# Weighted Differentiation Composition Operator from Logarithmic Bloch Spaces to Zygmund-Type Spaces

Authors: Huiying Qu; Yongmin Liu; Shulei Cheng
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

LetH
(
𝔻
) denote the space of all holomorphic functions on the unit disk 𝔻 of ℂ, u
∈
H
(
𝔻
) and let  n be a positive integer, φ a holomorphic self-map of 𝔻, and μ a weight. In this paper, we investigate the boundedness and compactness of a weighted differentiation composition operator 𝒟
φ
,
u
n
f
(
z
)
=
u
(
z
)
f
(
n
)
(
φ
(
z
)
)
,
f
∈
H
(
𝔻
), from the logarithmic Bloch spaces to the Zygmund-type spaces.


---

## Body

## 1. Introduction

Let𝔻 denote the open unit disk of the complex plane ℂ and H
(
𝔻
) the space of all analytic functions in 𝔻.The logarithmic Bloch space is defined as follows:(1)
ℬ
log
⁡
=
{
(
log
⁡
2
1
-
|
z
|
)
f
∈
H
(
𝔻
)
:
∥
f
∥
=
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
2
)
(
log
⁡
2
1
-
|
z
|
)
|
f
′
(
z
)
|
<
∞
}
.
The space ℬ
log
⁡ is a Banach space under the norm ∥
f
∥
ℬ
log
⁡
=
|
f
(
0
)
|
+
∥
f
∥. Let ℬ
log
⁡
,
0 denote the subspace of ℬ
log
⁡ consisting of those f
∈
ℬ
log
⁡ such that
(2)
lim
⁡
|
z
|
→
1
(
1
-
|
z
|
2
)
(
log
⁡
2
1
-
|
z
|
)
|
f
′
(
z
)
|
=
0
.
It is obvious that there are unbounded ℬ
log
⁡ functions. For example, consider the function f
(
z
)
=
log
⁡
log
⁡
(
e
/
(
1
-
z
)
). There are also bounded functions that do not belong to ℬ
log
⁡. In fact, the interpolating Blaschke products do not belong to ℬ
log
⁡. It is easily proved that, for 0
<
α
<
1, ℬ
α
⫋
ℬ
log
⁡
⫋
ℬ. ℬ
log
⁡ first appeared in the study of boundedness of the Hankel operators on the Bergman space. Attele in [1] proved that, for f
∈
L
a
2
(
𝒟
), the Hankel operator H
f
:
L
a
1
(
𝒟
)
→
L
1
(
𝒟
) is bounded if and only if ∥
f
∥
ℬ
log
⁡
<
∞, thus giving one reason, and not the only reason, why log-Bloch-type spaces are of interest. Ye in [2] proved that ℬ
log
⁡
,
0 is a closed subspace of ℬ
log
⁡. Galanopoulos in [3] characterized the boundedness and compactness of the composition operator C
φ
:
ℬ
log
⁡
→
Q
log
⁡
p and the boundedness and compactness of the weighted composition operator u
C
φ
:
ℬ
log
⁡
→
ℬ
log
⁡. Ye in [4] characterized the boundedness and compactness of the weighted composition operator u
C
φ between the logarithmic Bloch space ℬ
log
⁡ and the α-Bloch spaces ℬ
α on the unit disk and the boundedness and compactness of the weighted composition operator u
C
φ between the little logarithmic Bloch space ℬ
log
⁡
0 and the little α-Bloch spaces ℬ
0
α on the unit disk. Li in [5] characterized the boundedness and compactness of the weighted composition operator u
C
φ from Bergman spaces A
β
p into the logarithmic Bloch space ℬ
log
⁡ on the unit disk. Ye in [6] characterized the boundedness and compactness of the weighted composition operator u
C
φ from the general function space F
(
p
,
q
,
s
) into the logarithmic Bloch space ℬ
log
⁡ on the unit disk. Colonna and Li in [7] studied the boundedness and compactness of the weighted composition operators from Hardy space into the logarithmic Bloch space and the little logarithmic Bloch space. Petrov in [8] obtains sharp reverse estimates for the logarithmic Bloch spaces on the unit disk. Castillo et al. in [9] characterized the boundedness and compactness of the composition operator from the logarithmic Bloch spaces into weighted Bloch spaces. García Ortiz and Ramos-Fernández in [10] characterized the boundedness and compactness of the composition operators from logarithmic Bloch spaces into Bloch-type spaces.Letμ be a weight; that is, μ is a positive continuous function on 𝔻. The Zygmund-type space 𝒵
μ consists of all f
∈
H
(
𝔻
) such that
(3)
sup
⁡
z
∈
𝔻
μ
(
z
)
|
f
′′
(
z
)
|
<
∞
.
With the norm ∥
f
∥
𝒵
μ
=
|
f
(
0
)
|
+
|
f
′
(
0
)
|
+
sup
⁡
z
∈
𝔻
⁡
μ
(
z
)
|
f
′′
(
z
)
|, it becomes a Banach space. The little Zygmund-type space 𝒵
μ
,
0 is a subspace of 𝒵
μ consisting of those f
∈
𝒵
μ such that
(4)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
f
′′
(
z
)
|
=
0
.
When μ
(
z
)
=
1
-
|
z
|
2, the Zygmund-type space 𝒵
μ becomes the Zygmund space 𝒵 [11], while the little Zygmund-type space 𝒵
μ
,
0 becomes the little Zygmund space 𝒵
0.Let𝒟
=
𝒟
1 be the differentiation operator; that is, 𝒟
f
=
f
′. If n
∈
ℕ
0, then the operator 𝒟
n is defined by 𝒟
0
f
=
f
,
𝒟
n
f
=
f
(
n
)
,
f
∈
H
(
𝔻
).The weighted differentiation composition operator, denoted by𝒟
φ
,
u
n, is defined as follows [12, 13]:
(5)
𝒟
φ
,
u
n
f
(
z
)
=
u
(
z
)
f
(
n
)
(
φ
(
z
)
)
,
f
∈
H
(
𝔻
)
,
where u
∈
H
(
𝔻
) and φ is a nonconstant holomorphic self-map of 𝔻.Ifn
=
0, then 𝒟
φ
,
u
n becomes the weighted composition operator u
C
φ, defined by
(6)
u
C
φ
f
(
z
)
=
u
(
z
)
f
(
φ
(
z
)
)
,
z
∈
𝔻
,
which, for u
(
z
)
≡
1, is reduced to the composition operator C
φ for some recent articles on weighted composition operators on some H
∞-type spaces, for example, [14–16] and references therein. If n
=
1, u
(
z
)
=
φ
′
(
z
), then 𝒟
φ
,
u
n
=
𝒟
C
φ, which was studied in [17–21]. When n
=
1
,
u
(
z
)
≡
1, then 𝒟
φ
,
u
n
=
C
φ
𝒟, which was studied in [17, 19]. If n
=
1, φ
(
z
)
=
z, then 𝒟
φ
,
u
n
=
M
u
𝒟, that is, the product of differentiation operator and multiplication operator M
u defined by M
u
f
=
u
f. Zhu in [13] completely characterized the boundedness and compactness of linear operators which are obtained by taking products of differentiation, composition, and multiplication operators from Bergman type spaces to Bers spaces. Stević in [12] studied the boundedness and compactness of the weighted differentiation composition operator 𝒟
φ
,
u
n from mixed-norm spaces to weighted-type spaces or the little weighted-type space (see also [22–24]). Zhu in [25] studied the boundedness and compactness of the generalized weighted composition operator on weighted Bergman spaces. Yang in [21] studied the boundedness and compactness of the operator C
φ
𝒟 and 𝒟
C
φ from Q
K
(
p
,
q
) to ℬ
μ and ℬ
μ
,
0 spaces. Liu and Yu in [18] studied the boundedness and compactness of the operator 𝒟
C
φ between H
∞ and Zygmund spaces. Ye and Zhou in [26] studied the boundedness and compactness of the weighted composition operators from Hardy to Zygmund type spaces. Stević in [27] studied the boundedness and compactness of the generalized composition operator from mixed-norm space to the Bloch-type space, the little Bloch-type space, the Zygmund space, and the little Zygmund space. For other recently introduced products of operators on spaces of holomorphic functions see [13, 16]. Motivated by the results [12, 18, 23, 24, 27], we consider the boundedness and compactness of the operators 𝒟
φ
,
u
n from the logarithmic Bloch spaces to the Zygmund-type spaces and the little Zygmund-type spaces. For the proof, we need different test functions and some complex calculations kills.Throughout this paper, we will use the letterC to denote a positive constant that can change its value at each occurrence.


