# Cauchy and Poisson Integral of the Convolutor in Beurling Ultradistributions ofL

Authors: Byung Keun Sohn
Journal: International Journal of Mathematics and Mathematical Sciences
 (2014)

---

## Abstract

LetC be a regular cone in ℝ and let T
C
=
ℝ
+
i
C
⊂
ℂ be a tubular radial domain. Let U be the convolutor in Beurling ultradistributions of L
p-growth corresponding to T
C. We define the Cauchy and Poisson integral of U and show that the Cauchy integral of  U is analytic in T
C and satisfies a growth property. We represent  U as the boundary value of a finite sum of suitable analytic functions in tubes by means of the Cauchy integral representation of U. Also we show that the Poisson integral of U corresponding to T
C attains U as boundary value in the distributional sense.


---

## Body

## 1. Introduction

LetC be a regular cone in ℝ
n and let 𝒪
(
C
) denote its convex envelope. In [1] (or [2]) Carmichael defined the Cauchy and Poisson integrals for Schwartz distributions ∈
𝒟
L
p
′, 1
<
p
≤
2, corresponding to tubular domain T
𝒪
(
C
)
=
ℝ
n
+
i
C
⊂
ℂ
n. Carmichael obtained the boundary values of these integrals in the distributional sense on the boundary of T
𝒪
(
C
) and found the relation between analytic functions with a specific growth condition in T
𝒪
(
C
) and the Cauchy and Poisson integrals of their distributional boundary values. In [3] Pilipović defined ultradistributions 𝒟
′
(
(
M
p
)
,
L
2
) of L
2-growth, where M
p, p
=
0,1
,
2
,
…, is a certain sequence of positive numbers, and studied the Cauchy and Poisson integrals for elements of 𝒟
′
(
(
M
p
)
,
L
2
) in the case that the Cauchy and Poisson kernel functions are defined corresponding to the first quadrant {
y
∈
ℝ
n; y
j
>
0, j
=
1,2
,
…
,
n
} in ℝ
n. Pilipović showed that elements in 𝒟
′
(
(
M
p
)
,
L
2
) are boundary values of suitable analytic functions with a certain L
2-norm condition by means of the Cauchy integral representation and an analytic function with a certain L
2-norm condition determines, as a boundary value, an element from 𝒟
′
(
(
M
p
)
,
L
2
). In [4] Carmichael et al. defined ultradistributions of Beurling type 𝒟
′
(
(
M
p
)
,
L
s
) of L
s-growth and of Roumieu type 𝒟
′
(
{
M
p
}
,
L
s
) of L
s-growth, both of which generalize the Schwartz distributions 𝒟
L
p
′, and studied the Cauchy and Poisson integrals for elements of both 𝒟
′
(
(
M
p
)
,
L
s
) and 𝒟
′
(
{
M
p
}
,
L
s
) for 2
≤
s
≤
∞ corresponding to the arbitrary tubes T
C
=
ℝ
n
+
i
C
⊂
ℂ
n where C is an open connected cone in ℝ
n of which the quadrants are special cases. They showed that the Cauchy integral for elements of both 𝒟
′
(
(
M
p
)
,
L
s
) and 𝒟
′
(
{
M
p
}
,
L
s
) for 2
≤
s
≤
∞ corresponding to T
C is shown to be analytic in T
C, to satisfy a growth property and to obtain an ultradistribution boundary value, which leads to an analytic representation for the ultradistributions. They also showed that the Poisson integrals for elements of both 𝒟
′
(
(
M
p
)
,
L
s
) and 𝒟
′
(
{
M
p
}
,
L
s
) for 2
≤
s
≤
∞ corresponding to T
C are shown to have an ultradistribution boundary value. We can find the works of the Cauchy and Poisson integrals for ultradistributions of compact support in [5] and for various kinds of distributions in [2].In the meantime, Betancor et al. [6] introduced the spaces of Beurling ultradistributions of L
p-growth, 𝒟
L
p
,
(
ω
)
′
(
1
≤
p
≤
∞
), of which Schwartz distribution and ultradistributions of Beurling type are special cases. Here ω is a weight function in the sense of [7]. Betancor et al. defined the convolutors in 𝒟
L
p
,
(
ω
)
′, 1
≤
p
≤
∞; that is, the functionals U
∈
𝒟
L
p
,
(
ω
)
′ such that U
*
ϕ
∈
𝒟
L
p
,
(
ω
) for every ϕ
∈
𝒟
L
p
,
(
ω
), and studied the convolutors and the surjective convolution operators acting on 𝒟
L
p
,
(
ω
)
′ in [6].In this paper we define the Cauchy and Poisson integrals for the convolutors in𝒟
L
p
,
(
ω
)
′, 1
≤
p
≤
2. We will show that the Cauchy integrals for the convolutors in 𝒟
L
p
,
(
ω
)
′, 1
≤
p
≤
2, are analytic in a tubular domain T
C
=
ℝ
+
i
C
⊂
ℂ where C is a regular cone in ℝ and satisfies a certain boundedness condition. We will give the representation of the convolutors in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2, as boundary value of a finite sum of analytic functions in tubes. Also we will show that the Poisson integrals for the convolutor U in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2, attain U as boundary value in the distributional sense.Since𝒟
L
p
,
(
ω
)
′ is the natural generalization of the space 𝒟
L
p
′ from Lemma 4 (ii) and we can find a weight function κ such that 𝒟
′
(
(
M
p
)
,
L
s
)
=
𝒟
L
s
,
(
κ
)
′ from Remark 3.11 in [8], our results in this paper extend the results in [1, 2, 4] under a condition of U to be convolutor in 𝒟
L
p and 𝒟
(
(
M
p
)
,
L
s
), respectively.


