇 ∥𝑝′ ,𝜈𝛼 .
0⩽𝑘≤𝑚
Thus, from the relation (4.12) it follows that
∀𝑇 ∈ 𝐵 ′ , ∀𝑘 ∈ ℕ, 0 ⩽ 𝑘 ⩽ 𝑚;
∥𝑓𝑘,𝑇 ∥𝑝′ ,𝜈𝛼 ⩽ 𝐶.
(4.13)
In particular, for 𝜑 ∈ 𝒲𝑎𝑚 (ℝ2 ), we have
𝑚 ∫ +∞ ∫
∑
⟨𝔏𝑇 , 𝒜𝜑⟩ =
𝑓𝑘,𝑇 (𝑟, 𝑥)Δ𝑘𝛼 (𝜑)(𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥).
𝑘=0
0
ℝ
Using Hölder’s inequality and the relation (4.13), we get for all 𝑇 ∈ 𝐵 ′ and 𝜑 ∈
𝒲𝑎𝑚 (ℝ2 ),
(
)1/𝑝
⟨𝔏𝑇 , 𝒜𝜑⟩ ⩽ 𝐶(𝑚 + 1) 𝜈𝛼 (ℬ(0, 𝑎))
𝒩∞,𝑚 (𝜑),
this shows that the mapping 𝔏𝑇 𝑜𝒜 is a continuous extension of 𝑇 on 𝒲𝑎𝑚 (ℝ2 ), and
that the family {𝔏𝑇 𝑜𝒜}𝑇 ∈𝐵 ′ is equicontinuous, when applied to 𝒲𝑎𝑚 (ℝ2 ).
□
In the following, we will give a new characterization of the space M𝑝′ (ℝ2 ).
𝑝
Theorem 4.8. Let 𝑇 ∈ S∗′ (ℝ2 ), 𝑝 ∈ [1, +∞[, and 𝑝′ = 𝑝−1
. Then T belongs to
′
2
2
the space M𝑝 (ℝ ) if, and only if for every 𝜑 ∈ D∗ (ℝ ), the function 𝑇 ∗ 𝜑 belongs
′
to the space 𝐿𝑝 (𝑑𝜈𝛼 ), where
𝑇 ∗ 𝜑(𝑟, 𝑥) = ⟨𝑇, 𝜏(𝑟,−𝑥) (𝜑)⟩.
ˇ
Proof. ∙ Let 𝑇 ∈ M𝑝′ (ℝ2 ). From Theorem 4.1, there exist 𝑚 ∈ ℕ and 𝑓0 , ..., 𝑓𝑚 ∈
′
𝐿𝑝 (𝑑𝜈𝛼 ), such that
𝑚
∑
𝑇 =
Δ𝑘𝛼 (𝑇𝑓𝑘 ),
𝑘=0
34
C.BACCAR AND L.T.RACHDI
in M𝑝′ (ℝ2 ). Thus, for every 𝜑 ∈ D∗ (ℝ2 );
𝑇 ∗𝜑=
𝑚
∑
𝑇𝑓𝑘 ∗ Δ𝑘𝛼 (𝜑) =
𝑘=0
𝑚
∑
𝑓𝑘 ∗ Δ𝑘𝛼 (𝜑).
𝑘=0
𝑝′
Since, for all 𝑘 ∈ ℕ, 0 ⩽ 𝑘 ⩽ 𝑚, 𝑓𝑘 ∈ 𝐿 (𝑑𝜈𝛼 ) and Δ𝑘𝛼 (𝜑) ∈ 𝐿1 (𝑑𝜈𝛼 ); then from
′
the inequality (2.8), we deduce that 𝑓𝑘 ∗ Δ𝑘𝛼 (𝜑) ∈ 𝐿𝑝 (𝑑𝜈𝛼 ). This implies that the
′
function 𝑇 ∗ 𝜑 belongs to the space 𝐿𝑝 (𝑑𝜈𝛼 ).