## 2. Auxiliary Results

Here we prove and quote some auxiliary results which will be used in the proofs of the main results in this paper.Lemma 1.
Letn be a positive integer. Suppose f
∈
ℬ
log
⁡; there exists a constant C such that
(7)
|
f
(
n
)
(
z
)
|
≤
C
∥
f
∥
ℬ
log
⁡
(
1
-
|
z
|
2
)
n
log
⁡
(
2
/
(
1
-
|
z
|
)
)
,
z
∈
𝔻
,Proof.
We use induction onn. Using the definition of the logarithmic Bloch spaces we have
(8)
|
f
′
(
z
)
|
≤
C
∥
f
∥
ℬ
log
⁡
(
1
-
|
z
|
2
)
log
⁡
(
2
/
(
1
-
|
z
|
)
)
;
the case holds for n
=
1. Assume the case n
=
k holds; since
(9)
1
2
π
∫
0
2
π
‍
1
|
e
i
θ
-
z
|
2
d
θ
=
1
1
-
|
z
|
2
,
z
∈
𝔻
,
let ρ
=
(
1
+
|
z
|
)
/
2
<
1; then we have |
z
/
ρ
|
=
2
|
z
|
/
(
1
+
|
z
|
)
<
1, so
(10)
1
2
π
∫
0
2
π
‍
1
|
ρ
e
i
θ
-
z
|
2
d
θ
=
1
ρ
2
-
|
z
|
2
,
z
∈
𝔻
.
By the Cauchy integral formula we obtain
(11)
|
f
(
k
+
1
)
(
z
)
|
=
1
2
π
|
∫
0
2
π
‍
f
(
k
)
(
ρ
e
i
θ
)
(
ρ
e
i
θ
-
z
)
2
ρ
e
i
θ
d
θ
|
≤
1
2
π
∫
0
2
π
‍
|
f
(
k
)
(
ρ
e
i
θ
)
(
ρ
e
i
θ
-
z
)
2
ρ
e
i
θ
|
d
θ
≤
C
∥
f
∥
ℬ
log
⁡
(
1
-
ρ
2
)
k
(
log
⁡
(
2
/
(
1
-
ρ
)
)
)
×
1
2
π
∫
0
2
π
‍
ρ
|
ρ
e
i
θ
-
z
|
2
d
θ
=
C
∥
f
∥
ℬ
log
⁡
(
1
-
ρ
2
)
k
(
log
⁡
(
2
/
(
1
-
ρ
)
)
)
ρ
ρ
2
-
|
z
|
2
=
C
∥
f
∥
ℬ
log
⁡
(
1
-
ρ
2
)
k
(
log
⁡
(
2
/
(
1
-
ρ
)
)
)
ρ
(
ρ
+
|
z
|
)
(
ρ
-
|
z
|
)
≤
C
∥
f
∥
ℬ
log
⁡
(
1
-
ρ
2
)
k
(
log
⁡
(
2
/
(
1
-
ρ
)
)
)
1
ρ
-
|
z
|
=
C
∥
f
∥
ℬ
log
⁡
(
1
-
ρ
2
)
k
(
log
⁡
(
2
/
(
1
-
ρ
)
)
)
1
1
-
ρ
≤
C
∥
f
∥
ℬ
log
⁡
(
1
-
ρ
2
)
k
(
log
⁡
(
2
/
(
1
-
ρ
)
)
)
2
(
1
-
ρ
)
(
1
+
ρ
)
≤
2
C
∥
f
∥
ℬ
log
⁡
(
1
-
ρ
2
)
k
+
1
(
log
⁡
(
2
/
(
1
-
ρ
)
)
)
.
Note that
(12)
1
4
(
1
-
|
z
|
)
≤
1
-
ρ
=
1
2
(
1
-
|
z
|
)
≤
1
-
|
z
|
,
(13)
1
2
(
1
-
|
z
|
)
≤
1
-
ρ
2
=
(
1
+
ρ
)
(
1
-
ρ
)
≤
1
-
|
z
|
;
we have
(14)
|
f
(
k
+
1
)
(
z
)
|
≤
C
∥
f
∥
ℬ
log
⁡
(
1
-
|
z
|
2
)
k
+
1
log
⁡
(
2
/
(
1
-
|
z
|
)
)
,
for every z
∈
𝔻. Hence the case n
=
k
+
1 holds. The desired result follows. The proof of this lemma is complete.Lemma 2 (see [4, 28]).
Let(15)
g
t
(
z
)
=
(
1
-
|
z
|
)
log
⁡
⁡
(
2
/
(
1
-
|
z
|
)
)
(
1
-
|
t
z
|
)
log
⁡
(
2
/
(
1
-
|
t
z
|
)
)
,
t
∈
[
0,1
]
,
z
∈
𝔻
;
then |
g
t
(
z
)
|
<
2.The following criterion for the compactness is a useful tool and it follows from standard arguments (e.g., [29, Proposition 3.11] or [30, Lemma 2.10]).Lemma 3.
Letu
∈
H
(
𝔻
), and let n be a nonnegative integer, φ a holomorphic self-map of 𝔻, and μ a weight. Then 𝒟
φ
,
u
n
:
ℬ
log
⁡
(
ℬ
log
⁡
,
0
)
→
𝒵
μ is compact if and only if 𝒟
φ
,
u
n
:
ℬ
log
⁡
(
ℬ
log
⁡
,
0
)
→
𝒵
μ is bounded and, for any bounded sequence {
f
k
} in ℬ
log
⁡
(
ℬ
log
⁡
,
0
) which converges to zero uniformly on compact subsets of 𝔻 as k
→
∞, we have ∥
𝒟
φ
,
u
n
f
k
∥
𝒵
μ
→
0 as k
→
∞.Lemma 4.
A closed setK in 𝒵
μ
,
0 is compact if and only if K is bounded and satisfies
(16)
lim
⁡
|
z
|
→
1
sup
⁡
f
∈
K
μ
(
z
)
|
f
′′
(
z
)
|
=
0
.The proof is similar to that of Lemma 1 in [31]; hence we omit it.


