# Structure Theorem for Functionals in the Space𝔖

Authors: Hamed M. Obiedat; Wasfi A. Shatanawi; Mohd M. Yasein
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

---

## Abstract

We introduce the space𝔖
ω
1
,
ω
2 of all C
∞ functions ϕ such that sup
|
α
|
≤
m
∥
e
k
ω
1
∂
α
ϕ
∥
∞ and sup
|
α
|
≤
m
∥
e
k
ω
2
∂
α
ϕ
^
∥
∞ are finite for all k
∈
ℕ
0, α
∈
ℕ
0
n, where ω
1 and ω
2 are two weights satisfying
the classical Beurling conditions. Moreover, we give a topological
characterization of the space 𝔖
ω
1
,
ω
2 without conditions on the derivatives. For functionals in the dual space  𝔖
ω
1
,
ω
2
′, we prove a structure
theorem by using the classical Riesz representation thoerem.


---

## Body

## 1. Introduction

The theory of ultradistributions introduced by
Beurling [1] was to find an appropriate context for his work on almost
holomorphic extensions. Beurling proved that ultradistributions are limits of
holomorphic functions in the upper and lower half-planes. Björck [2] studied and expanded the theory of Beurling on ultradistributions to extend the work of
Hörmander [3] on existence, nonexistence, and regularity of solutions of constant coefficient linear partial differential equations.The Beurling-Björck space𝔖
w, as defined in [2], consists of C
∞ functions such that the functions and their Fourier transform jointly with all their
derivatives decay ultrarapidly at infinity.In this paper, we introduce the space𝔖
w
1
,
w
2 of C
∞ functions such
that the functions and their Fourier transform jointly with all their
derivatives decay ultrarapidly at infinity. Moreover, we give a
characterization of the space 𝔖
w
1
,
w
2 and its
dual 𝔖
w
1
,
w
2
′
.The main difference between the Beurling-Björck space𝔖
w and the space 𝔖
w
1
,
w
2 is that the
decay of the functions in 𝔖
w and their
Fourier transform are measured by the same submultiplicative function e
k
w, k
≥
0. Whereas the decay
of the functions in 𝔖
w
1
,
w
2 and their
Fourier transform are measured by two different submultiplicative functions e
k
w
1 and e
k
w
2
,
k
≥
0.This paper is organized in three sections. In Section2, we give preliminary definitions and results and introduce the space 𝔖
w
1
,
w
2
. In Section 3,
we give a topological characterization of the space 𝔖
w
1
,
w
2 without conditions
on the derivatives. In Section 4, we use the topological characterization of
the space 𝔖
w
1
,
w
2 that is given
in Section 3 to prove a representation theorem for functionals in the dual space 𝔖
w
1
,
w
2
′ of the space 𝔖
w
1
,
w
2
.The symbolsC
∞, C
0
∞, L
p, and so forth indicate the usual spaces of functions defined
on ℝ
n, with complex values. We denote by |
⋅
| the Euclidean
norm on ℝ
n, while ∥
⋅
∥
∞ indicates the
norm in the space L
∞ . When we do not work on the general Euclidean space ℝ
n, we will write L
p
(
ℝ
), and so forth as appropriate. Partial derivatives will be  denoted by
∂
α, where α is a
multi-index (
α
1
,
…
,
α
n
). If it is necessary to indicate on which variables we
are taking the derivative, we will do so by attaching subindexes. We will use
the standard abbreviations |
α
|
=
α
1
+
⋯
+
α
n, x
α
=
x
1
α
1
,
…
,
x
n
α
n. With α
≤
β, we mean that α
j
≤
β
j for every j. The Fourier transform of a function g will be denoted
by ℱ
(
g
) or g
^ and it will be
defined as ∫
ℝ
n
e
−
2
π
i
x
ξ
g
(
x
)
d
x. The inverse Fourier transform is then ℱ
−
1
(
g
)
=
∫
ℝ
n
e
2
π
i
x
ξ
g
(
ξ
)
d
ξ. The letter C will indicate a
positive constant, that may be different at different occurrences. If it is
important to indicate that a constant depends on certain parameters, we will do
so by attaching subindexes to the constant. We will not indicate the dependence
of constants on the dimension n or other fixed
parameters.