′
2
∙ Conversely, let 𝑇 ∈ S∗ (ℝ ) such that for every 𝜑 ∈ D∗ (ℝ2 )) the function 𝑇 ∗ 𝜑
′
belongs to the space 𝐿𝑝 (𝑑𝜈𝛼 ). For 𝜑, 𝜓 in D∗ (ℝ2 ), we have
ˇ = ⟨𝑇, 𝜓 ∗ 𝜑⟩
⟨𝑇𝑇 ∗𝜑 , 𝜓⟩ = ⟨𝑇, 𝜑 ∗ 𝜓⟩
˘ = ⟨𝑇𝑇 ∗𝜓 , 𝜑⟩.
Thus, from Hlder’s inequality and using the hypothesis, we obtain
∣⟨𝑇𝑇 ∗𝜑 , 𝜓⟩∣ ⩽ ∥𝑇 ∗ 𝜓∥𝑝′ ,𝜈𝛼 ∥𝜑∥𝑝,𝜈𝛼 ,
from which, we deduce that the set
{
}
𝐵 ′ = 𝑇𝑇 ∗𝜑 , 𝜑 ∈ D∗ (ℝ2 ); ∥𝜑∥𝑝,𝜈𝛼 ⩽ 1 ,
is bounded in D∗′ (ℝ2 ). Now, using Theorem 4.7, it follows that for all 𝑎 > 0 there
exists 𝑚 ∈ ℕ, such that for all 𝜑 ∈ D∗ (ℝ2 ); ∥𝜑∥𝑝,𝜈𝛼 ⩽ 1, the mapping 𝑇𝑇 ∗𝜑 can
be continuously extended to the space 𝒲𝑎𝑚 (ℝ2 ) and the family of these extensions
is equicontinuous, which means that there exists 𝐶 > 0, such that for all 𝜑 ∈
D∗ (ℝ2 ), ∥𝜑∥𝑝,𝜈𝛼 ⩽ 1, and 𝜓 ∈ 𝒲𝑎𝑚 (ℝ2 ),
∣⟨𝑇𝑇 ∗𝜑 , 𝜓⟩∣ ⩽ 𝐶𝒩∞,𝑚 (𝜓).
This involves that for all 𝜑 ∈ D∗ (ℝ2 ), for all 𝜓 ∈ 𝒲𝑎𝑚 (ℝ2 ),
∣⟨𝑇𝑇 ∗𝜑 , 𝜓⟩∣ ⩽ 𝐶𝒩∞,𝑚 (𝜓)∥𝜑∥𝑝,𝜈𝛼 .
On the other hand, we have for all 𝜑 ∈ D∗ (ℝ ), and 𝜓 ∈
2
(4.14)
𝒲𝑎𝑚 (ℝ2 ),
⟨𝑇𝑇 ∗𝜑 , 𝜓⟩ = ⟨𝑇 ∗ 𝑇𝜓 , 𝜑⟩,
ˇ
(4.15)
where, for all 𝜑 ∈ S∗ (ℝ2 ),
⟨𝑇 ∗ 𝑇𝜓 , 𝜑⟩ = ⟨𝑇, 𝑇𝜓 ∗ 𝜑⟩ = ⟨𝑇, 𝜓 ∗ 𝜑⟩.
The relations (4.14) and (4.15) lead to, for all 𝜑 ∈ D∗ (ℝ2 ),
∣⟨𝑇 ∗ 𝑇𝜓 , 𝜑⟩∣ ⩽ 𝐶𝒩∞,𝑚 (𝜓)∥𝜑∥𝑝,𝜈𝛼 .
This last inequality shows that the functional 𝑇 ∗ 𝑇𝜓 can be continuously extended
′
to the space 𝐿𝑝 (𝑑𝜈𝛼 ) and from Riesz theorem there exists 𝑔 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ), such that
𝑇 ∗ 𝑇𝜓 = 𝑇𝑔
(4.16)
Furthermore, from proposition 4.5, there exist 𝑛 ∈ ℕ, 𝜓𝑛 ∈ 𝒲𝑎𝑚 (ℝ2 ), and 𝜑𝑛 ∈
D∗,𝑎 (ℝ2 ) satisfying
𝛿 = (𝐼 + 𝐿)𝑛 𝑇𝜓𝑛 + 𝑇𝜑𝑛 ,
then,
𝑇 = (𝐼 + 𝐿)𝑛 (𝑇 ∗ 𝑇𝜓𝑛 ) + 𝑇 ∗ 𝑇𝜑𝑛 = (𝐼 + 𝐿)𝑛 (𝑇 ∗ 𝑇𝜓𝑛 ) + 𝑇𝑇 ∗𝜑𝑛 .