## 2. Beurling Ultradistributions ofL
p-Growth

In this section we will review Beurling ultradistributions ofL
p-growth in ℝ which is introduced by Betancor et al. in [6] and established some of their properties which will be needed later on. Firstly, we will review Beurling ultradistributions which are introduced by Braun et al. in [7].Definition 1.
Aweight function is an increasing continuous function ω
:
[
0
,
∞
)
→
[
0
,
∞
) with the following properties:
(
α
)
there existsL
≥
0 with ω
(
2
t
)
≤
L
(
ω
(
t
)
+
1
) for all t
≥
0,
(
β
)
∫
1
∞
‍
(
ω
(
t
)
/
t
2
)
d
t
<
∞,
(
γ
)
log
⁡
(
t
)
=
o
(
ω
(
t
)
) as t tends to ∞,
(
δ
)
ψ
:
t
→
ω
(
e
t
) is convex.By(
β
),Lemma 2.
Consider the following:(1)
lim
⁡
t
→
∞
ω
(
t
)
t
=
0
.By(
δ
), ψ
(
0
)
=
0 and lim
⁡
x
→
∞
x
/
ψ
(
x
)
=
0. Then we can define the Young conjugate ψ
* of ψ by
(2)
ψ
*
:
[
0
,
∞
)
⟶
ℝ
,
ψ
*
(
y
)
=
sup
⁡
x
≥
0
(
x
y
-
ψ
(
x
)
)
.Obviously we have the following:Lemma 3 (see [7]).
(i)ψ
* is convex and increasing and satisfies ψ
*
(
0
)
=
0.
(ii)ψ
*
(
y
)
/
y is increasing and lim
⁡
y
→
∞
y
/
ψ
*
(
y
)
=
0.
(iii)(
ψ
*
)
*
=
ψ.
(iv) There existsL
≥
1 such that L
ψ
*
(
t
/
L
)
+
t
≤
ψ
*
(
t
)
+
L
,
t
≥
0.Letω be a weight function. For a compact set K
⊂
ℝ, we define the following:
(3)
𝒟
(
ω
)
(
K
)
=
{
f
∈
𝒟
(
K
)
:
∥
f
∥
K
,
λ
∞
<
for every
λ
>
0
}
,
where ∥
f
∥
K
,
λ
=
sup
⁡
x
∈
K
sup
⁡
n
∈
ℕ
0
|
f
(
n
)
(
x
)
|
exp
⁡
(
-
λ
ψ
*
(
n
/
λ
)
).𝒟
(
ω
)
=
in
d
n
𝒟
(
ω
)
(
[
-
n
,
n
]
). The elements of 𝒟
(
ω
)
′ are called ultradistributions of Beurling type.We denote byℰ
(
ω
) the set of all C
∞ functions f such that ∥
f
∥
K
,
λ
<
∞ for every compact K and every λ
>
0. For more details about 𝒟
(
ω
) and ℰ
(
ω
), we refer to [7].A functionϕ
∈
C
∞
(
ℝ
) is in the space 𝒮
(
ω
) when, for every m
,
n
∈
ℕ,
(4)
p
m
,
n
=
sup
⁡
x
∈
ℝ
e
m
ω
(
x
)
|
ϕ
(
n
)
(
x
)
|
<
∞
,
π
m
,
n
=
sup
⁡
x
∈
ℝ
e
m
ω
(
x
)
|
ϕ
^
(
n
)
(
x
)
|
<
∞
.𝒮
(
ω
) is endowed with the topology generated by the family {
p
m
,
n
,
π
m
,
n
}, where m
,
n
∈
ℕ, of seminorms. Thus 𝒮
(
ω
) is a Fréchet space and the Fourier transform ℱ defines an automorphism of 𝒮
(
ω
). 𝒟
(
ω
) is a dense subspace of 𝒮
(
ω
). For more details about 𝒮
(
ω
), we refer to [9].For every1
≤
p
≤
∞, k
∈
ℕ and ϕ
∈
C
∞
(
ℝ
), and γ
k
,
p
(
ϕ
) is defined as follows:
(5)
γ
k
,
p
(
ϕ
)
=
sup
⁡
α
∈
ℕ
∥
ϕ
(
α
)
∥
p
e
-
k
ψ
*
(
α
/
k
)
,
where ∥
·
∥
p denotes the usual norm in L
p
(
ℝ
). (∥
f
∥
∞ means ess sup |
f
(
t
)
|.) If 1
≤
p
<
∞, the space 𝒟
L
p
,
(
ω
) is the set of all C
∞ functions ϕ on ℝ such that γ
k
,
p
(
ϕ
)
<
∞ for each k
∈
ℕ. We denote by 𝒟
L
∞
,
(
ω
) the set of all bounded C
∞ functions ϕ on ℝ such that γ
k
,
∞
(
ϕ
)
<
∞. The topology of 𝒟
L
p
,
(
ω
), 1
≤
p
≤
∞, is generated by the family {
γ
k
,
p
(
ϕ
)
}
k
∈
ℕ of seminorms.The dual of𝒟
L
p
,
(
ω
) will be denoted by 𝒟
L
p
,
(
ω
)
′ and it will be endowed with the strong topology. The elements of 𝒟
L
p
,
(
ω
)
′ are called the Beurling ultradistributions of L
p-growth. For more details about 𝒟
L
p
,
(
ω
), we refer to [6].Lemma 4.
(i)𝒟
L
p
,
(
ω
), 1
≤
p
≤
∞, is Fréchet spaces.
(ii)𝒟
L
p
,
(
ω
) is continuously contained in the Schwartz distributions 𝒟
L
p, 1
≤
p
≤
∞.
(iii)𝒟
L
p
,
(
ω
) is continuously contained in 𝒟
L
q
,
(
ω
), when 1
≤
p
≤
q
≤
∞.
(iv)𝒟
(
ω
)
⊂
𝒟
L
p
,
(
ω
)
⊂
ℰ
(
ω
), 1
≤
p
≤