## 2. Preliminary Definitions and Results

In this section, we give definitions and results which we will use later.Definition 2.1 (see [2]).
Withℳ
c, we denote the space of functions w
:
ℝ
n
→
ℝ of the form w
(
x
)
=
Ω
(
|
x
|
), where
(1)
Ω
:
[
0
,
∞
)
→
[
0,
∞
) is increasing,
continuous, and concave,
(2)
Ω
(
0
)
=
0,
(3)
∫
ℝ
Ω
(
(
t
)
/
(
1
+
t
2
)
)
d
t
<
∞,
(4)
Ω
(
t
)
≥
a
+
b
ln
(
1
+
t
) for some a
∈
ℝ and some b
>
0.Standard classes of functionsw in ℳ
c are given by
(2.1)
w
(
x
)
=
|
x
|
d
for
0
<
d
<
1
,
w
(
x
)
=
p
ln
(
1
+
|
x
|
)
for
p
>
0.Remark 2.2.
Let us
observe for future use that if we take an integerN
>
(
n
/
b
)
, then
(2.2)
C
N
=
∫
ℝ
n
e
−
N
w
(
x
)
d
x
<
∞
,
∀
w
∈
ℳ
c
,
where b is the constant
in condition 4 of Definition 2.1.The following lemma was observed in [2] without proof. Our proof is an adaptation of [4,Proposition 4.6].Lemma 2.3.
Conditions 1 and 2 in Definition2.1 imply
that w
is subadditive for all w
∈
ℳ
c.Proof.
Let0
<
k
<
1. Since Ω is increasing, we obtain
(2.3)
w
(
x
+
y
)
≤
Ω
(
k
k
|
x
|
+
1
−
k
1
−
k
|
y
|
)
≤
max
{
Ω
(
|
x
|
k
)
,
Ω
(
|
y
|
1
−
k
)
}
.
Since Ω is concave on [
0
,
∞
) and Ω
(
0
)
=
0, we have
(2.4)
Ω
(
k
k
|
x
|
)
≥
k
Ω
(
|
x
|
k
)
,
Ω
(
|
y
|
1
−
k
)
≥
1
1
−
k
Ω
(
|
y
|
)
.
If we take
(2.5)
k
=
Ω
(
|
x
|
)
Ω
(
|
x
|
)
+
Ω
(
|
y
|
)
,
then we have
(2.6)
w
(
x
+
y
)
≤
max
{
Ω
(
|
x
|
k
)
,
Ω
(
|
y
|
1
−
k
)
}
≤
w
(
x
)
+
w
(
y
)
.
This completes the proof of Lemma 2.3.We now recall a topological characterization of the
Beurling-Björck space𝔖
w of test
functions for tempered ultradistributions.Theorem 2.4 (see [5]).
Givenw
∈
ℳ
c, the space  𝔖
w
can be described both as a set and as a topology by
(2.7)
𝔖
w
=
{
φ
:
ℝ
n
ℂ
:
φ
i
s
c
o
n
t
i
n
u
o
u
s
a
n
d
f
o
r
a
l
l
k
=
0
,
1
,
2
,
…
,
p
k
,0
(
φ
)
<
∞
,
p
k
,0
∘
ℱ
(
φ
)
<
∞
}
,
where p
k
,0
(
φ
)
=
∥
e
k
w
φ
∥
∞ and p
k
,0
∘
ℱ
(
φ
)
=
∥
e
k
w
φ
^
∥
∞.We observe that𝔖
w becomes the
Schwartz space 𝔖 when
(2.8)
w
(
x
)
=
ln
(
1
+
|
x
|
)
.
For α
,
β
>
0, the Gelfand-Shilov space S
α
β of type S is
characterized in [6] by the space of all C
∞ functions φ
:
ℝ
n
→
ℂ for which the
seminorms
(2.9)
∥
e
k
|
x
|
1
/
α
φ
∥
∞
,
∥
e
m
|
x
|
1
/
β
φ
^
∥
∞
are finite for some k
,
m
∈
ℕ
0.Definition 2.5.
Givenw
1
,
w
2
∈
ℳ
c
,  the space  𝔖
w
1,
w
2
is the space of