We complete the proof by using the hypothesis, the relation (4.16) and Theorem
4.1.
□
SPACES OF 𝐷𝐿𝑝 -TYPE
35
In the following, we will give a characterization of the bounded subsets of
M𝑝′ (ℝ2 ).
Theorem 4.9. Let 𝑝 ∈ [1, ∞[ and let 𝐵 ′ be a subset of M𝑝′ (ℝ2 ). The following
assertions are equivalent
(i) the subset 𝐵 ′ is weakly bounded in M𝑝′ (ℝ2 ),
(ii) there exist 𝐶 > 0 and 𝑚 ∈ ℕ, such that for every 𝑇 ∈ 𝐵 ′ , it is possible to
′
find 𝑓0 , ..., 𝑓𝑚 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ) satisfying
𝑇 =
𝑚
∑
Δ𝑘𝛼 (𝑇𝑓𝑘 ) with max
0⩽𝑘⩽𝑚
𝑘=0
∥𝑓𝑘 ∥𝑝′ ,𝜈𝛼 ⩽ 𝐶,
{
}
′
(iii) for every 𝜑 ∈ D∗ (ℝ2 ), the set 𝑇 ∗ 𝜑, 𝑇 ∈ 𝐵 ′ is bounded in 𝐿𝑝 (𝑑𝜈𝛼 ).
Proof. (1) Suppose that the subset 𝐵 ′ is weakly bounded in M𝑝′ (ℝ2 ), then from
[27] 𝐵 ′ is equicontinuous. There exist 𝐶 > 0 and 𝑚 ∈ ℕ, such that
∀ 𝑇 ∈ 𝐵 ′ , ∀ 𝑓 ∈ M𝑝 (ℝ2 ),
∣⟨𝑇, 𝑓 ⟩∣ ⩽ 𝐶𝛾𝑚,𝑝 (𝑓 ).
(4.17)
As in the proof of theorem 4.7, we consider the mappings
(
)𝑚+1
𝒜 : M𝑝 (ℝ2 ) −→ 𝐿𝑝 (𝑑𝜈𝛼 )
,
𝑓 7−→ (𝑓, Δ𝛼 (𝑓 ), ..., Δ𝑚
𝛼 (𝑓 )),
and for all 𝑇 ∈ 𝐵 ′ ,
𝔏𝑇 : 𝒜(M𝑝 (ℝ2 )) −→ ℂ;
⟨𝔏𝑇 , 𝒜(𝑓 )⟩ = ⟨𝑇, 𝑓 ⟩.
Then, the relation (4.17) implies that for all 𝜑 ∈ M𝑝 (ℝ2 ),
𝔏𝑇 (𝒜𝜑) ⩽ 𝐶∥𝒜𝜑∥(𝐿𝑝 (𝑑𝜈 ))𝑚+1 .
𝛼
Using Hahn Banach theorem and Riesz theorem, we deduce that 𝔏𝑇 can be con(
)𝑚+1
tinuously extended to 𝐿𝑝 (𝑑𝜈𝛼 )
, denoted again by 𝔏𝑇 , and that there exists
(
)𝑚+1
′
𝑝
(𝑓𝑘 )0⩽𝑘⩽𝑚 ⊂ 𝐿𝑝 (𝑑𝜈𝛼 ), 𝑝′ = 𝑝−1
, verifying for all 𝜓 = (𝜓0 , ..., 𝜓𝑚 ) ∈ 𝐿𝑝 (𝑑𝜈𝛼 )
,
𝑚 ∫
∑
⟨𝔏𝑇 , 𝜓⟩ =
𝑘=0
with 𝔏𝑇 (𝐿𝑝 (𝑑𝜈𝛼 ))𝑚+1 =
+∞
0
∫
𝑓𝑘 (𝑟, 𝑥)𝜓𝑘 (𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥),
ℝ
max ∥𝑓𝑘 ∥𝑝′ ,𝜈𝛼 ⩽ 𝐶.