∞, with continuous and dense inclusions.
(v)𝒮
ω is continuously contained in 𝒟
L
1
,
(
ω
).Proof.
(i), (iii), and (iv) can be found in Proposition 2.1 of [6]. (ii) is obvious and (v) can be found in the proof of [6, Proposition 2.9].Definition 5.
The convolutors in𝒟
L
p
,
(
ω
), 1
≤
p
≤
∞, are the functionals T
∈
𝒟
L
p
,
(
ω
)
′ such that T
*
ϕ
∈
𝒟
L
p
,
(
ω
) for every ϕ
∈
𝒟
L
p
,
(
ω
).Remark 6.
From Proposition 3.2, Theorem 3, and Proposition 3.6 in [6], we can find a necessary condition for an ultradistribution to be a convolutor.
IfT
∈
𝒟
L
2
,
(
ω
)
′ and there exists k
∈
ℕ such that T
^
e
-
k
ω
∈
L
∞, then T is a convolutor in 𝒟
L
p
,
(
ω
) for 1
/
2
≤
p
≤
2.
In particular, ifT
∈
𝒟
L
1
,
(
ω
)
′
∩
𝒟
L
∞
,
(
ω
)
′, then T is a convolutor in 𝒟
L
1
,
(
ω
).
We will consider the convolutors in𝒟
L
p
,
(
ω
) for 1
≤
p
≤
2 in Sections 4 and 5.
Assume thatG is an entire function such that log
⁡
|
G
(
z
)
|
=
O
(
ω
(
|
z
|
)
), as |
z
|
→
∞. The functional T
G on ℰ
(
ω
) is defined by
(6)
〈
T
G
,
ϕ
〉
=
∑
α
=
0
∞
‍
(
-
1
)
α
G
(
α
)
(
0
)
α
!
ϕ
(
α
)
(
0
)
,
ϕ
∈
ℰ
(
ω
)
.
The operatorG
(
D
) defined on 𝒟
(
ω
)
′ by
(7)
G
(
D
)
:
𝒟
(
ω
)
⟶
𝒟
(
ω
)
,
μ
⟶
G
(
D
)
μ
=
μ
*
T
G
,
is called an ultradistributional operator of (
ω
)-class. When G
(
D
) is restricted to ℰ
(
ω
), G
(
D
) is a continuous operator from ℰ
(
ω
) into ℰ
(
ω
) and, for every ϕ
∈
ℰ
(
ω
),
(8)
(
G
(
D
)
ϕ
)
(
x
)
=
∑
α
=
0
∞
‍
(
-
1
)
α
G
(
α
)
(
0
)
α
!
ϕ
(
α
)
(
x
)
,
x
∈
ℝ
.Definition 7.
An ultradistributional operatorG
(
D
) of (
ω
)-class is said to bestrongly elliptic if there exist M
>
0 and l
>
0 such that |
G
(
z
)
|
≥
M
e
l
ω
(
|
z
|
) when |
Im
⁡
z
|
<
M
|
Re
z
|.Now we will obtain the characterization of convolutors in𝒟
L
p
,
(
ω
), 1
≤
p
≤
∞.Lemma 8.
LetT be a convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
∞. Then there exists a strongly elliptic ultradistributional operator G
(
D
) of (
ω
)-class and f
,
g
∈
𝒟
L
p
,
(
ω
) such that T
=
G
(
D
)
f
+
g.Proof.
Let1
≤
p
≤
∞. Since T is a convolutor in 𝒟
L
p
,
(
ω
), T
*
ϕ
∈
𝒟
L
p
,
(
ω
)
⊂
𝒟
L
p
⊂
L
p for every ϕ
∈
𝒟
(
ω
). By the proof of Proposition 2.3 in [6], there exists a strongly elliptic ultradistributional operator G
(
D
) of (
ω
)-class and χ
∈
𝒟
(
ω
) and Γ
∈
𝒟
(
ω
) such that T
=
G
(
D
)
(
T
*
χ
)
+
(
T
*
Γ
). If we let f
=
T
*
χ and g
=
T
*
Γ, then f
,
g
∈
T
*
𝒟
(
ω
)
⊂
T
*
𝒟
L
p
,
(
ω
)
⊂
𝒟
L
p
,
(
ω
) since T is a convolutor in 𝒟
L
p
,
(
ω
).Only by replacing∥
h
*
g
∥
p
≤
C
∥
h
∥
p in the second to the last line of the proof of Lemma 2.4 in [6] by
(9)
γ
k
,
p
(
h
*
g
)
=
sup
⁡
α
∈
ℕ
∥
(
h
*
g
)
(
α
)
∥
p
e
-
k
ψ
*
(
α
/
k
)
≤
C
sup
⁡
α
∈
ℕ
∥
h
(
α
)
∥
p
e
-
k
ψ
*
(
α
/
k
)
=
C
γ
k
,
p
(
h
)
that we have the following:Lemma 9.
Let1
≤
p
≤
∞. If G
(
D
) is a strongly elliptic ultradistributional operator of (
ω
)-class, then 𝒟
L
p
,
(
ω
) is contained in G
(
D
)
(
𝒟
L
p
,
(
ω
)
).Combining Lemmas8 and 9 and given the fact that 𝒟
L
p
,
(
ω
)
⊂
ℰ
(
ω
), we have the characterization of the convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
∞.Theorem 10.
LetT be a convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
∞. Then there exists a strongly elliptic ultradistributional operator G
(
D
) of (
ω
)-class and f
∈
𝒟
L
p
,
(
ω
) such that
(10)
T
(
x
)
=
G
(
D
)
f
(
x
)
=
∑
α
=
0
∞
‍
(
-
1
)
α
G
(
α
)
(
0
)
α
!
f
(
α
)
(
x
)
.