all  C
∞  functions  φ
:
ℝ
n
→
ℂ  for which the
seminorms
(2.10)
p
k
,
m
(
φ
)
=
sup
|
β
|
≤
m
∥
e
k
w
1
∂
β
φ
∥
∞
,
π
k
,
m
(
φ
)
=
sup
|
β
|
≤
m
∥
e
k
w
2
∂
β
φ
^
∥
∞
are finite, for k
,
m
∈
ℕ
0 and β
∈
ℕ
0
n.We can assign to𝔖
w
1,
w
2 a structure to
Fréchet space by means of the countable family of seminorms(2.11)
S
=
{
p
k
,
m
,
π
k
,
m
}
k
,
m
=
0
∞
.Sincep
k
,
m
(
φ
)
<
∞ for all k
=
0
,
1
,
2
,
…
,
φ is integrable,
so φ
^ is well defined
and the formulation of the condition π
k
,
m
(
φ
) makes sense for
all k
=
0
,
1
,
2
,
….The space𝔖
w
1,
w
2, equipped with the family of seminorms
(2.12)
S
=
{
p
k
,
m
,
π
k
,
m
:
k
,
m
∈
ℕ
0
}
,
is a Fréchet space.We observe that the space𝔖
w
1,
w
2 becomes the
Beurling-Björck space 𝔖
w
1, when w
1
=
w
2. When w
2
(
x
)
=
ln
(
1
+
|
x
|
), the space of C
∞ functions with
compact support 𝔇 is dense
subspace of 𝔖
w
1,
w
2 for all w
1
∈
ℳ
c. The conditions imposed on the function w assure that the
space 𝔖
w
1,
w
2 satisfies the
properties expected from a space of testing functions. For instance, the
operators of differentiation and multiplication by x
α are continuous
from 𝔖
w
1,
w
2 intothemselves, the space 𝔖
w
1,
w
2 is a
topological algebra under pointwise multiplication and convolution.
Unfortunately, the Fourier transformation on 𝔖
w
1,
w
2 is not a
topological isomorphism from 𝔖
w
1,
w
2 into itself for
some w
1, w
2
∈
ℳ
c. For Example, if we take w
1
(
x
)
=
|
x
|
1
/
2, w
2
(
x
)
=
ln
(
1
+
|
x
|
), and f
∈
𝔇
∖
𝔇
w
1, then f
∈
𝔖
w
1,
w
2 but f
^
∉
𝔖
w
1
,
w
2; see [1, 2].Theorem 2.6  (Riesz Representation Theorem [7]).
Given a functionalL in the topological dual of the space 𝒞
0, there exists a unique regular complex Borel measure μ such that
(2.13)
L
(
φ
)
=
∫
ℝ
n
φ
d
μ
.
Moreover, the norm of the functional L is equal to the
total variation |
μ
| of the measure μ. Conversely, any such measure μ defines a
continuous linear functional on 𝒞
0.We conclude this section with Lemma2.7 [8], the version
of which is due to Hadamard [9], see also [10].Lemma 2.7 (See [8, 10]).
Letf
:
ℝ
→
ℝ be a continuous function with continuous derivatives of order ≤
2. Assume that there exist P
,
Q
≥
0 such that
(2.14)
|
f
(
x
)
|
≤
P
,
|
f
′
′
(
x
)
|
≤
Q
,
for all x
∈
ℝ. Then
(2.15)
|
f
′
(
x
)
|
≤
2
P
Q
for all x
∈
ℝ.