0⩽𝑘≤𝑚
In particular, if 𝜓 = 𝒜(𝑓 ), 𝑓 ∈ M𝑝 (ℝ2 ),
⟨𝔏𝑇 , 𝒜(𝑓 )⟩ = ⟨𝑇, 𝑓 ⟩ =
𝑚
∑
⟨Δ𝑘𝛼 (𝑇𝑓𝑘 ), 𝑓 ⟩,
𝑘=0
this proves that (i) implies (ii).
(2) Suppose that there exist 𝐶 > 0 and 𝑚 ∈ ℕ, such that for every 𝑇 ∈ 𝐵 ′ one
′
can find 𝑓0 , ..., 𝑓𝑚 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ), satisfying
𝑇 =
𝑚
∑
𝑘=0
Δ𝑘𝛼 (𝑇𝑓𝑘 ),
max ∥𝑓𝑘 ∥𝑝′ ,𝜈𝛼 ⩽ 𝐶,
0⩽𝑘⩽𝑚
36
C.BACCAR AND L.T.RACHDI
then, for all 𝑓 ∈ M𝑝 (ℝ2 ) and 𝑇 ∈ 𝐵 ′ ,
𝑚 ∫ +∞ ∫
∑
⟨𝑇, 𝑓 ⟩ =
𝑓𝑘 (𝑟, 𝑥)𝑔𝑘 (𝑟, 𝑥)𝑑𝜈𝛼 (𝑟, 𝑥);
𝑘=0
0
ℝ
consequently,
∣⟨𝑇, 𝑓 ⟩∣ ⩽ 𝐶(𝑚 + 1)𝛾𝑚,𝑝 (𝑓 ),
which means that the set 𝐵 ′ is weakly bounded in M ′ 𝑝 (ℝ2 ) and proves that (ii)
implies (i).
(3) Suppose that (ii) holds. Let 𝜑 ∈ D∗ (ℝ2 ), then from Theorem 4.8 we know
′
that for all 𝑇 ∈ 𝐵 ′ the function 𝑇 ∗ 𝜑 belongs to the space 𝐿𝑝 (𝑑𝜈𝛼 ). But
𝑇 ∗𝜑=
𝑚
∑
𝑇𝑓𝑘 ∗ Δ𝑘𝛼 (𝜑)
𝑘=0
′
thus, for all 𝑇 ∈ 𝐵 ,
𝑇 ∗ 𝜑 ′
⩽ 𝐶(𝑚 + 1)𝛾𝑚,𝑝 (𝜑).
𝑝 ,𝜈𝛼
{
}
′
This shows that the set 𝑇 ∗ 𝜑, 𝑇 ∈ 𝐵 ′ is bounded in 𝐿𝑝 (𝑑𝜈𝛼 ) and therefore (ii)
involves (iii).
(4) Suppose that (iii) holds and let 𝑇 ∈ 𝐵 ′ . For all 𝜑, 𝜓 ∈ D∗ (ℝ2 ), we have
∣⟨𝑇𝑇 ∗𝜑 , 𝜓⟩∣ = ∣⟨𝑇𝑇 ∗𝜓 , 𝜑⟩∣ ⩽ ∥𝑇 ∗ 𝜓∥𝑝′ ,𝜈𝛼 ∥𝜑∥𝑝,𝜈𝛼 ,
from which, we deduce that the set
{
}
𝑇𝑇 ∗𝜑 , 𝑇 ∈ 𝐵 ′ , 𝜑 ∈ D∗ (ℝ2 ); ∥𝜑∥𝑝,𝜈𝛼 ⩽ 1 ,
is bounded in D∗′ (ℝ2 ).
Now, using Theorem 4.7, it follows that for all 𝑎 > 0 there exists 𝑚 ∈ ℕ, such
that for all 𝜑 ∈ D∗ (ℝ2 ); ∥𝜑∥𝑝,𝜈𝛼 ⩽ 1, and 𝑇 ∈ 𝐵 ′ , the mapping 𝑇𝑇 ∗𝜑 can be
continuously extended on the space 𝒲𝑎𝑚 (ℝ2 ) and the family of these extensions
is equicontinuous. This means that there exists 𝐶 > 0 such that for all 𝑇 ∈ 𝐵 ′ ,
𝜑 ∈ D∗ (ℝ2 ), and 𝜓 ∈ 𝒲𝑎𝑚 (ℝ2 ), the inequality (4.14) holds. Using the relations
(4.14) and (4.15), we deduce that the functional 𝑇 ∗𝑇𝜓 can be continuously extended
′
on the space 𝐿𝑝 (𝑑𝜈𝛼 ) and from Riesz theorem there exists 𝑔𝑇,𝜓 ∈ 𝐿𝑝 (𝑑𝜈𝛼 ), such
that
𝑇 ∗ 𝑇𝜓 = 𝑇𝑔𝑇 ,𝜓 .