## 3. Boundedness and Compactness of𝒟
φ
,
u
n from ℬ
log
⁡
(
ℬ
log,0
) to 𝒵
μ
(
𝒵
μ
,
0
) Spaces

In this section, we study the boundedness and compactness of𝒟
φ
,
u
n
:
ℬ
log
⁡
(
ℬ
log
⁡
,
0
)
→
𝒵
μ
(
𝒵
μ
,
0
).Theorem 5.
Letu
∈
H
(
𝔻
), and let n be a nonnegative integer, φ a holomorphic self-map of 𝔻, and μ a weight. Then the following statements are equivalent:(1)
𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ is bounded;
(2)
𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ is bounded;
(3)
(17)
sup
⁡
z
∈
𝔻
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
∞
,
(18)
sup
⁡
z
∈
𝔻
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
1
log
⁡
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
∞
,
(19)
sup
⁡
z
∈
𝔻
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
∞
.Proof.
(
3
)
⇒
(
1
). Suppose that (17), (18), and (19) hold. Then, for every z
∈
𝔻 and f
∈
ℬ
log
⁡, by Lemma 1, we have
(20)
μ
(
z
)
|
(
𝒟
φ
,
u
n
f
)
′′
(
z
)
|
=
μ
(
z
)
|
(
φ
′
(
z
)
)
2
u
′′
(
z
)
f
(
n
)
(
φ
(
z
)
)
=
μ
(
z
)
+
(
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
)
f
(
n
+
1
)
(
φ
(
z
)
)
=
μ
(
z
)
+
u
(
z
)
(
φ
′
(
z
)
)
2
f
(
n
+
2
)
(
φ
(
z
)
)
|
≤
μ
(
z
)
|
u
′′
(
z
)
|
|
f
(
n
)
(
φ
(
z
)
)
|
+
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
|
f
(
n
+
1
)
(
φ
(
z
)
)
|
+
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
|
f
(
n
+
2
)
(
φ
(
z
)
)
|
≤
C
∥
f
∥
ℬ
log
⁡
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
+
C
∥
f
∥
ℬ
log
⁡
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
+
C
∥
f
∥
ℬ
log
⁡
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
+
C
∥
f
∥
ℬ
log
⁡
.
On the other hand, we have
(21)
|
(
𝒟
φ
,
u
n
f
)
(
0
)
|
=
|
u
(
0
)
f
(
n
)
(
φ
(
0
)
)
|
≤
C
|
u
(
0
)
|
(
1
-
|
φ
(
0
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
0
)
|
)
)
∥
f
∥
ℬ
log
⁡
,
|
(
𝒟
φ
,
u
n
f
)
′
(
0
)
|
=
|
u
(
0
)
′
f
(
n
)
(
φ
(
0
)
)
+
u
(
0
)
f
(
n
+
1
)
(
φ
(
0
)
)
φ
′
(
0
)
|
≤
C
|
u
′
(
0
)
|
(
1
-
|
φ
(
0
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
0
)
|
)
)
∥
f
∥
ℬ
log
⁡
+
C
|
u
(
0
)
φ
′
(
0
)
|
(
1
-
|
φ
(
0
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
0
)
|
)
)
∥
f
∥
ℬ
log
⁡
.
Applying conditions (20) and (21), we deduce that the operator 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ is bounded.
(
1
)
⇒
(
2
). This implication is clear.
(
2
)
⇒
(
3
). Assume that 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ is bounded; that is, there exists a constant C, such that
(22)
∥
𝒟
φ
,
u
n
f
∥
𝒵
μ
≤
C
∥
f
∥
ℬ
log
⁡
,
for all f
∈
ℬ
log
⁡
,
0. For f
(
z
)
=
z
n
/
n
!
∈
ℬ
log
⁡
,
0, we have that
(23)
K
1
≔
sup
⁡
z
∈
𝔻
μ
(
z
)
|
u
′′
(
z
)
|
<
∞
.
Taking f
(
z
)
=
z
n
+
1
/
(
n
+
1
)
!
∈
ℬ
log
⁡
,
0; we have that
(24)
sup
⁡
z
∈
𝔻
μ
(
z
)
|
u
′′
(
z
)
φ
(
z
)
+
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
<
∞
.
By (23), (24), and the boundedness of the function φ
(
z
), we get
(25)
K
2
:
=
sup
⁡
z
∈
𝔻
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
<
∞
.
In the same way, taking f
(
z
)
=
z
n
+
2
/
(
n
+
2
)
!
∈
ℬ
log
⁡
,
0, we have that
(26)
sup
⁡
z
∈
𝔻
μ
(
z
)
|
(
φ
′
(
z
)
)
2
u
′′
(
z
)
(
φ
(
z
)
)
2
sup
⁡
z
∈
𝔻
μ
(
z
)
+
2
(
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
)
φ
(
z
)
sup
⁡
z
∈
𝔻
μ
(
z
)
+
2
u
(
z
)
(
φ
′
(
z
)
)
2
|
<
∞
.
By (23), (25), (26), and the boundedness of the function φ
(
z
), we have that
(27)
K
3
:
=
sup
⁡
z
∈
𝔻
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
<
∞
.
For a fixed ω
∈
𝔻, set
(28)
f
ω
(
z
)
=
(
n
+
2
)
(
n
+
3
)
1
-
|
φ
(
ω
)
|
2
(
1
-
z
φ
(
ω
)
¯
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
2
(
n
+
3
)
(
1
-
|
φ
(
ω
)
|
2
)
2
(
1
-
z
φ
(
ω
)
¯
)
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
2
(
1
-
|
φ
(
ω
)
|
2
)
3
(
1
-
z
φ
(
ω
)
¯
)
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
We get that
(29)
f
ω
(
n
)
(
z
)
=
(
n
+
3
)
!
n
+
1
(
1
-
|
φ
(
ω
)
|
2
)
(
φ
(
ω
)
¯
)
n
(
1
-
z
φ
(
ω
)
¯
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
2
(
n
+
3
)
·
(
n
+
1
)
!
(
1
-
|
φ
(
ω
)
|
2
)
2
(
φ
(
ω
)
¯
)
n
(
1
-
z
φ
(
ω
)
¯
)
n
+
2
log
⁡
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
(
n
+
2
)
!
(
1
-
|
φ
(
ω
)
|
2
)
3
(
φ
(
ω
)
¯
)
n
(
1
-
z
φ
(
ω
)
¯
)
n
+
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
;
f
ω
(
n
+
1
)
(
z
)
=
(
n
+
3
)
!
(
1
-
|
φ
(
ω
)
|
2
)
(
φ
(
ω
)
¯
)
n
+
1
(
1
-
z
φ
(
ω
)
¯
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
2
·
(
n
+
3
)
!
(
1
-
|
φ
(
ω
)
|
2
)
2
(
φ
(
ω
)
¯
)
n
+
1
(
1
-
z
φ
(
ω
)
¯
)
n
+
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
(
n
+
3
)
!
(
1
-
|
φ
(
ω
)
|
2
)
3
(
φ
(
ω
)
¯
)
n
+
1
(
1
-
z
φ
(
ω
)
¯
)
n
+
4
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
;
f
ω
(
n
+
2
)
(
z
)
=
(
n
+
2
)
·
(
n
+
3
)
!
(
1
-
|
φ
(
ω
)
|
2
)
(
φ
(
ω
)
¯
)
n
+
2
(
1
-
z
φ
(
ω
)
¯
)
n
+
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
2
(
n
+
3
)
·
(
n
+
3
)
!
(
1
-
|
φ
(
ω
)
|
2
)
2
(
φ
(
ω
)
¯
)
n
+
2
(
1
-
z
φ
(
ω
)
¯
)
n
+
4
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
(
n
+
4
)
!
(
1
-
|
φ
(
ω
)
|
2
)
3
(
φ
(
ω
)
¯
)
n
+
2
(
1
-
z
φ
(
ω
)
¯
)
n
+
5
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
By Lemma 2 we have
(30)
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
2
)
(
log
⁡
2
1
-
|
z
|
)
|
f
ω
′
(
z
)
|
≤
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
2
)
(
log
⁡
2
1
-
|
z
|
)
×
|
(
n
+
2
)
(
n
+
3
)
(
1
-
|
φ
(
ω
)
|
2
)
φ
(
ω
)
¯
(
1
-
z
φ
(
ω
)
¯
)
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
|
+
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
2
)
(
log
⁡
2
1
-
|
z
|
)
×
|
4
(
n
+
3
)
(
1
-
|
φ
(
ω
)
|
2
)
2
φ
(
ω
)
¯
(
1
-
z
φ
(
ω
)
¯
)
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
|
+
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
2
)
(
log
⁡
2
1
-
|
z
|
)
×
|
6
(
1
-
|
φ
(
ω
)
|
2
)
3
φ
(
ω
)
¯
(
1
-
z
φ
(
ω
)
¯
)
4
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
|
≤
4
(
n
+
2
)
(
n
+
3
)
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
(
log
⁡
2
1
-
|
z
|
)
×
(
1
-
|
φ
(
ω
)
|
)
(
1
-
|
φ
(
ω
)
¯
|
)
(
1
-
|
z
φ
(
ω
)
¯
|
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
32
(
n
+
3
)
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
(
log
⁡
2
1
-
|
z
|
)
×
(
1
-
|
φ
(
ω
)
|
)
2
(
1
-
|
φ
(
ω
)
¯
|
)
2
(
1
-
|
z
φ
(
ω
)
¯
|
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
96
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
(
log
⁡
2
1
-
|
z
|
)
×
(
1
-
|
φ
(
ω
)
|
)
3
(
1
-
|
φ
(
ω
)
¯
|
)
3
(
1
-
|
z
φ
(
ω
)
¯
|
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
=
4
(
n
+
2
)
(
n
+
3
)
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
(
log
⁡
(
2
/
(
1
-
|
z
|
)
)
)
(
1
-
|
z
φ
(
ω
)
¯
|
)
(
log
⁡
⁡
(
2
/
(
1
-
|
z
φ
(
ω
)
¯
|
)
)
)