## 3. The Cauchy and Poisson Kernel Functions

LetC be a regular cone in ℝ, that is, an open convex cone, such that C
¯ does not contain any straight line. O
(
C
) will denote the convex hull (envelop) of C, and T
C
=
ℝ
+
i
C
⊂
ℂ is a tube in ℂ. If C is open, T
C is called a tubular cone. If C is open and connected, T
C is called a tubular radial domain. The set C
*
=
{
t
∈
ℝ
:
〈
t
,
y
〉
≥
0
for
all
y
∈
C
} is the dual cone of the cone C. We will give a very important lemma concerning cones and their dual cones, which is proved in Lemma 2 in page 223 of [10].Lemma 11.
LetC be an open cone in ℝ and let y
∈
O
(
C
). Then there exist a δ
=
δ
y
>
0 depending on y such that
(11)
〈
y
,
t
〉
≥
δ
|
y
|
|
t
|
,
t
∈
C
*
.Definition 12.
LetC be a regular cone in ℝ. The Cauchy kernel K
(
z
-
t
)
,
z
∈
T
C
=
ℝ
+
i
C
,
t
∈
ℝ, corresponding to the tube T
C is
(12)
K
(
z
-
t
)
=
∫
C
*
‍
exp
⁡
(
2
π
i
〈
z
-
t
,
η
〉
)
d
η
,
z
∈
T
C
,
t
∈
ℝ
.
The Poisson kernelP
(
z
;
t
) corresponding to the tube T
C is
(13)
P
(
z
;
t
)
=
K
(
z
-
t
)
K
(
z
-
t
)
¯
K
(
2
i
y
)
=
|
K
(
z
-
t
)
|
2
K
(
2
i
y
)
,
h
h
h
h
h
h
h
z
=
x
+
i
y
∈
T
C
,
t
∈
ℝ
.We note thatK
(
2
i
y
)
>
0, y
∈
C, by Lemma 1 in page 222 of [10]. For a regular cone C, K
(
z
-
t
) and P
(
z
;
t
) are well defined for z
∈
T
C and t
∈
ℝ ([1], Section 3).In this section we will prove thatK
(
z
-
t
) and P
(
z
;
t
) are elements of 𝒟
L
q
,
(
ω
), 2
≤
q
≤
∞, as a function of t
∈
ℝ for z
∈
T
C.Theorem 13.
LetC be a regular cone in ℝ and let 2
≤
p
≤
∞. Then K
(
z
-
t
)
∈
𝒟
L
q
,
(
ω
) as a function of t
∈
ℝ for z
∈
T
C.Proof.
Letz
=
x
+
i
y
∈
T
C be arbitrary but fixed and let α
∈
ℕ. Let I
C
* denote the characteristic function of C
*. By the proof of Theorem 4.4
.
1 in [2], K
(
z
-
t
)
∈
C
∞, I
C
*
(
η
)
η
α
e
2
π
i
〈
z
,
η
〉
∈
L
1
∩
L
p, 1
<
p
≤
2, as a function of t
∈
ℝ for z
∈
T
C, and
(14)
D
t
α
K
(
z
-
t
)
=
ℱ
-
1
[
I
C
*
(
η
)
η
α
e
2
π
i
〈
z
,
η
〉
;
t
]
,
h
h
h
h
h
h
h
h
h
z
∈
T
C
,
t
∈
ℝ
,
where the inverse transform ℱ
-
1 can be interpreted in both L
1 and L
p sense. If 1
<
p
≤
2 and 1
/
p
+
1
/
q
=
1, we have from (14) and the Parseval inequality the following:
(15)
∥
D
t
α
K
(
z
-
t
)
∥
q
≤
∥
I
C
*
(
η
)
η
α
e
2
π
i
〈
z
,
η
〉
∥
p
<
∞
.
Using (15) and Lemmas 3 (iii), 11, and 2, we get that, for every k
∈
ℕ and z
∈
T
C, there exist δ
>
0 such that
(16)
sup
⁡
α
∈
ℕ
∥
D
t
α
K
(
z
-
t
)
∥
q
e
-
k
ψ
*
(
α
/
k
)
≤
sup
⁡
α
∈
ℕ
(
∫
C
*
‍
|
η
|
p
α
e
-
2
π
p
〈
y
,
η
〉
d
η
)
1
/
p
e
-
k
ψ
*
(
α
/
k
)
≤
sup
⁡
α
∈
ℕ
(
∫
C
*
‍
e
p
α
log
⁡
|
η
|
e
-
p
k
ψ
*
(
α
/
k
)
e
-
2
π
p
〈
y
,
η
〉
d
η
)
1
/
p
≤
(
∫
C
*
‍
e
p
k
ψ
*
*
(
log
⁡
|
η
|
)
e
-
2
π
p
〈
y
,
η
〉
d
η
)
1
/
p
≤
(
∫
C
*
‍
e
p
k
ψ
(
log
⁡
|
η
|
)
e
-
2
π
p
δ
|
y
∥
η
|
d
η
)
1
/
p
≤
(
∫
C
*
‍
e
p
|
η
|
(
k
·
(
ω
(
η
)
/
|
η
|
)
-
2
π
δ
|
y
|
)
d
η
)
1
/
p
<
∞
.
Letp
=
1. We get from the same method of estimation of integrand in (16) that, for every k
∈
ℕ and z
∈
T
C, there exist δ
>
0 such that
(17)
sup
⁡
α
∈
ℕ
∥
D
t
α
(
z
-
t
)
∥
∞
e
-
k
ψ
*
(
α
/
k
)
≤
sup
⁡
α
∈
ℕ
∫
C
*
‍
|
η
|
α
e
-
2
π
p
〈
y
,
η
〉
e
-
k
ψ
*
(
α
/
k
)
d
η
≤
∫
C
*
‍
e
|
η
|
(
k
·
(
ω
(
η
)
/
|
η
|
)
-
2
π
δ
|
y
|
)
d
η
<
∞
.
From (16) and (17), the proof is complete.Theorem 14.
LetC be a regular cone in ℝ and let 2
≤
q
≤
∞. Then P
(
z
;
t
)
∈
𝒟
L
q
,
(
ω
) as a function of t
∈
ℝ for z
∈
T
C.Proof.
By Theorem4.4
.
4 in [2], P
(
z
;
t
)
∈
C
∞ as a function of t
∈
ℝ for z
∈
T
C. Since K
(
z
-
t
) is contained in the spaces of infinitely differentiable functions which vanish at infinity together with each of their derivatives by Theorem 4.4
.
1 in [2], D
t
γ
K
(
z
-
t
)
¯ is bounded on ℝ. Hence, we get from Lemma 3 and Theorem 13 that, for every k
∈
ℕ, z
∈
T
C, and 2
≤
q
≤
∞,
(18)
γ
k
,
q
(
P
(
z
;
t
)
)
=
sup
⁡
α
∈
ℕ
∥
D
t
α
P
(
z
-
t
)
∥
q
e
-
k
ψ
*
(
α
/
k
)
=
sup
⁡
α
∈
ℕ
∥
1
K
(
2
i
y
)
∑
β
+
γ
=
α
‍
α
!
β
!
γ
!
D
t
β
K
(
z
-
t
)
D
t
γ
K
(
z
-
t
)
¯
∥
q
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
e
-
k
ψ
*
(
α
/
k
)
≤
1
K
(
2
i
y
)
sup
⁡
α
∈
ℕ
∑
β
+
γ
=
α
‍
C
γ
,
K
α
!
β
!
γ
!
∥
D
t
β
K
(
z
-
t
)
∥
q
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
e
-
k
ψ
*
(
β
/
k
)
=
1
K
(
2
i
y
)
∑
β
+
γ
=
α
‍
C
γ
,
K
α
!
β
!
γ
!
γ
k
,
q
(
K
(
z
-
t
)
)
<
∞
.