(4.18)
′
Applying again the relations (4.14) and (4.15), we deduce that for all 𝑇 ∈ 𝐵 ,
∥𝑔𝑇,𝜓 ∥𝑝′ ,𝜈𝛼 ⩽ 𝐶𝒩∞,𝑚 (𝜓).
Again by Proposition 4.5, it follows that there exist 𝑛 ∈ ℕ, 𝜓𝑛 ∈
𝜑𝑛 ∈ D∗,𝑎 (ℝ2 ) verifying for all 𝑇 ∈ 𝐵 ′ ,
(4.19)
𝒲𝑎𝑚 (ℝ2 )
and
𝑇 = 𝑇 ∗ 𝛿 = (𝐼 + Δ𝛼 )𝑛 (𝑇 ∗ 𝑇𝜓𝑛 ) + 𝑇𝑇 ∗𝜑𝑛 ,
and by the relation (4.18), we get
𝑇 = (𝐼 + Δ𝛼 )𝑛 𝑇𝑔𝑇 ,𝑛 + 𝑇𝑇 ∗𝜑𝑛 .
(4.20)
On the other hand, from the hypothesis there exists 𝐶1 > 0, such that
∀𝑇 ∈ 𝐵 ′ ,
∥𝑇 ∗ 𝜑𝑛 ∥𝑝′ ,𝜈𝛼 ⩽ 𝐶1 ,
(4.21)
SPACES OF 𝐷𝐿𝑝 -TYPE
37
However, by the relation (4.19), we have
∀ 𝑇 ∈ 𝐵′,
∥𝑔𝑇,𝑛 ∥𝑝′ ,𝜈𝛼 ⩽ 𝐶2 𝒩∞,𝑚 (𝜑𝑛 ).
(4.22)
′
The relations (4.2), (4.20), (4.21) and (4.22) show that the set 𝐵 is bounded in
M𝑝′ (ℝ2 ).
□
5. Convolution product on the space M𝑝′ (ℝ2 ) × ℳ𝑟 (ℝ2 )
In this section, we define and study a convolution product on the space M𝑝′ (ℝ2 )×
ℳ𝑟 (ℝ2 ), 1 ⩽ 𝑟 ⩽ 𝑝 < +∞, where ℳ𝑟 (ℝ2 ) is the closure of the space S∗ (ℝ2 ) in
M𝑟 (ℝ2 ).
Proposition 5.1. Let 𝑝 ∈ [1, +∞[. For every (𝑟, 𝑥) ∈ [0, +∞[×ℝ, the translation
operator 𝜏(𝑟,𝑥) given by Definition 2.1 (i), is a continuous mapping from M𝑝 (ℝ2 )
into itself. Moreover, for all 𝑓 ∈ M𝑝 (ℝ2 ) and 𝑘 ∈ ℕ, we have
Δ𝑘𝛼 (𝜏(𝑟,𝑥) (𝑓 ))
= 𝜏(𝑟,𝑥) (Δ𝑘𝛼 (𝑓 )),
(5.1)
where
Δ𝑘𝛼 (𝑇𝑓 ) = 𝑇Δ𝑘𝛼 (𝑓 ) .