×
log
⁡
⁡
(
2
/
(
1
-
|
z
φ
(
ω
)
¯
|
)
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
32
(
n
+
3
)
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
(
log
⁡
⁡
(
2
/
(
1
-
|
z
|
)
)
)
(
1
-
|
z
φ
(
ω
)
¯
|
)
(
log
⁡
(
2
/
(
1
-
|
z
φ
(
ω
)
¯
|
)
)
)
×
log
⁡
(
2
/
(
1
-
|
z
φ
(
ω
)
¯
|
)
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
96
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
(
log
⁡
(
2
/
(
1
-
|
z
|
)
)
)
(
1
-
|
z
φ
(
ω
)
¯
|
)
(
log
⁡
(
2
/
(
1
-
|
z
φ
(
ω
)
¯
|
)
)
)
×
log
⁡
(
2
/
(
1
-
|
z
φ
(
ω
)
¯
|
)
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
4
(
n
+
2
)
(
n
+
3
)
×
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
(
log
⁡
(
2
/
(
1
-
|
z
|
)
)
)
(
1
-
|
z
φ
(
ω
)
¯
|
)
(
log
⁡
(
2
/
(
1
-
|
z
φ
(
ω
)
¯
|
)
)
)
×
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
¯
|
)
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
32
(
n
+
3
)
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
(
log
⁡
(
2
/
(
1
-
|
z
|
)
)
)
(
1
-
|
z
φ
(
ω
)
¯
|
)
(
log
⁡
⁡
(
2
/
(
1
-
|
z
φ
(
ω
)
¯
|
)
)
)


×
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
¯
|
)
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
96
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
(
log
⁡
(
2
/
(
1
-
|
z
|
)
)
)
(
1
-
|
z
φ
(
ω
)
¯
|
)
(
log
⁡
(
2
/
(
1
-
|
z
φ
(
ω
)
¯
|
)
)
)
×
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
¯
|
)
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
8
(
n
+
2
)
(
n
+
3
)
+
64
(
n
+
3
)
+
192
=
84
n
2
+
104
n
+
432
.
Hence, f
ω
∈
ℬ
log
⁡ and sup
⁡
ω
∈
𝔻
⁡
∥
f
ω
∥
ℬ
log
⁡
≤
C.
On the other hand for each fixω
∈
𝔻, by (30), we obtain that
(31)
(
1
-
|
z
|
2
)
(
log
⁡
2
1
-
|
z
|
)
|
f
ω
′
(
z
)
|
⟶
0
,
(
as
|
z
|
⟶
1
)
;
it follows that f
ω
∈
ℬ
log
⁡
,
0 for each fix ω
∈
𝔻. From (29), we have f
ω
(
n
+
1
)
(
φ
(
ω
)
)
=
f
ω
(
n
+
2
)
(
φ
(
ω
)
)
=
0 and
(32)
f
ω
(
n
)
(
φ
(
ω
)
)
=
2
·
n
!
(
φ
(
ω
)
¯
)
n
(
1
-
|
φ
(
ω
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
Hence
(33)
C
≥
∥
𝒟
φ
,
u
n
f
ω
∥
𝒵
μ
≥
sup
⁡
z
∈
𝔻
μ
(
z
)
|
(
𝒟
φ
,
u
n
f
ω
)
′′
(
z
)
|
=
sup
⁡
z
∈
𝔻
μ
(
z
)
|
(
φ
′
(
z
)
)
2
u
′′
(
z
)
f
ω
(
n
)
(
φ
(
z
)
)
=
sup
⁡
z
∈
𝔻
μ
(
z
)
+
(
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
)
f
ω
(
n
+
1
)
(
φ
(
z
)
)
=
sup
⁡
z
∈
𝔻
μ
(
z
)
+
u
(
z
)
(
φ
′
(
z
)
)
2
f
ω
(
n
+
2
)
(
φ
(
z
)
)
|
≥
μ
(
ω
)
|
(
φ
′
(
ω
)
)
2
u
′′
(
ω
)
f
ω
(
n
)
(
φ
(
ω
)
)
≥
μ
(
ω
)
+
(
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
)
f
ω
(
n
+
1
)
(
φ
(
ω
)
)
+
≥
μ
(
ω
)
+
u
(
ω
)
(
φ
′
(
ω
)
)
2
f
ω
(
n
+
2
)
(
φ
(
ω
)
)
|
=
2
·
n
!
μ
(
ω
)
|
u
′′
(
ω
)
|
|
φ
(
ω
)
¯
|
n
(
1
-
|
φ
(
ω
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
By (33), we obtain that
(34)
sup
⁡
1
/
2
<
|
φ
(
ω
)
|
<
1
μ
(
ω
)
|
u
′′
(
ω
)
|
(
1
-
|
φ
(
ω
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
2
n
+
1
·
n
!
sup
⁡
1
/
2
<
|
φ
(
ω
)
|
<
1
μ
(
ω
)
|
u
′′
(
ω
)
|
|
φ
(
ω
)
¯
|
n
(
1
-
|
φ
(
ω
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)