## 4. The Cauchy Integral of the Convolutor in𝒟
L
p
,
(
ω
)
,
1
≤
p
≤
2

In this section we will define the Cauchy integral of the convolutor in𝒟
L
p
,
(
ω
), 1
≤
p
≤
2, and show that the Cauchy integral of the convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2, is analytic in a tubular domain T
C and satisfies certain growth properties.LetC be a regular cone in ℝ and 1
≤
p
≤
2. By Theorem 13, K
(
z
-
t
)
∈
𝒟
L
q
,
(
ω
), 1
/
p
+
1
/
q
=
1, as a function of t
∈
ℝ for z
∈
T
C. Hence 〈
U
t
,
K
(
z
-
t
)
〉 is well defined for U
∈
𝒟
L
p
,
(
ω
)
′, 1
≤
p
≤
2.Definition 15.
LetC be a regular cone in ℝ and let U be a convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2. The Cauchy integral C
(
U
;
z
) of U corresponding to C is
(19)
C
(
U
;
z
)
=
〈
U
t
,
K
(
z
-
t
)
〉
,
z
∈
T
C
.Theorem 16.
LetC be a regular cone in ℝ and let U be a convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2. The Cauchy integral C
(
U
;
z
) of U corresponding to C is analytic in T
C and for any compact subset C
′
⊂
C and z
=
x
+
i
y
∈
T
C
′, there exists a constant M
(
p
,
C
′
) depending on p and C
′ such that
(20)
|
C
(
U
;
z
)
|
≤
M
(
p
,
C
′
)
|
y
|
-
1
/
p
.Proof.
By Theorem10, there exists a strongly elliptic ultradistributional operator G
(
D
) of (
ω
)-class and f
∈
𝒟
L
p
,
(
ω
)
⊂
ℰ
(
ω
), 1
≤
p
≤
2, such that U
=
G
(
D
)
f; that is,
(21)
U
(
x
)
=
(
G
(
D
)
f
)
(
x
)
=
∑
α
=
0
∞
‍
i
α
G
(
α
)
(
0
)
α
!
f
(
α
)
(
x
)
,
x
∈
ℝ
.
Then(22)
C
(
U
;
z
)
=
〈
G
(
D
)
f
(
t
)
,
K
(
z
-
t
)
〉
=
∫
‍
∑
α
=
0
∞
‍
i
α
G
(
α
)
(
0
)
α
!
f
(
α
)
(
t
)
∫
C
*
‍
e
2
π
i
〈
z
-
t
,
η
〉
d
η
d
t
;
hence
(23)
D
z
β
C
(
U
;
z
)
=
(
-
1
)
β
〈
G
(
D
)
f
(
t
)
,
D
z
β
K
(
z
-
t
)
〉
=
(
-
1
)
β
∫
‍
∑
α
=
0
∞
‍
i
α
G
(
α
)
(
0
)
α
!
f
(
α
)
(
t
)
h
h
h
h
h
h
h
h
h
h
h
h
×
∫
C
*
‍
η
β
e
2
π
i
〈
z
-
t
,
η
〉
d
η
d
t
.
Now we will apply the method in the proof of Proposition 2.4 of [11] to L
p-norm estimation of the integral with respect to t in (23). Since log
⁡
|
G
(
z
)
|
=
O
(
ω
(
|
z
|
)
), as |
z
|
→
∞, there exists k
>
0 such that
(24)
log
⁡
|
G
(
z
)
|
≤
k
(
1
+
ω
(
|
z
|
)
)
,
z
∈
ℂ
.
If we letC
R
=
{
z
;
|
z
|
=
e
r
,
r
>
0
}, we get that, by Cauchy's theorem and (24),
(25)
|
G
(
α
)
(
0
)
α
!
|
=
1
2
π
|
∫
C
R
‍
G
(
z
)
z
α
+
1
d
z
|
≤
1
2
π
∫
C
R
‍
e
k
e
k
ω
(
|
z
|
)
|
z
|
α
+
1
|
d
z
|
=
1
2
π
e
k
e
k
ω
(
e
r
)
e
α
(
r
+
1
)
·
e
r
=
(
2
π
)
-
1
e
k
·
e
k
ψ
(
r
)
-
α
r
.
Letm
∈
ℕ and let L be as in Lemma 3 (iv). Take h
≥
max
⁡
{
k
L
,
m
}. By Lemma 4 (iii), we have the following:
(26)
f
∈
𝒟
p
,
(
ω
)
⊂
𝒟
h
,
(
ω
)
,
1
≤
p
≤
2
.
Sinceψ
*
(
t
)
/
t is increasing by Lemma 3 (ii), we get that, for every α
∈
ℕ
0,
(27)
α
+
h
ψ
*
(
α
h
)
≤
α
+
k
L
ψ
*
(
α
k
L
)
≤
k
ψ
*
(
α
h
)
+
k
L
.
Using (25), (26), and (27),
(28)
∑
α
=
0
∞
‍
|
G
(
α
)
(
0
)
α
!
|
p
∫
‍
|
f
(
α
)
(
t
)
|
p
d
t
=
∑
α
=
0
∞
‍
(
2
π
)
-
1
e
p
k
e
p
(
k
ψ
(
r
)
-
α
r
)
γ
h
,
p
(
f
)
e
h
ψ
*
(
α
/
h
)
≤
∑
α
=
0
∞
‍
γ
h
,
p
(
f
)
(
2
π
)
-
1
e
p
k
e
-
k
ψ
*
(
α
/
k
)
e
h
ψ
*
(
α
/
h
)
≤
∑
α
=
0
∞
‍
γ
h
,
p
(
f
)
(
2
π
)
-
1
e
p
k
e
k
L
-
α
≤
C
h
,
k
,
p
,
L
γ
h
,
p
(
f
)
.
We will consider theL
q-norm estimation of the integral with respect to η in (23). Using (14), (15), and (5.63) in the proof of Theorem 5.5
.
1 in [2], there exist δ
>
0 as in Lemma 11 such that if 1
<
p
≤
2 and 1
/
p
+
1
/
q
=
1,
(29)
∥
D
t
β
K
(
z
-
t
)
∥
q
=
(
∫
C
*
‍
|
η
|
β
q
e
2
π
q
〈
z
-
t
,
η
〉
d
η
)
1
/
q
=
∥
ℱ
-
1
[
I
C
*
(
η
)
η
β
e
2
π
i
〈
z
,
η
〉
;
t
]
∥
q
≤
∥
I
C
*
(
η
)
η
β
e
2
π
i
〈
z
,
η
〉
∥
p
≤
(
2
Γ
(
p
β
+
1
)
)
1
/
p
(
2
π
p
δ
|
y
|
)
(
-
p
β
-
1
)
/
p
.
LetK be an arbitrary compact subset of T
C. For z
∈
K
⊂
T
C, y
=
Im
⁡
(
z
)
∈
C
′ for some fixed compact subcone C
′
⊂
C and y is bounded away from 0 by l
>
0 for some l. For this compact subcone C
′
⊂
C and z
=
x
+
i
y
∈
K
⊂
T
C, if 1
<
p
≤
2 and 1
/
p
+
1
/
q
=
1, we get from (29) that there exist δ
>
0 as in Lemma 11 such that
(30)
(
∫
C
*
|
η
|
β
q
e
2
π
q
〈
z
-
t
,
η
〉