Proof. Let 𝑓 ∈ M𝑝 (ℝ2 ). Since for all (𝑟, 𝑥) ∈ [0, +∞[×ℝ, the translation operator
𝜏(𝑟,𝑥) is continuous from 𝐿𝑝 (𝑑𝜈𝛼 ) into itself; then the function 𝜏(𝑟,𝑥) (𝑓 ) belongs to
the space 𝐿𝑝 (𝑑𝜈𝛼 ). Moreover; for all 𝜑 ∈ S∗ (ℝ2 ) and 𝑘 ∈ ℕ; we have
⟨Δ𝑘𝛼 (𝑇𝜏(𝑟,𝑥) (𝑓 ) ), 𝜑⟩ = ⟨𝑇𝜏(𝑟,𝑥) (𝑓 ) , Δ𝑘𝛼 (𝜑)⟩
∫ +∞ ∫
=
𝑓 (𝑠, 𝑦)𝜏(𝑟,−𝑥) (Δ𝑘𝛼 (𝜑))(𝑠, 𝑦)𝑑𝜈𝛼 (𝑠, 𝑦)
0
ℝ
∫ +∞ ∫
=
𝑓 (𝑠, 𝑦)Δ𝑘𝛼 (𝜏(𝑟,−𝑥) (𝜑))(𝑠, 𝑦)𝑑𝜈𝛼 (𝑠, 𝑦)
0
ℝ
= ⟨𝑇𝑓 , Δ𝑘𝛼 (𝜏(𝑟,−𝑥) (𝜑))⟩
= ⟨Δ𝑘𝛼 (𝑇𝑓 ), 𝜏(𝑟,−𝑥) (𝜑)⟩
= ⟨𝑇Δ𝑘𝛼 (𝑓 ) , 𝜏(𝑟,−𝑥) (𝜑)⟩
= ⟨𝑇𝜏(𝑟,𝑥) (Δ𝑘𝛼 (𝑓 )) , 𝜑⟩.
Since the operator 𝜏(𝑟,𝑥) is continuous from 𝐿𝑝 (𝑑𝜈𝛼 ) into itself, we deduce that for
all 𝑓 ∈ M𝑝 (ℝ2 ) and (𝑟, 𝑥) ∈ [0, +∞[×ℝ, the function 𝜏(𝑟,𝑥) (𝑓 ) belongs to the space
M𝑝 (ℝ2 ) and that for all 𝑘 ∈ ℕ, Δ𝑘𝛼 (𝜏(𝑟,𝑥) (𝑓 )) = 𝜏(𝑟,𝑥) (Δ𝑘𝛼 (𝑓 )). Moreover, from the
relations (2.7) and (5.1) , we have
𝛾𝑚,𝑝 (𝜏(𝑟,𝑥) (𝑓 )) = max 𝜏(𝑟,𝑥) (Δ𝑘𝛼 (𝑓 ))𝑝,𝜈 ⩽ max ∥Δ𝑘𝛼 (𝑓 )∥𝑝,𝜈𝛼 = 𝛾𝑚,𝑝 (𝑓 ),
0⩽𝑘⩽𝑚
𝛼
0⩽𝑘⩽𝑚
which shows that the operator 𝜏(𝑟,𝑥) is continuous from M𝑝 (ℝ2 ) into itself.
□
The precedent proposition allows us to define the coming convolution product
Definition 5.2. The convolution product of 𝑇 ∈ M𝑝′ (ℝ2 ) and 𝑓 ∈ M𝑝 (ℝ2 ) is
defined by
∀(𝑟, 𝑥) ∈ [0, +∞[×ℝ;
𝑇 ∗ 𝑓 (𝑟, 𝑥) = ⟨𝑇, 𝜏(𝑟,−𝑥) (𝑓ˇ)⟩.
38
C.BACCAR AND L.T.RACHDI
Let 𝑇 ∈ M𝑝′ (ℝ2 ) and 𝜑 ∈ M𝑝 (ℝ2 ). From Theorem 4.1, there exist 𝑚 ∈ ℕ and
′
{𝑓0 , ..., 𝑓𝑚 } ⊂ 𝐿𝑝 (𝑑𝜈𝛼 ), such that
𝑇 =
𝑚
∑
Δ𝑘𝛼 (𝑇𝑓𝑘 ).