≤
2
n
+
1
·
n
!
sup
⁡
ω
∈
𝔻
μ
(
ω
)
|
u
′′
(
ω
)
|
|
φ
(
ω
)
¯
|
n
(
1
-
|
φ
(
ω
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
C
2
n
<
∞
.
And from (23), we have
(35)
sup
⁡
|
φ
(
ω
)
|
≤
1
/
2
μ
(
ω
)
|
u
′′
(
ω
)
|
(
1
-
|
φ
(
ω
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
sup
⁡
|
φ
(
ω
)
|
≤
1
/
2
μ
(
ω
)
|
u
′′
(
ω
)
|
(
1
-
|
φ
(
ω
)
|
2
)
n
log
⁡
2
≤
(
4
3
)
n
1
log
⁡
2
sup
⁡
|
φ
(
ω
)
|
≤
1
/
2
μ
(
ω
)
|
u
′′
(
ω
)
|
≤
(
4
3
)
n
K
1
log
⁡
2
<
∞
.
Thus combining (35) with (34) we get the condition (17).
For a fixedω
∈
𝔻, set
(36)
g
ω
(
z
)
=
(
n
+
1
)
(
n
+
3
)
1
-
|
φ
(
ω
)
|
2
(
1
-
z
φ
(
ω
)
¯
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
(
2
n
+
5
)
(
1
-
|
φ
(
ω
)
|
2
)
2
(
1
-
z
φ
(
ω
)
¯
)
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
2
(
1
-
|
φ
(
ω
)
|
2
)
3
(
1
-
z
φ
(
ω
)
¯
)
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
It is easy to see that
(37)
g
ω
(
n
)
(
z
)
=
(
n
+
3
)
·
(
n
+
1
)
!
(
1
-
|
φ
(
ω
)
|
2
)
(
φ
(
ω
)
¯
)
n
(
1
-
z
φ
(
ω
)
¯
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
(
2
n
+
5
)
·
(
n
+
1
)
!
(
1
-
|
φ
(
ω
)
|
2
)
2
(
φ
(
ω
)
¯
)
n
(
1
-
z
φ
(
ω
)
¯
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
(
n
+
2
)
!
(
1
-
|
φ
(
ω
)
|
2
)
3
(
φ
(
ω
)
¯
)
n
(
1
-
z
φ
(
ω
)
¯
)
n
+
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
;
g
ω
(
n
+
1
)
(
z
)
=
(
n
+
1
)
·
(
n
+
3
)
!
(
n
+
2
)
(
1
-
|
φ
(
ω
)
|
2
)
(
φ
(
ω
)
¯
)
n
+
1
(
1
-
z
φ
(
ω
)
¯
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
(
2
n
+
5
)
·
(
n
+
2
)
!
(
1
-
|
φ
(
ω
)
|
2
)
2
(
φ
(
ω
)
¯
)
n
+
1
(
1
-
z
φ
(
ω
)
¯
)
n
+
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
(
n
+
3
)
!
(
1
-
|
φ
(
ω
)
|
2
)
3
(
φ
(
ω
)
¯
)
n
+
1
(
1
-
z
φ
(
ω
)
¯
)
n
+
4
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
;
g
ω
(
n
+
2
)
(
z
)
=
(
n
+
1
)
·
(
n
+
3
)
!
(
1
-
|
φ
(
ω
)
|
2
)
(
φ
(
ω
)
¯
)
n
+
2
(
1
-
z
φ
(
ω
)
¯
)
n
+
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
(
2
n
+
5
)
·
(
n
+
3
)
!
(
1
-
|
φ
(
ω
)
|
2
)
2
(
φ
(
ω
)
¯
)
n
+
2
(
1
-
z
φ
(
ω
)
¯
)
n
+
4
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
(
n
+
4
)
!
(
1
-
|
φ
(
ω
)
|
2
)
3
(
φ
(
ω
)
¯
)
n
+
2
(
1
-
z
φ
(
ω
)
¯
)
n
+
5
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
Using Lemma 2, we easily get that g
ω
∈
ℬ
log
⁡
,
0 and sup
⁡
ω
∈
𝔻
⁡
∥
g
ω
∥
ℬ
log
⁡
≤
C with a direct calculation. From (37), we have g
ω
(
n
)
(
φ
(
ω
)
)
=
g
ω
(
n
+
2
)
(
φ
(
ω
)
)
=
0,
(38)
g
ω
(
n
+
1
)
(
φ
(
ω
)
)
=
-
(
n
+
1
)
!
(
φ
(
ω
)
¯
)
n
+
1
(
1
-
|
φ
(
ω
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
Hence
(39)
C
≥
∥
𝒟
φ
,
u
n
g
ω
∥
𝒵
μ
≥
sup
⁡
z
∈
𝔻
μ
(
z
)
|
(
𝒟
φ
,
u
n
g
ω
)
′′
(
z
)
|
=
sup
⁡
z
∈
𝔻
μ
(
z
)
|
(
φ
′
(
z
)
)
2
u
′′
(
z
)
g
ω
(
n
)
(
φ
(
z
)
)
=
sup
⁡
z
∈
𝔻
μ
(
z
)
+
(
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
)
g
ω
(
n
+
1
)
(
φ
(
z
)
)
=
sup
⁡
z
∈
𝔻
μ
(
z
)
+
u
(
z
)
(
φ
′
(
z
)
)
2
g
ω
(
n
+
2
)
(
φ
(
z
)
)
|
≥
μ
(
ω
)
|
(
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
)
(
φ
′
(
ω
)
)
2
u
′′
(
ω
)
g
ω
(
n
)
(
φ
(
ω
)
)
≥
μ
(
ω
)
+
(
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
)
g
ω
(
n
+
1
)
(
φ
(
ω
)
)
≥
μ
(
ω
)
+
u
(
ω
)
(
φ
′
(
ω
)
)
2
g
ω
(
n
+
2
)
(
φ
(
ω
)
)
(
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
)
|
=
(
n
+
1
)
!
μ
(
ω
)
|
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
|
|
φ
(
ω
)
¯
|
n
+
1
(
1
-
|
φ
(
ω
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
From (39), we obtain that
(40)
sup
⁡
1
/
2
<
|
φ
(
ω
)
|
<
1
μ
(
ω
)
|
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
|
(
1
-
|
φ
(
ω
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
2
n
+
1
sup
⁡
1
/
2
<
|
φ
(
ω
)
|
<
1
(
n
+
1
)
!
×
μ
(
ω
)
|