d
η
‍
)
1
/
q
≤
(
2
Γ
(
p
β
+
1
)
)
1
/
p
(
2
π
p
δ
|
l
|
)
(
-
p
β
-
1
)
/
p
,
where Γ
(
x
) is the gamma function. By (23), (28), and (30), consider the following:
(31)
|
D
z
β
C
(
U
;
z
)
|
≤
∥
∑
α
=
0
∞
‍
G
(
α
)
(
0
)
α
!
f
(
α
)
(
t
)
∥
p
(
∫
C
*
‍
|
η
|
β
q
e
2
π
q
〈
z
-
t
,
η
〉
d
η
)
1
/
q
≤
C
h
,
k
,
p
,
L
γ
h
,
p
(
f
)
(
2
Γ
(
p
β
+
1
)
)
1
/
p
(
2
π
p
δ
l
)
(
-
p
β
-
1
)
/
p
for compact subcone C
′
⊂
C and z
=
x
+
i
y
∈
K
⊂
T
C. Equation (31) shows that any derivatives of C
(
U
;
z
) with respect to z converge absolutely and uniformly for all z
∈
K
⊂
T
C; hence C
(
U
;
z
) is analytic in T
C.
It remains to prove the analyticity ofC
(
U
;
z
) for p
=
1. By Proposition 2.4 in [11], G
(
D
)
f
∈
𝒟
L
1
,
(
ω
) for f
∈
𝒟
L
1
,
(
ω
). Since 𝒟
L
1
,
(
ω
)
⊂
𝒟
L
1
⊂
L
1, the analyticity of C
(
U
;
z
) for p
=
1 in this theorem can be obtained from the analyticity of C
(
U
;
z
) for p
=
1 in the case of U
∈
𝒟
L
1 of Theorem 5.5
.
1 in [2].
The growth ofC
(
U
;
z
) remains to be proved. If 1
<
p
≤
2, we get that by, (29) with β
=
0,
(32)
|
C
(
U
;
z
)
|
≤
C
h
,
k
,
p
,
L
γ
h
,
p
(
f
)
(
2
Γ
(
1
)
)
1
/
p
(
2
π
p
δ
)
-
1
/
p
|
y
|
-
1
/
p
for compact subcone C
′
⊂
C, z
=
x
+
i
y
∈
T
C
′, and δ
=
δ
(
C
′
). If p
=
1, we get that, by (29) with p
=
1 and β
=
0,
(33)
|
∫
‍
I
C
*
e
2
π
i
〈
z
-
t
,
η
〉
d
η
|
≤
2
Γ
(
1
)
(
2
π
δ
)
-
1
|
y
|
-
1
;
hence
(34)
|
C
(
U
;
z
)
|
≤
C
h
,
k
,
1
,
L
γ
h
,
1
(
f
)
2
Γ
(
1
)
(
2
π
δ
)
-
1
|
y
|
-
1
for compact subcone C
′
⊂
C, z
=
x
+
i
y
∈
T
C
′, and δ
=
δ
(
C
′
). From (32) and (34) we get that there is a constant M
(
p
,
C
′
) such that
(35)
|
C
(
U
;
z
)
|
≤
M
(
p
,
C
′
)
|
y
|
-
1
/
p
for compact subcone C
′
⊂
C and z
=
x
+
i
y
∈
T
C
′.Now we will represent a convolutorU in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2, as the boundary value of analytic functions in tube. We need several lemmas.Lemma 17.
LetC be a regular cone in ℝ and let U be a convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2. Let φ
∈
𝒮
(
ω
). For fixed y
=
Im
⁡
(
z
)
∈
C,
(36)
〈
C
(
U
;
z
)
,
φ
(
x
)
〉
=
〈
U
,
〈
K
(
z
-
t
)
,
φ
(
x
)
〉
〉
.Proof.
Let1
≤
p
≤
2. Since K
(
z
-
t
)
∈
𝒟
L
p
,
(
ω
) by Theorem 13,
(37)
K
y
=
K
(
x
+
i
y
)
=
∫
C
*
‍
e
2
π
i
〈
x
+
i
y
,
η
〉
d
η
∈
𝒟
L
p
,
(
ω
)
,
y
∈
C
.
SinceU is a convolutor in 𝒟
L
p
,
(
ω
), (
U
;
z
)
=
(
U
*
K
y
)
(
x
)
∈
𝒟
L
p
,
(
ω
). Since 𝒟
L
p
,
(
ω
)
⊂
𝒟
L
p
,
(
ω
)
′ and 𝒮
(
ω
)
⊂
𝒟
L
p
,
(
ω
), 〈
C
(
U
;
z
)
,
φ
(
x
)
〉
=
〈
(
U
*
K
y
)
(
x
)
,
φ
(
x
)
〉
=
〈
〈
U
,
K
(
z
-
t
)
〉
,
φ
(
x
)
〉 is well defined for φ
∈
𝒮
(
ω
). We get from Theorem 10, (28), and Fubini's theorem that there exists a strongly elliptic ultradifferential operator G
(
D
) of (
ω
)-class and f
∈
𝒟
L
p
,
(
ω
)
⊂
ℰ
(
ω
), 1
≤
p
≤
2, such that
(38)
〈
C
(
U
;
z
)
,
φ
(
x
)
〉
=
〈
〈
U
,
K
(
z
-
t
)
〉
,
φ
(
x
)
〉
=
∬
∑
α
=
0
∞
‍
(
-
1
)
α
G
(
α
)
(
0
)
α
!
f
(
α
)
(
x
)
K
(
z
-
t
)
d
t
φ
(
x
)
d
x
=
∬
∑
α
=
0
∞
‍
(
-
1
)
α
G
(
α
)
(
0
)
α
!
f
(
α
)
(
x
)
K
(
z
-
t
)
φ
(
x
)
d
x
d
t
=
∫
‍
∑
α
=
0
∞
‍
(
-
1
)
α
G
(
α
)
(
0
)
α
!
f
(
α
)
(
x
)
∫
‍
K
(
z
-
t
)
φ
(
x
)
d
x
d
t
=
〈
U
,
〈
K
(
z
-
t
)
,
φ
(
x
)
〉
〉
for y
=
Im
⁡
(
z
)
∈
C.Remark 18.
We note that the reason for working withφ
∈
𝒮
(
ω
) in Lemma 17 is that the symmetry of 𝒮
(
ω
) under the Fourier and inverse Fourier transform will be needed later on.Lemma 19.
LetC be a regular cone in ℝ and φ
∈
𝒮
(
ω
). Then
(39)
lim
⁡
y
→
0
〈
K
(
x
+
i
y
-
t
)
,
φ
(
x
)
〉
=
ℱ
-
1
[
I
C
*
(
η
)
φ
^
(
η
)
;
t
]
,
h
h
h
h
h
h
h
h
h
h
h
h
y
∈
C
,
in 𝒟
L
q
,
(
ω
), 1
≤
p
≤
2, 1
/
p
+
1
/
q
=
1, where I
C
*
(
η
) is the characteristic function in C
*.Proof.
Let1
<
p
≤
2 and let 1
/
p
+
1
/
q
=
1. For an arbitrary α
∈
ℕ, we get from (14) and (15) that
(40)
∥
D
t
α
〈
K
(
z
-
t
)
,
φ
(
x
)
〉
-
D
t
α
ℱ
-
1
[
I
C
*
(
η
)
φ
^
(
η
)
;
t
]
∥
q
=
∥
ℱ
-
1
[
I
C
*
(
η
)
η
α
φ
^
(
η
)
(
e
-
2
π
〈
y
,
η
〉
-
1
)
;
t
]
∥
q
≤
∥
I
C
*
(
η
)
η
α
φ
^
(
η
)
(
e
-
2
π
〈
y
,
η
〉
-
1
)
∥
p
.
For anyk