## 3. Topological Characterization of the Space𝔖
w
1
,
w
2

In this section, we present the following
characterization of the space𝔖
w
1
,
w
2, which imposes no conditions on the derivative.Theorem 3.1.
Givenw
1
,
w
2
∈
ℳ
c, the space 𝔖
w
1
,
w
2 can be
described as a set and as a topology by
(3.1)
𝔖
w
1
,
w
2
=
{
φ
:
ℝ
n
→
ℂ
:
φ
is continuous and  for all
k
=
0
,
1
,
2
,
…
,
p
k
,
0
(
φ
)
<
∞
,
π
k
,0
(
φ
)
<
∞
}
,
where p
k
,0
(
φ
)
=
∥
e
k
w
1
φ
∥
∞
,
π
k
,0
(
φ
)
=
∥
e
k
w
2
φ
^
∥
∞
.Proof.
Let us denote by𝔅
w
1
,
w
2 the space
defined in (3.1). The conditions p
k
,0
(
φ
) and π
k
,0
(
φ
) imply the
smoothness of φ and φ
^
. The space 𝔅
w
1
,
w
2 becomes a
Fréchet space with respect to the family of norms
(3.2)
B
=
{
p
k
,0
,
π
k
,0
}
k
=
0
∞
.
From these  definitions, it
is clear that 𝔖
w
1
,
w
2
⊆
𝔅
w
1
,
w
2 and that the
inclusion is continuous. To prove the converse, we use the induction on |
β
| and the general
idea of Landau's inequality. Fix φ
∈
𝔅
w
1
,
w
2
∖
{
0
}. We want to show that ∥
e
k
w
1
(
x
)
∂
β
φ
∥
∞ and ∥
e
k
w
2
(
ξ
)
∂
β
φ
^
∥
∞ are finite, for
every k
=
0
,
1
,
2
,
… and every
multi-index β, which is true for all k, when β
=
0. We assume that it is true for all k, when |
β
|
≤
m, and we want to prove it for all k and for |
β
|
=
m
+
1.
We start with ∥
e
k
w
1
∂
β
φ
∥
∞. Assume that β
=
(
β
1
+
1,
β
2
,
…
,
β
n
) with β
1
+
β
2
+
⋯
+
β
n
=
m, m
=
0
,
1
,
2
,
…
. We also indicate β
′
=
(
β
1
,
β
2
,
…
,
β
n
), ∂
β
φ
=
∂
x
1
∂
β
′
φ, f
x
′
(
x
1
)
=
∂
β
′
φ
(
x
1
,
x
′
) for x
′
=
(
x
2
,...,
x
n
) fixed, ∂
β
φ
(
x
)
=
f
x
′
′
(
x
1
). Moreover, if h
≠
0, we have
(3.3)
f
x
′
(
x
1
+
h
)
=
f
x
′
(
x
1
)
+
f
x
′
′
(
x
1
)
h
+
1
2
f
x
′
′
′
(
y
)
h
2
,
where y is a number
between x
1 and x
1
+
h. Thus,
(3.4)
|
f
x
′
′
(
x
1
)
|
≤
|
f
x
′
(
x
1
+
h
)
|
+
|
f
x
′
(
x
1
)
|
|
h
|
+
|
h
|
2
|
f
x
′
′
′
(
y
)
|
.
We can write
(3.5)
|
e
k
w
1
(
x
1
+
h
,
x
′
)
f
x
′
(
x
1
+
h
)
|
≤
|
e
k
w
1
(
x
1
+
h
,
x
′
)
∂
β
′
φ
(
x
1
,
x
′
)
|
≤
q
k
,
m
(
φ
)
,
|
e
k
w
1
(
x
)
f
x
′
(
x
)
|
≤
q
k
,
m
(
φ
)
.
If we take h with the same
sign as x
1, we have
(3.6)
w
1
(
x
)
≤
w
1
(
x
1
+
h
,
x
′
)
.
That is,
(3.7)
|
f
x
′
(
x
1
+
h
)
|
+
|
f
x
′
(
x
1
)
|
≤
C
m
p
k
,
m
(
φ
)
e
−
k
w
1
(
x
)
.
To estimate f
x
′
′
′
(
y
)
=
∂
x
1
∂
β
φ
(
y
), we write
(3.8)
|
∂
x
1
∂
β
φ
(
y
)
|
=
|
∂
x
1
∂
β
φ
^
^
(
y
)
|
≤
∫
ℝ
n
|
2
π
i
ξ
1
(
2
π
i
ξ
)
β
φ
^
(
ξ
)
|
d
ξ
≤
C
β