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
|
|
φ
(
ω
)
¯
|
n
+
1
(
1
-
|
φ
(
ω
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
2
n
+
1
(
n
+
1
)
!
×
sup
⁡
ω
∈
𝔻
μ
(
ω
)
|
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
|
|
φ
(
ω
)
¯
|
n
+
1
(
1
-
|
φ
(
ω
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
C
2
n
+
1
<
∞
.
By (25), we have
(41)
sup
⁡
|
φ
(
ω
)
|
≤
1
/
2
μ
(
ω
)
|
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
|
(
1
-
|
φ
(
ω
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
sup
⁡
|
φ
(
ω
)
|
≤
1
/
2
μ
(
ω
)
|
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
|
(
1
-
|
φ
(
ω
)
|
2
)
n
+
1
log
⁡
2
≤
(
4
3
)
n
+
1
1
log
⁡
2
sup
⁡
|
φ
(
ω
)
|
≤
1
/
2
μ
(
ω
)
|
2
u
′
(
ω
)
φ
′
(
ω
)
≤
(
4
3
)
n
+
1
1
log
⁡
2
sup
⁡
|
φ
(
ω
)
|
≤
(
1
/
2
)
μ
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
|
≤
(
4
3
)
n
+
1
K
2
log
⁡
2
<
∞
.
Thus combining (40) with (41) we get the condition (18).
Next, we prove (19). To see this, for a fixed ω
∈
𝔻, put
(42)
h
ω
(
z
)
=
(
n
+
1
)
(
n
+
2
)
1
-
|
φ
(
ω
)
|
2
(
1
-
z
φ
(
ω
)
¯
)
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
2
(
n
+
2
)
(
1
-
|
φ
(
ω
)
|
2
)
2
(
1
-
z
φ
(
ω
)
¯
)
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
2
(
1
-
|
φ
(
ω
)
|
2
)
3
(
1
-
z
φ
(
ω
)
¯
)
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
It is easy to see that
(43)
h
ω
(
n
)
(
z
)
=
(
n
+
2
)
!
(
1
-
|
φ
(
ω
)
|
2
)
(
φ
(
ω
)
¯
)
n
(
1
-
z
φ
(
ω
)
¯
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
2
·
(
n
+
2
)
!
(
1
-
|
φ
(
ω
)
|
2
)
2
(
φ
(
ω
)
¯
)
n
(
1
-
z
φ
(
ω
)
¯
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
(
n
+
2
)
!
(
1
-
|
φ
(
ω
)
|
2
)
3
(
φ
(
ω
)
¯
)
n
(
1
-
z
φ
(
ω
)
¯
)
n
+
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
;
h
ω
(
n
+
1
)
(
z
)
=
(
n
+
1
)
·
(
n
+
2
)
!
(
1
-
|
φ
(
ω
)
|
2
)
(
φ
(
ω
)
¯
)
n
+
1
(
1
-
z
φ
(
ω
)
¯
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
2
(
n
+
2
)
·
(
n
+
2
)
!
(
1
-
|
φ
(
ω
)
|
2
)
2
(
φ
(
ω
)
¯
)
n
+
1
(
1
-
z
φ
(
ω
)
¯
)
n
+
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
(
n
+
3
)
!
(
1
-
|
φ
(
ω
)
|
2
)
3
(
φ
(
ω
)
¯
)
n
+
1
(
1
-
z
φ
(
ω
)
¯
)
n
+
4
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
;
h
ω
(
n
+
2
)
(
z
)
=
(
n
+
1
)
(
n
+
2
)
·
(
n
+
2
)
!
(
1
-
|
φ
(
ω
)
|
2
)
(
φ
(
ω
)
¯
)
n
+
2
(
1
-
z
φ
(
ω
)
¯
)
n
+
3
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
-
2
(
n
+
2
)
·
(
n
+
3
)
!
(
1
-
|
φ
(
ω
)
|
2
)
2
(
φ
(
ω
)
¯
)
n
+
2
(
1
-
z
φ
(
ω
)
¯
)
n
+
4
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
+
(
n
+
4
)
!
(
1
-
|
φ
(
ω
)
|
2
)
3
(
φ
(
ω
)
¯
)
n
+
2
(
1
-
z
φ
(
ω
)
¯
)
n
+
5
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
From Lemma 2 we obtain that h
ω
∈
ℬ
log
⁡
,
0 and sup
⁡
ω
∈
𝔻
⁡
∥
h
ω
∥
ℬ
log
⁡
≤
C with a direct calculation. From (43), we have h
ω
(
n
)
(
φ
(
ω
)
)
=
h
ω
(
n
+
1
)
(
φ
(
ω
)
)
=
0,
(44)
h
ω
(
n
+
2
)
(
φ
(
ω
)
)
=
2
·
(
n
+
2
)
!
(
φ
(
ω
)
¯
)
n
+
2
(
1
-
|
φ
(
ω
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
Hence
(45)
C
≥
∥
𝒟
φ
,
u
n
h
ω
∥
𝒵
μ
≥
sup
⁡
z
∈
𝔻
μ
(
z
)
|
(
𝒟
φ
,
u
n
h
ω
)
′′
(
z
)
|
=
sup
⁡
z
∈
𝔻
μ
(
z
)
|
(
φ
′
(
z
)
)
2
u
′′
(
z
)
h
ω
(
n
)
(
φ
(
z
)
)
=
sup
⁡
z
∈
𝔻
μ
(
z
)
+
(
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
)
h
ω
(
n
+
1
)
(
φ
(
z
)
)
=
sup
⁡
z
∈
𝔻
μ
(
z
)
+
u
(
z
)
(
φ
′
(
z
)
)
2
h
ω
(
n
+
2
)
(
φ
(
z
)
)
|
≥
μ
(
ω
)
|
(
φ
′
(
ω
)
)
2
u
′′
(
ω
)
h
ω
(
n
)
(
φ
(
ω
)
)
≥
μ
(
ω
)
+
(
2
u
′
(
ω
)
φ
′
(
ω
)
+
u
(
ω
)
φ
′′
(
ω
)
)
h
ω
(
n
+
1
)
(
φ
(
ω
)
)
≥
μ
(
ω
)
+
u
(
ω
)
(
φ
′
(
ω
)
)
2
h
ω
(
n
+
2
)
(
φ
(
ω
)
)
|
=
2
·
(
n
+
2
)
!
μ
(
ω
)
|
u
(
ω
)
(
φ
′
(
ω
)
)
2
|
|
φ
(
ω
)
¯
|
n
+
2
(
1
-
|
φ
(
ω
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
.
By (45), we obtain that