∈
ℕ and q
,
1
/
p
+
1
/
q
=
1, we get that by (40)
(41)
γ
k
,
q
(
〈
K
(
z
-
t
)
,
φ
(
x
)
〉
-
ℱ
-
1
[
I
C
*
(
η
)
φ
^
(
η
)
;
t
]
)
=
sup
⁡
α
∈
ℕ
∥
D
t
α
〈
K
(
z
-
t
)
,
φ
(
x
)
〉
-
D
t
α
ℱ
-
1
[
I
C
*
(
η
)
φ
^
(
η
)
;
t
]
∥
q
×
e
-
k
ψ
*
(
α
/
k
)
≤
sup
⁡
α
∈
ℕ
∥
I
C
*
(
η
)
|
η
|
α
φ
^
(
η
)
(
e
-
2
π
〈
y
,
η
〉
-
1
)
∥
p
e
-
k
ψ
*
(
α
/
k
)
=
sup
⁡
α
∈
ℕ
∥
I
C
*
(
η
)
e
α
log
⁡
|
η
|
e
-
k
ψ
*
(
α
/
k
)
φ
^
(
η
)
(
e
-
2
π
〈
y
,
η
〉
-
1
)
∥
p
=
∥
I
C
*
(
η
)
e
k
ψ
*
*
(
log
⁡
|
η
|
)
φ
^
(
η
)
(
e
-
2
π
〈
y
,
η
〉
-
1
)
∥
p
=
∥
I
C
*
(
η
)
e
k
ω
(
η
)
φ
^
(
η
)
(
e
-
2
π
〈
y
,
η
〉
-
1
)
∥
p
.
Since(42)
|
I
C
*
(
η
)
e
k
ω
(
η
)
φ
^
(
η
)
(
e
-
2
π
〈
y
,
η
〉
-
1
)
|
≤
|
e
k
ω
(
η
)
φ
^
(
η
)
|
and φ
^
(
η
)
∈
𝒮
(
ω
),
(43)
I
C
*
(
η
)
e
k
ω
(
η
)
φ
^
(
η
)
(
e
-
2
π
〈
y
,
η
〉
-
1
)
∈
L
1
∩
L
p
.
Hence we have from Lebesgue’s dominated convergence theorem that the last line of (41) converges to 0 as y
→
0 for y
∈
C.
It remains to prove the convergence forp
=
1, that is, in 𝒟
L
∞
,
(
ω
). By the same method of estimation of integrand in (41), we get that
(44)
γ
k
,
∞
(
〈
K
(
z
-
t
)
,
φ
(
x
)
〉
-
ℱ
-
1
[
I
C
*
(
η
)
φ
^
(
η
)
;
t
]
)
=
sup
⁡
α
∈
ℕ
∥
D
t
α
〈
K
(
z
-
t
)
,
φ
(
x
)
〉
-
D
t
α
ℱ
-
1
[
I
C
*
(
η
)
φ
^
(
η
)
;
t
]
n
n
n
n
n
n
n
n
n
-
D
t
α
ℱ
-
1
[
I
C
*
(
η
)
φ
^
(
η
)
;
t
]
∥
∞
e
-
k
ψ
*
(
α
/
k
)
=
sup
⁡
α
∈
ℕ
ℱ
-
1
[
I
C
*
(
η
)
η
α
φ
^
(
η
)
(
e
-
2
π
〈
y
,
η
〉
-
1
)
;
t
]
e
-
k
ψ
*
(
α
/
k
)
=
sup
⁡
α
∈
ℕ
∫
‍
e
i
〈
η
,
t
〉
I
C
*
(
t
)
t
α
φ
^
(
t
)
(
e
-
2
π
〈
y
,
t
〉
-
1
)
d
t
e
-
k
ψ
*
(
α
/
k
)
=
∫
I
C
*
(
t
)
e
k
ω
(
t
)
φ
^
(
t
)
(
e
-
2
π
〈
y
,
t
〉
-
1
)
d
t
.
By (42), (43), and Lebesgue’s dominated convergence theorem, the last line of (44) converges to 0 as y
→
0 for y
∈
C.Combining Lemmas17 and 19 and using the continuity of a convolutor U in 𝒟
L
p
,
(
ω
), consider the following:Lemma 20.
LetC be a regular cone in ℝ and let U be a convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2. If φ
∈
𝒮
(
ω
), one has the following:
(45)
lim
⁡
y
→
0
〈
C
(
U
;
z
)
,
φ
(
x
)
〉
=
〈
U
,
ℱ
-
1
[
I
C
*
(
η
)
φ
^
(
η
)
;
t
]
〉
,
h
h
h
h
h
h
h
h
h
y
=
Im
⁡
(
z
)
∈
C
.Now we will give the representation of a convolutorU in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2, as the distributional limit of a finite number of functions analytic in tubes.Theorem 21.
LetU be a convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2, and let φ
∈
𝒮
(
ω
). Let C
j, j
=
1
,
…
,
r, be a finite number of regular cones whose dual cone satisfies the property that
(46)
ℝ
∖
∪
j
=
1
r
C
j
*
,
C
j
*
∩
C
k
*
,
j
≠
k
,
j
,
k
=
1
,
…
,
r
,
are sets of Lebesgue’s measure 0. Then there exist functions U
j which are analytic in ℝ
+
i
C
j, j
=
1
,
…
,
r, and satisfy
(47)
|
U
j
(
z
)
|
≤
M
(
p
,
C
j
′
)
|
y
|
-
1
/
p
for an arbitrary compact subcone C
j
′
⊂
C
j and a constant M
(
p
,
C
j
′
) depending on p and C
j
′ and
(48)
〈
U
,
φ
〉
=
∑
j
=
r
r
‍
lim
⁡
y
→
0
〈
U
j
(
x
+
i
y
)
,
φ
(
x
)
〉
,
y
∈
C
j
.Proof.
Let(49)
U
j
=
〈
U
,
∫
C
j
*
‍
e
2
π
i
〈
z
-
t
,
η
〉
d
η
〉
,
z
=
x
+
i
y
∈
ℝ
+
i
C
j
.
The analyticity and growth ofU
j
(
z
), j
=
1
,
…
,
r, are followed by Theorem 16. Using Lemma 20, the linearity of U, and (46), if y
∈
C
j,
(50)
∑
j
=
r
r
‍
lim
⁡
y
→
0
〈
U
j
(
x
+
i
y
)
,
φ
(
x
)
〉
=
∑
j
=
r
r
〈
U
,
ℱ
-
1
[
I
C
j
*
(
η
)
φ
^
(
η
)
;
t
]
〉
‍
=
〈
U
,
∑
j
=
r
r
‍
ℱ
-
1
[
I
C
j
*
(
η
)
φ
^
(
η
)
;
t
]
〉
=
〈
U
,
ℱ
-
1
[
φ
^
(
η
)
;
t
]
〉
=
〈
U
,
φ
〉
.
The proof is complete.Remark 22.
As mentioned in page 246 of [2], under certain conditions on U
∈
𝒟
L
p
′, the Cauchy integral of U does have U as boundary value. But we see from Lemma 20 that the Cauchy integral of U
∈
𝒟
L
p
,
(
ω
)
′, 1
≤
p
≤
2, corresponding to a regular cone C in ℝ
n does not attain U as boundary value as y
→
0
¯
,
y
∈
C. We cannot find the conditions on U
∈
𝒟
L
p
,
(
ω
)
′, 1
≤
p
≤
2, such that the Cauchy integral of U
∈
𝒟
L
p
,
(
ω
)
′, 1
≤
p
≤
2, does attain U as boundary value.