𝑘=0
Thus,
𝑇 ∗ 𝜑(𝑟, 𝑥)
𝑚
∑
=
ˇ
⟨Δ𝑘𝛼 (𝑇𝑓𝑘 ), 𝜏(𝑟,−𝑥) (𝜑)⟩
𝑘=0
𝑚
∑
⟨𝑇𝑓𝑘 , 𝜏(𝑟,−𝑥) (Δ𝑘𝛼ˇ(𝜑))⟩
=
𝑘=0
𝑚
∑
=
(
)
𝑓𝑘 ∗ Δ𝑘𝛼 (𝜑) (𝑟, 𝑥)
𝑘=0
Using the relation (2.8) and the fact that 𝜑 ∈ M𝑝 (ℝ2 ) we deduce that the function
𝑇 ∗ 𝜑 belongs to 𝐿∞ (𝑑𝜈𝛼 ) and
(𝑚
)
∑
∣∣𝑇 ∗ 𝜑∣∣∞,𝜈𝛼 ⩽ 𝛾𝑚,𝑝 (𝜑)
∣∣𝑓𝑘 ∣∣𝑝′ ,𝜈𝛼
(5.2)
𝑘=0
Let 𝑇 ∈ M ′ 𝑝 (ℝ2 ), 𝑇 =
𝑚
∑
′
Δ𝑘𝛼 (𝑇𝑓𝑘 ) with {𝑓𝑘 }0⩽𝑘⩽𝑚 ⊂ 𝐿𝑝 (𝑑𝜈𝛼 ) and 𝜙 ∈ ℳ𝑟 (ℝ2 ),
𝑘=0
1 ⩽ 𝑟 ⩽ 𝑝. From the inequality (2.8), it follows that for 0 ⩽ 𝑘 ⩽ 𝑚 the function
𝑓𝑘 ∗ Δ𝑘𝛼 (𝜙) belongs to the space 𝐿𝑞 (𝑑𝜈𝛼 ) with, 1/𝑞 = 1/𝑟 + 1/𝑝′ − 1 = 1/𝑟 − 1/𝑝
and by using the density of S∗ (ℝ2 ) in ℳ𝑟 (ℝ2 ), we deduce that the expression
𝑚
∑
𝑓𝑘 ∗ Δ𝑘𝛼 (𝜙) is independent of the sequence {𝑓𝑘 }0⩽𝑘⩽𝑚 . Then, we put
𝑘=0
𝑇 ∗𝜙=
𝑚
∑
𝑓𝑘 ∗ Δ𝑘𝛼 (𝜙).
𝑘=0
Again, from the relation (2.8), we deduce that the function 𝑇 ∗ 𝜙 belongs to the
space 𝐿𝑞 (𝑑𝜈𝛼 ) and
∥𝑇 ∗ 𝜙∥𝑞,𝜈𝛼
⩽ 𝛾𝑚,𝑟 (𝜙)(
𝑚
∑
∥𝑓𝑘 ∥𝑝′ ,𝜈𝛼 )
(5.3)
𝑘=0
This allows us to say that
M𝑝′ (ℝ2 ) ∗ ℳ𝑟 (ℝ2 ) ⊂ 𝐿𝑞 (𝑑𝜈𝛼 ).
Lemma 5.3. Let 1 ⩽ 𝑟 ⩽ 𝑝 < ∞. Then
i) The operator Δ𝛼 is continuous from ℳ𝑟 (ℝ2 ) into itself.
ii) For all 𝑇 ∈ M ′ 𝑝 (ℝ2 ) and 𝜙 ∈ ℳ𝑟 (ℝ2 ), the function 𝑇 ∗ 𝜙 belongs to the space
M𝑞 (ℝ2 ) and we have
∀𝑘 ∈ ℕ,
Δ𝑘𝛼 (𝑇 ∗ 𝜙) = 𝑇 ∗ Δ𝑘𝛼 (𝜙).
Proof. i) Let 𝑓 ∈ ℳ𝑟 (ℝ2 ). There exists (𝑓𝑘 )𝑘 ⊂ S∗ (ℝ2 ) such that
∀𝑚 ∈ ℕ,
lim 𝛾𝑚,𝑟 (𝑓𝑘 − 𝑓 ) = 0.