(46)
sup
⁡
1
/
2
<
|
φ
(
ω
)
|
<
1
μ
(
ω
)
|
u
(
ω
)
(
φ
′
(
ω
)
)
2
|
(
1
-
|
φ
(
ω
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
2
n
+
2
sup
⁡
1
/
2
<
|
φ
(
ω
)
|
<
1
2
·
(
n
+
2
)
!
μ
(
ω
)
|
u
(
ω
)
(
φ
′
(
ω
)
)
2
|
|
φ
(
ω
)
¯
|
n
+
2
(
1
-
|
φ
(
ω
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
2
n
+
2
sup
⁡
ω
∈
𝔻
2
·
(
n
+
2
)
!
μ
(
ω
)
|
u
(
ω
)
(
φ
′
(
ω
)
)
2
|
|
φ
(
ω
)
¯
|
n
+
2
(
1
-
|
φ
(
ω
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
C
2
n
+
2
<
∞
.
By (27), we have
(47)
sup
⁡
|
φ
(
ω
)
|
≤
1
/
2
μ
(
ω
)
|
u
(
ω
)
(
φ
′
(
ω
)
)
2
|
(
1
-
|
φ
(
ω
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
ω
)
|
)
)
≤
sup
⁡
|
φ
(
ω
)
|
≤
1
/
2
μ
(
ω
)
|
u
(
ω
)
(
φ
′
(
ω
)
)
2
|
(
1
-
|
φ
(
ω
)
|
2
)
n
+
2
log
⁡
2
≤
(
4
3
)
n
+
2
1
log
⁡
2
sup
⁡
|
φ
(
ω
)
|
≤
1
/
2
μ
(
ω
)
|
u
(
ω
)
(
φ
′
(
ω
)
)
2
|
≤
(
4
3
)
n
+
2
K
3
log
⁡
2
<
∞
.
Thus combining (47) with (46) we get the condition (19), finishing the proof of the theorem.Theorem 6.
Letu
∈
H
(
𝔻
), and let n be a nonnegative integer, φ a holomorphic self-map of 𝔻, and μ a weight. Then the following statements are equivalent:(1)
𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ is compact;
(2)
𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ is compact;
(3)
𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ is bounded and
(48)
lim
⁡
|
φ
(
z
)
|
→
1
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
=
0
,
(49)
lim
⁡
|
φ
(
z
)
|
→
1
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
=
0
,
(50)
lim
⁡
|
φ
(
z
)
|
→
1
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
=
0
.Proof.
(
3
)
⇒
(
1
). Assume that 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ is bounded and that conditions (48), (49), and (50) hold. For any bounded sequence {
f
k
} in ℬ
log
⁡ which converges to zero uniformly on compact subsets of 𝔻. To establish the assertion, it suffices, in view of Lemma 3, to show that
(51)
∥
𝒟
φ
,
u
n
f
k
∥
𝒵
μ
⟶
0
,
as
k
⟶
∞
.
We assume that ∥
f
k
∥
ℬ
log
⁡
≤
1. From (48), (49), and (50) we have that, for any ɛ
>
0, there exists ρ
∈
(
0,1
); when ρ
<
|
φ
(
z
)
|
<
1, we have
(52)
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
ɛ
,
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
ɛ
,
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
ɛ
.
From the boundedness of 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ by Theorem 5, we see that (23), (25), and (27) hold. Since f
k
→
0 uniformly on compact subsets of 𝔻, Cauchy's estimate gives that f
k
(
n
), f
k
(
n
+
1
), and f
k
(
n
+
2
) converge to 0 uniformly on compact subsets of 𝔻; there exists a K
0
∈
ℕ such that k
>
K
0 implies that
(53)
|
(
𝒟
φ
,
u
n
f
k
)
(
0
)
|
+
|
(
𝒟
φ
,
u
n
f
k
)
′
(
0
)
|
+
sup
⁡
|
φ
(
z
)
|
≤
ρ
μ
(
z
)
|
(
𝒟
φ
,
u
n
f
k
)
′′
(
z
)
|
≤
|
u
(
0
)
|
|
f
k
(
n
)
(
φ
(
0
)
)
|
+
|
u
′
(
0
)
|
|
f
k
(
n
)
(
φ
(
0
)
)
|
+
|
u
(
0
)
|
|
f
k
(
n
+
1
)
(
φ
(
0
)
)
|
|
φ
′
(
0
)
|
+
sup
⁡
|
φ
(
z
)
|
≤
ρ
μ
(
z
)
|
u
′′
(
z
)
|
|
f
k
(
n
)
(
φ
(
z
)
)
|
+
sup
⁡
|
φ
(
z
)
|
≤
ρ
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
|
f
k
(
n
+
1
)
(
φ
(
z
)
)
|
+
sup
⁡
|
φ
(
z
)
|
≤
ρ
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
|
f
k
(
n
+
2
)
(
φ
(
z
)
)
|
≤
|
u
(
0
)
|
|
f
k
(
n
)
(
φ
(
0
)
)
|
+
|
u
′
(
0
)
|
|
f
k
(
n
)
(
φ
(
0
)
)
|
+
|
u
(
0
)
|
|
φ
′
(
0
)
|
|
f
k
(
n
+
1
)
(
φ
(
0
)
)
|
+
K
1
sup
⁡
|
φ
(
z
)
|
≤
ρ
|
f
k
(
n
)
(
φ
(
z
)
)
|
+
K
2
sup
⁡
|
φ
(
z
)
|
≤
ρ
|
f
k
(
n
+
1
)
(
φ
(
z
)
)
|
+
K
3
sup
⁡
|
φ
(
z
)
|
≤
ρ
|
f
k
(
n
+
2
)
(
φ
(
z
)
)
|
<
C
ɛ
.
From (52) and (53) and Lemma 1 we have
(54)
∥
𝒟
φ
,
u
n
f
k
∥
𝒵
μ
=
|
(
𝒟
φ
,
u
n
f
k
)
(
0
)
|
+
|
(
𝒟
φ
,
u
n
f
k
)
′
(
0
)
|
+
sup
⁡
z
∈
𝔻
μ
(
z
)
|
(
𝒟
φ
,
u
n
f
k
)
′′
(
z
)
|