## 5. Poisson Integral of the Convolutor in𝒟
L
p
,
(
ω
)
,
1
≤
p
≤
2

LetC be a regular cone in ℝ and let 2
≤
q
≤
∞. In Theorem 14, we showed that P
(
z
;
t
)
∈
𝒟
L
q
,
(
ω
) as a function of t
∈
ℝ for z
∈
T
C. Hence, for U
∈
𝒟
L
p
,
(
ω
)
′, 1
≤
p
≤
2, 〈
U
t
,
P
(
z
;
t
)
〉 is a well-defined function of t
∈
ℝ for z
∈
T
C.Definition 23.
LetC be a regular cone in ℝ and let U be a convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2. The Poisson integral PI
(
U
;
z
) of U corresponding to C is
(51)
PI
(
U
;
z
)
=
〈
U
t
,
P
(
z
;
t
)
〉
,
z
∈
T
C
.In this section we will show that, for the convolutorU in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2, the Poisson integral PI
(
U
;
z
)
,
z
∈
T
C, has U as boundary value distributionally as y
→
0
,
y
∈
C. We need several lemmas.Definition 24 (see [6], Definition 3).
LetS
∈
𝒟
L
p
,
(
ω
)
′ and T
∈
𝒟
L
q
,
(
ω
)
′ be given, 1
/
p
+
1
/
q
=
1. Then the convolution S
*
T is the ultradistribution S
*
T
∈
𝒟
(
ω
) given by 〈
S
*
T
,
ϕ
〉
=
〈
S
,
T
~
*
ϕ
〉, where T
~
(
ϕ
)
=
T
(
ϕ
~
).Lemma 25 (see [2], Lemma 5.5
.
9).
LetC be a regular cone and g
∈
L
p, 1
≤
p
<
∞. One has
(52)
lim
⁡
y
→
0
∫
‍
g
(
t
)
P
(
x
+
i
y
;
t
)
d
t
=
g
(
t
)
,
y
∈
C
,
in L
p.Lemma 26.
LetC be a regular cone in ℝ and let U be a convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2. For φ
∈
𝒟
(
ω
),
(53)
〈
PI
(
U
;
z
)
,
φ
(
x
)
〉
=
〈
U
,
〈
P
(
z
;
t
)
,
φ
(
x
)
〉
〉
,
h
h
h
h
h
y
=
Im
⁡
(
z
)
∈
C
.Proof.
Forφ
∈
𝒟
(
ω
),
(54)
∫
‍
P
(
x
+
i
y
;
t
)
φ
(
x
)
d
x
=
∫
‍
P
(
x
,
y
)
φ
(
x
+
t
)
d
x
,
y
∈
C
,
where
(55)
P
(
x
,
y
)
=
K
(
x
+
i
y
)
K
(
x
+
i
y
)
¯
K
(
2
i
y
)
,
x
∈
ℝ
,
y
∈
C
.
SinceP
(
x
,
y
)
∈
𝒟
L
q
,
(
ω
)
⊂
𝒟
L
q
,
(
ω
)
′, 2
≤
q
≤
∞, as a function of x
∈
ℝ and y
∈
C, we get from Definition 24 that, for a convolutor U in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2, U
*
P
(
x
,
y
)
∈
𝒟
(
ω
). Hence, for φ
∈
𝒟
(
ω
), 〈
U
*
P
(
x
,
y
)
,
φ
〉 is well defined and by the definition of convolution
(56)
〈
U
*
P
(
x
,
y
)
,
φ
〉
=
〈
U
,
〈
P
(
x
,
y
)
,
φ
(
x
+
t
)
〉
〉
,
y
∈
C
.
From (54) and (56),
(57)
〈
U
,
〈
P
(
z
;
t
)
,
φ
(
x
)
〉
〉
=
〈
U
,
〈
P
(
x
,
y
)
,
φ
(
x
+
t
)
〉
〉
=
〈
U
*
P
(
x
,
y
)
,
φ
〉
for φ
∈
𝒟
(
ω
). Hence 〈
U
,
〈
P
(
z
;
t
)
,
φ
(
x
)
〉
〉 is well defined for y
∈
C. Using Theorem 10 and (28), we get from change of order of integration that, for y
∈
C and φ
∈
𝒟
(
ω
), there exists a strongly elliptic ultradifferential operator G
(
D
) of (
ω
)-class and f
∈
𝒟
L
p
,
(
ω
)
⊂
ℰ
(
ω
), 1
≤
p
≤
2, such that(58)
〈
U
,
〈
P
(
z
;
t
)
,
φ
(
x
)
〉
〉
=
∫
‍
∑
α
=
0
∞
‍
(
-
1
)
α
G
(
α
)
(
0
)
α
!
f
(
α
)
(
x
)
∫
‍
P
(
z
;
t
)
φ
(
x
)
d
x
d
t
=
∫
‍
φ
(
x
)
∫
‍
∑
α
=
0
∞
‍
(
-
1
)
α
G
(
α
)
(
0
)
α
!
f
(
α
)
(
x
)
P
(
z
;
t
)
d
t
d
x
=
〈
PI
(
U
;
z
)
,
φ
(
x
)
〉
.Lemma 27.
LetC be a regular cone in ℝ and φ
∈
𝒟
(
ω
). Consider the following:
(59)
lim
⁡
y
→
0
∫
‍
P
(
x
+
i
y
;
t
)
φ
(
x
)
d
x
=
φ
(
t
)
,
y
∈
C
,
in the topology of 𝒟
L
q
,
(
ω
), 1
≤
q
≤
∞.Proof.
Letα be an arbitrary integer. By (ii) and (iv) in Lemma 4, φ
∈
𝒟
(
ω
)
⊂
𝒟
L
1
,
(
ω
)
⊂
𝒟
L
1
⊂
L
1. We get from (5.92) in the proof of Lemma 5.5
.
11 of [2] that, for 1
≤
p
≤
∞,
(60)
lim
⁡
y
→
0
∥
D
t
α
∫
‍
P
(
x
+
i
y
;
t
)
φ
(
x
)
d
x
-
D
t
α
φ
(
t
)
∥
q
=
0
,
y
∈
C
.
Thus for anyk
∈
ℕ and 2
≤
q
≤
∞
(61)
lim
⁡
y
→
0
sup
⁡
α
∈
ℕ
∥
D
t
α
∫
‍
P
(
x
+
i
y
;
t
)
φ
(
x
)
d
x
-
D
t
α
φ
(
t
)
∥
q
×
e
-
k
ψ
*
(
k
/
α
)
=
0
,
y
∈
C
.Combining Lemmas26 and 27, and the continuity of U
∈
𝒟
L
p
,
(
ω
)
′, we can obtain the convergence of PI
(
U
;
z
) to U distributionally as y
→
0
,
y
∈
C, as follows.Theorem 28.
LetC be a regular cone in ℝ and let U be a convolutor in 𝒟
L
p
,
(
ω
), 1
≤
p
≤
2. For φ
∈
𝒟
(
ω
),
(62)
lim
⁡
y
→
0
〈
PI
(
U
;
z
)
,
φ
(
x
)
〉
=
〈
U
,
φ
〉
,
y
∈
C
.



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*Source: 926790-2014-04-09.xml*