,
m
∫
ℝ
n
(
1
+
|
ξ
|
)
m
+
2
e
−
r
w
2
(
ξ
)
e
r
w
2
(
ξ
)
|
φ
^
(
ξ
)
|
d
ξ
,
where r
>
(
m
+
n
+
2
)
/
b is an integer
and b is the constant
in condition 4 of Definition 2.1:
(3.9)
|
∂
x
1
∂
β
φ
(
y
)
|
≤
C
m
π
r
,0
(
φ
)
.
Thus, we have
(3.10)
|
∂
x
1
∂
β
φ
(
y
)
|
≤
C
m
π
r
,0
(
φ
)
,
that is,
(3.11)
|
∂
β
φ
(
x
)
|
≤
C
m
[
1
t
p
k
,
m
(
φ
)
e
−
k
w
1
(
x
)
+
t
π
r
,0
(
φ
)
]
f
o
r
a
l
l
t
>
0. As a function of t, the right side of (3.11) has a global minimum at
(3.12)
t
=
(
p
k
,
m
(
φ
)
e
−
k
w
1
(
x
)
)
1
/
2
(
π
r
,0
(
φ
)
)
−
1
/
2
.
Thus, we obtain the inequality
(3.13)
|
∂
β
φ
(
x
)
|
≤
C
m
(
p
k
,
m
(
φ
)
)
1
/
2
(
π
r
,0
(
φ
)
)
1
/
2
e
(
−
k
/
2
)
w
1
(
x
)
,
that is,
(3.14)
|
e
k
w
1
(
x
)
∂
β
φ
(
x
)
|
≤
C
m
(
p
2
k
,
m
(
φ
)
)
1
/
2
(
π
r
,0
(
φ
)
)
1
/
2
.
An argument, similar to the one leading to (3.14),
produces
(3.15)
|
e
k
w
2
(
ξ
)
∂
β
φ
^
(
ξ
)
|
≤
C
m
(
π
2
k
,
m
(
φ
)
)
1
/
2
(
p
r
,0
(
φ
)
)
1
/
2
.
Combining (3.14), (3.15), the inductive hypothesis
implies that φ
∈
𝔖
w. The open mapping theorem can provide once again the
continuity of the inclusion.   However, solving the recursive inequalities (3.14), (3.15) , we obtain
(3.16)
|
e
k
w
(
x
)
∂
β
φ
(
x
)
|
≤
C
m
(
p
2
m
+
1
k
,0
(
φ
)
)
2
−
m
−
1
(
π
r
,0
(
φ
)
)
1
−
2
−
m
−
1
,
|
e
k
w
(
ξ
)
∂
β
φ
^
(
ξ
)
|
≤
C
m
(
π
2
m
+
1
k
,0
∘
ℱ
(
φ
)
)
2
−
m
−
1
(
p
r
,0
(
φ
)
)
1
−
2
−
m
−
1
.
This completes the proof of Theorem 3.1.Whenw
1
(
x
)
=
w
2
(
x
), the characterization of 𝔖
w
1
,
w
2 given by
Theorem 3.1 reduces to the characterization of Beurling-Björck space 𝔖
w
1 given by
Theorem 2.4. In particular, when w
1
(
x
)
=
w
2
(
x
)
=
ln
(
1
+
|
x
|
), the characterization of 𝔖
w
1
,
w
2 reduces to the
characterization of Schwartz space 𝔖
.Remark 3.2.
The Fourier transform is a topological isomorphism between𝔖
w
1
,
w
2 and 𝔖
w
2
,
w
1. As a consequence, the Fourier transform is also a
topological isomorphism between the dual spaces 𝔖
w
1
,
w
2
′ and 𝔖
w
2
,
w
1
′.Note that the dual spaces𝔖
w
1
,
w
2
′ and 𝔖
w
2
,
w
1
′ are assigned
to the weak topologies. For different pairs of admissible functions, the space 𝔖
w
1
,
w
2 has the
following embedding properties.Lemma 3.3.
For everyw
1
<
w
1
′ and w
2
<
w
2
′, one has
(3.17)
𝔖
w
1
′
,
w
2
′
↪
𝔖
w
1
,
w
2
.Lemma 3.4.
Forα
,
β
>
1, one has 𝔖
|
x
|
1
/
α
,
|
x
|
1
/
β
⊆
S
α
β. As a consequence, (
S
α
β
)
′
⊆
𝔖
|
x
|
1
/
α
,
|
x
|
1
/
β
′.