𝑘→+∞
SPACES OF 𝐷𝐿𝑝 -TYPE
39
However,
𝛾𝑚,𝑟 (Δ𝛼 (𝑓𝑘 ) − Δ𝛼 (𝑓 )) ⩽ 𝛾𝑚+1,𝑟 (𝑓𝑘 − 𝑓 ),
thus, the sequence (Δ𝛼 (𝑓𝑘 ))𝑘 of S∗ (ℝ2 ) converges to Δ𝛼 (𝑓 ) in ℳ𝑟 (ℝ2 ), which
shows that the function Δ𝛼 (𝑓 ) belongs to the space ℳ𝑟 (ℝ2 ).
ii) If 𝜙 ∈ S∗ (ℝ2 ), then the function 𝑇 ∗ 𝜙 is infinitely differentiable, and we have
Δ𝑘𝛼 (𝑇𝑇 ∗𝜙 ) = 𝑇Δ𝑘𝛼 (𝑇 ∗𝜙) = 𝑇𝑇 ∗Δ𝑘𝛼 (𝜙) ,
therefore, the result follows from the density of S∗ (ℝ2 ) in ℳ𝑟 (ℝ2 ), the relation
(5.3) and the fact that the operator Δ𝛼 is continuous from ℳ𝑟 (ℝ2 ) into itself. □
Proposition 5.4. Let 1 ⩽ 𝑟 ⩽ 𝑝 < ∞ and 𝑞 ∈ [1, +∞], such that
1 1
1
= − .
𝑞
𝑟
𝑝
(5.4)
Then for every 𝑇 ∈ M ′ 𝑝 (ℝ2 ), the mapping
𝜙 −→ 𝑇 ∗ 𝜙
is continuous from ℳ𝑟 (ℝ ) into M𝑞 (ℝ2 ).
2
Proof. Let 𝑇 ∈ M ′ 𝑝 (ℝ2 ); 𝑇 =
𝑚
∑
Δ𝑘𝛼 (𝑇𝑓𝑘 ) and 𝜙 ∈ ℳ𝑟 (ℝ2 ). From Lemma 5.3,
𝑘=0
the function 𝑇 ∗ 𝜙 belongs to the space M𝑞 (ℝ2 ) and for all 𝑙 ∈ ℕ
𝛾𝑙,𝑞 (𝑇 ∗ 𝜙) =
max ∥Δ𝑘𝛼 (𝑇 ∗ 𝜙)∥𝑞,𝜈𝛼 =
0⩽𝑘⩽𝑙
max ∥𝑇 ∗ Δ𝑘𝛼 (𝜙)∥𝑞,𝜈𝛼
0⩽𝑘⩽𝑙
According to the relation (5.3), it follows that
𝑚
(∑
)
∥𝑓𝑘 ∥𝑝′ ,𝜈𝛼 max 𝛾𝑚,𝑟 (Δ𝑘𝛼 (𝜙))
𝛾𝑙,𝑞 (𝑇 ∗ 𝜙) ⩽
0⩽𝑘⩽𝑙
𝑘=0
⩽
𝑚
(∑
)
∥𝑓𝑘 ∥𝑝′ ,𝜈𝛼 𝛾𝑚+𝑙,𝑟 (𝜙).
𝑘=0
□
Definition 5.5. Let 1 ⩽ 𝑝, 𝑞, 𝑟 < +∞, such that (5.4) holds. The convolution
product of 𝑇 ∈ M𝑝′ (ℝ2 ) and 𝑆 ∈ M ′ 𝑞 (ℝ2 ) is defined for all 𝜙 ∈ ℳ𝑟 (ℝ2 ), by
⟨𝑆 ∗ 𝑇, 𝜙⟩ = ⟨𝑆, 𝑇 ∗ 𝜙⟩.
From this Definition and Proposition 5.4, we deduce the following result
Proposition 5.6. Let 1 ⩽ 𝑝, 𝑞, 𝑟 < +∞ such that (5.4) holds. Then, for all
𝑇 ∈ M𝑝′ (ℝ2 ) and 𝑆 ∈ M ′ 𝑞 (ℝ2 ), the functional 𝑆 ∗ 𝑇 is continuous on ℳ𝑟 (ℝ2 ).
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Lakhdar Tannech Rachdi
Department of Mathematics, Faculty of Sciences of Tunis, 2092 Elmanar II, Tunisia
E-mail address: lakhdartannech.rachdi@fst.rnu.tn
SPACES OF 𝐷𝐿𝑝 -TYPE
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Cyrine Baccar
Département de mathématiques appliquées, ISI, 2, rue Abourraihan Al Bayrouni 2080,
Ariana, Tunis,
E-mail address: cyrine.baccar@isi.rnu.tn