## 4. A Representation Theorem for Functionals in the Space𝔖
w
1
,
w
2
′

From Theorem3.1, we can write
(4.1)
𝔖
w
1
,
w
2
=
{
φ
:
ℝ
n
→
ℂ
:
φ
is continuous and
f
o
r
a
l
l
k
=
0
,
1
,
2
,
…
,
𝒩
k
,
(
φ
)
<
∞
}
,
where 𝒩
k
(
φ
)
=
∥
e
k
w
1
φ
∥
∞
+
∥
e
k
w
2
φ
^
∥
∞.Theorem 4.1.
GivenL
:
𝔖
w
1
,
w
2
→
ℂ, the following statements are equivalent:
(i)
L
∈
𝔖
w
1
,
w
2
′;
(ii)
there exist two
regular complex Borel measuresμ
1 and μ
2 of finite total
variation and k
∈
{
0,1,2,
…
} such that
(4.2)
L
=
e
k
w
1
μ
1
+
ℱ
[
e
k
w
2
μ
2
]
,
in the sense of 𝔖
w
1
,
w
2
′.Proof.
(
i
)
⇒
(
i
i
)
. Given L
∈
𝔖
w
1
,
w
2
′, according to (4.1) there exist k and C so that
(4.3)
L
(
φ
)
≤
C
(
∥
e
k
w
1
φ
∥
∞
+
∥
e
k
w
2
φ
^
∥
∞
)
for all φ
∈
𝔖
w
1
,
w
2. Moreover, the map
(4.4)
𝔖
w
1
,
w
2
→
𝒞
0
×
𝒞
0
,
φ
→
(
e
k
w
1
φ
,
e
k
w
2
φ
^
)
is well defined, linear, continuous, and injective. Let ℛ be the range of this map, on which we define the map
(4.5)
l
1
(
f
,
g
)
=
L
(
φ
)
,
where f
=
e
k
w
1
φ, g
=
e
k
w
2
φ
^ for a unique φ
∈
𝔖
w
1
,
w
2. The map l
1
:
ℛ
→
ℂ is linear and continuous. By the Hahn-Banach theorem, there exists a functional L
1 in the topological dual (
𝒞
0
×
𝒞
0
)
′ of 𝒞
0
×
𝒞
0 such that ∥
L
1
∥
=
∥
l
1
∥ and the restriction of L
1 to ℛ is l
1.
Since the spaces(
𝒞
0
×
𝒞
0
)
′ and 𝒞
0
′
×
𝒞
0
′ are isomorphic as Banach spaces, we can write L
1
(
f
,
g
)
=
L
1
(
f
,0
)
+
L
1
(
0,
g
). Using Theorem 2.6, there exist regular complex Borel
measures μ
1 and μ
2 of finite total variation such that
(4.6)
L
1
(
f
,
g
)
=
∫
ℝ
n
f
d
μ
1
+
∫
ℝ
n
g
d
μ
2
for all (
f
,
g
)
∈
𝒞
0
×
𝒞
0. If (
f
,
g
)
∈
ℛ, then we conclude that
(4.7)
L
(
φ
)
=
∫
ℝ
n
e
k
w
1
φ
d
μ
1
+
∫
ℝ
n
e
k
w
2
φ
^
d
μ
2
for all φ
∈
𝔖
w
1
,
w
2. In the sense of 𝔖
w
1
,
w
2
′,
(4.8)
L
=
e
k
w
1
μ
1
+
ℱ
[
e
k
w
2
μ
2
]
.
(
i
i
)
⇒
(
i
). If μ
1 and μ
2 are two regular complex Borel measures satisfying (
i
i
) and φ
∈
𝔖
w
1
,
w
2, then
(4.9)
L
(
φ
)
=
∫
ℝ
n
e
k
w
1
φ
d
μ
1
+
∫
ℝ
n
e
k
w
2
φ
^
d
μ
2
.
This implies that
(4.10)
|
L
(
φ
)
|
≤
|
∫
ℝ
n
e
k
w
1
φ
d
μ
1
|
+
|
∫
ℝ
n
e
k
w
2
φ
^
d
μ
2
|
≤
|
μ
1
|
(
ℝ
n
)
∥
e
k
w
1
φ
∥
∞
+
|
μ
2
|
(
ℝ
n
)
∥
e
k
w
2
φ
^
∥
∞
≤
C
(
∥
e
k
w
1
φ
∥
∞
+
∥
e
k
w
2
φ
^
∥
∞
)
.
It may be noted that μ
1 and μ
2
, employed to
obtain the above inequality, are of finite total variations. This completes the
proof of Theorem 4.1
.Remark 4.2.
Whenw
1
(
x
)
=
w
2
(
x
)
=
(
1
+
|
x
|
)
k, (4.2) becomes
(4.11)
L
=
(
1
+
|
x
|
)
k
μ
1
+
ℱ
[
(
1
+
|
ξ
|
)
k
μ
2
]
,
which gives a representation for the tempered
distributions.As consequence of Lemma3.4
, we can view the functionals
in (
S
a
b
)
′ as functionals
in the space 𝔖
w
1
,
w
2
′. Then as a result we can characterize (
S
α
β
)
′ using Theorem 4.1.Corollary 4.3.
Letα
,
β
>
1. Then any L
∈
(
S
α
β
)
′
can be written
as
(4.12)
L
=
e
k
|
x
|
1
/
α
μ
1
+
ℱ
[
e
k
|
ξ
|
1
/
β
μ
2
]
which characterizes the dual space (
S
α
β
)
′.



---

*Source: 756834-2008-02-20.xml*