≤
(
+
sup
⁡
|
φ
(
z
)
|
≤
ρ
|
(
𝒟
φ
,
u
n
f
k
)
(
0
)
|
+
|
(
𝒟
φ
,
u
n
f
k
)
′
(
0
)
|
+
sup
⁡
|
φ
(
z
)
|
≤
ρ
μ
(
z
)
|
(
𝒟
φ
,
u
n
f
k
)
′′
(
z
)
|
)
+
sup
⁡
ρ
<
|
φ
(
z
)
|
<
1
μ
(
z
)
|
(
𝒟
φ
,
u
n
f
k
)
′′
(
z
)
|
<
C
ɛ
+
sup
⁡
ρ
<
|
φ
(
z
)
|
<
1
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
C
∥
f
∥
ℬ
log
⁡


+
sup
⁡
ρ
<
|
φ
(
z
)
|
<
1
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
C
∥
f
∥
ℬ
log
⁡
+
sup
⁡
ρ
<
|
φ
(
z
)
|
<
1
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
C
∥
f
∥
ℬ
log
⁡
<
4
C
ɛ
,
when K
>
K
0. It follows that the operator 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ is compact.
(
1
) ⇒ (
2
). It is obvious.
(
2
) ⇒ (
3
). Assume that 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ is compact. Then it is clear that 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ is bounded. By Theorem 5 we get that 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ is bounded. Let {
z
k
} be a sequence in 𝔻 such that |
φ
(
z
k
)
|
→
1 as k
→
∞. We can use the test functions
(55)
f
k
(
z
)
=
f
z
k
(
z
)
=
(
n
+
2
)
(
n
+
3
)
1
-
|
φ
(
z
k
)
|
2
(
1
-
z
φ
(
z
k
)
¯
)
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
-
2
(
n
+
3
)
(
1
-
|
φ
(
z
k
)
|
2
)
2
(
1
-
z
φ
(
z
k
)
¯
)
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
+
2
(
1
-
|
φ
(
z
k
)
|
2
)
3
(
1
-
z
φ
(
z
k
)
¯
)
3
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
.
Note that
(56)
|
f
k
(
z
)
|
≤
|
(
n
+
2
)
(
n
+
3
)
(
1
-
|
φ
(
z
k
)
|
2
)
(
1
-
z
φ
(
z
k
)
¯
)
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
|
+
|
2
(
n
+
3
)
(
1
-
|
φ
(
z
k
)
|
2
)
2
(
1
-
z
φ
(
z
k
)
¯
)
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
|
+
|
2
(
1
-
|
φ
(
z
k
)
|
2
)
3
(
1
-
z
φ
(
z
k
)
¯
)
3
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
|
≤
(
n
+
2
)
(
n
+
3
)
(
1
+
|
φ
(
z
k
)
|
)
(
1
-
|
φ
(
z
k
)
|
)
(
1
-
|
φ
(
z
k
)
|
)
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
+
2
(
n
+
3
)
(
1
+
|
φ
(
z
k
)
|
)
2
(
1
-
|
φ
(
z
k
)
|
)
2
(
1
-
|
φ
(
z
k
)
|
)
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
+
2
(
1
+
|
φ
(
z
k
)
|
)
3
(
1
-
|
φ
(
z
k
)
|
)
3
(
1
-
|
φ
(
z
k
)
|
)
3
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
≤
2
(
n
+
2
)
(
n
+
3
)
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
+
8
(
n
+
3
)
log
⁡
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
+
16
log
⁡
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
=
2
n
2
+
18
n
+
52
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
⟶
0
,
(
k
⟶
∞
)
,
for |
z
|
<
1. We see that f
k converges to 0 uniformly on 𝔻; hence, f
k converges to 0 uniformly on compact subsets of 𝔻 and from (30) and (33) we have sup
⁡
k
∈
ℕ
⁡
∥
f
k
∥
ℬ
log
⁡
≤
C, f
k
∈
ℬ
log
⁡
,
0. Then f
k is a bounded sequence in ℬ
log
⁡
,
0 which converges to 0 uniformly on compact subsets of 𝔻. By Lemma 3, we have
(57)
lim
⁡
k
→
∞
∥
𝒟
φ
,
u
n
f
k
∥
𝒵
μ
=
0
.
Note that
(58)
f
k
(
n
+
1
)
(
φ
(
z
k
)
)
=
f
k
(
n
+
2
)
(
φ
(
z
k
)
)
=
0
,
f
k
(
n
)
(
φ
(
z
k
)
)
=
2
·
n
!
(
φ
(
z
k
)
¯
)
n
(
1
-
|
φ
(
z
k
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
.
From (33) and using the compactness of 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ we obtain
(59)
2
·
n
!
μ
(
z
k
)
|
u
′′
(
z
k
)
|
|
φ
(
z
k
)
¯
|
n
(
1
-
|
φ
(
z
k
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
≤
∥
𝒟
φ
,
u
n
f
k
∥
𝒵
μ
⟶
0
,
as
k
⟶
∞
.
From (59) and |
φ
(
z
k
)
|
→
1, it follows that
(60)
lim
⁡
k
→
∞
μ
(
z
k
)
|
u
′′
(
z
k
)
|
(
1
-
|
φ
(
z
k
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
=
0
and consequently (48) holds.
Next, let(61)
g
k
(
z
)
=
g
z
k
(
z
)
=
(
n
+
1
)
(
n
+
3
)
1
-
|
φ
(
z
k
)
|
2
(
1
-
z
φ
(
z
k
)
¯
)
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
-
(
2
n
+
5
)
(
1
-
|
φ
(
z
k
)
|
2
)
2
(
1
-
z
φ
(
z
k
)
¯
)
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
+
2
(
1
-
|
φ
(
z
k
)
|
2
)
3
(
1
-
z
φ
(
z
k
)
¯
)
3
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
.
By a direct calculation, we obtain that g
k converges to 0 uniformly on compact subsets of 𝔻, g
k
∈
ℬ
log
⁡
,
0, and sup
⁡
k
∈
ℕ
⁡
∥
g
k
∥
ℬ
log
⁡
≤
C. By Lemma 3, we have
(62)
lim
⁡
k
→
∞
∥
𝒟
φ
,
u
n
g
k
∥
𝒵
μ
=
0
.
Note that
(63)
g
k
(
n
+
1
)
(
φ
(
z
k
)
)
=
-
(
n
+
1
)
!
(
φ
(
z
k
)
¯
)
n
+
1
(
1
-
|
φ
(
z
k
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
,
g
k
(
n
)
(
φ
(
z
k
)
)
=
g
ω
(
n
+
2
)
(
φ
(
z
k
)
)
=
0
.
From (39) and using the compactness of 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ we obtain
(64)
(
n
+
1
)
!
μ
(
z
k
)
|
2
u
′
(
z
k
)
φ
′
(
z
k
)
+
u
(
z
k
)
φ
′′
(
z
k
)
|
|
φ
(
z
k
)
¯
|
n
+
1
(
1
-
|
φ
(
z
k
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
≤
∥
𝒟
φ
,
u
n
g
k
∥
𝒵
μ
⟶
0
,
as
k
⟶
∞
.
From (64) and |
φ
(
z
k
)
|
→
1, we have
(65)
lim
⁡
k
→
∞
μ
(
z
k
)
|
2
u
′
(
z
k
)
φ
′
(
z
k
)
+
u
(
z
k
)
φ
′′
(
z
k
)
|
(
1
-
|
φ
(
z
k
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
=
0
;
it implies that (49) holds.
In order to prove (50), choose
(66)
h
k
(
z
)
=
h
z
k
(
z
)
=
(
n
+
1
)
(
n
+
2
)
1
-
|
φ
(
z
k
)
|
2
(
1
-
z
φ
(
z
k
)
¯
)
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
-
2
(
n
+
2
)
(
1
-
|
φ
(
z
k
)
|
2
)
2
(
1
-
z
φ
(
z
k
)
¯
)
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
+
2
(
1
-
|
φ
(
z
k
)
|
2
)
3
(
1
-
z
φ
(
z
k
)
¯
)
3
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
.
By a direct calculation, we may easily prove that h
k converges to 0 uniformly on compact subsets of 𝔻, h
k
∈
ℬ
log
⁡
,
0, and sup
⁡
k
∈
ℕ
⁡
∥
h
k
∥
ℬ
log
⁡
≤
C. By Lemma 3, we have
(67)
lim
⁡
k
→
∞
∥
𝒟
φ
,
u
n
h
k
∥
𝒵
μ
=
0
.
Note that
(68)
h
k
(
n
+
2
)
(
φ
(
z
k
)
)
=
2
·
(
n
+
2
)
!
(
φ
(
z
k
)
¯
)
n
+
2
(
1
-
|
φ
(
z
k
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
,
h
k
(
n
)
(
φ
(
z
k
)
)
=
h
k
(
n
+
1
)
(
φ
(
z
k
)
)
=
0
.
From (45) and using the compactness of 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ we obtain
(69)
2
·
(
n
+
2
)
!
μ
(
z
k
)
|
u
(
z
k
)
(
φ
′
(
z
k
)
)
2
|
|
φ
(
z
k
)
¯
|
n
+
2
(
1
-
|
φ
(
z
k
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
≤
∥
𝒟
φ
,
u
n
h
k
∥
𝒵
μ
⟶
0
,
as
k
⟶
∞
.
From (69) and |
φ
(
z
k
)
|
→
1, it follows that
(70)
lim
⁡
k
→
∞
μ
(
z
k
)
|
u
(
z
k
)
(
φ
′
(
z
k
)
)
2
|
(
1
-
|
φ
(
z
k
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
k
)
|
)
)
=
0
,
and consequently (50) holds, finishing the proof of the theorem.Theorem 7.
Letu
∈
H
(
𝔻
), and let n be a nonnegative integer, φ a holomorphic self-map of 𝔻, and μ a weight. Then 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ
,
0 is a bounded operator provided that the following conditions are satisfied:
(71)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
=
0
,
(72)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
=
0
,
(73)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
=
0
.Proof.
Suppose that (71), (72), and (73) hold. It is clear that (17), (18), and (19) hold. By Theorem 5 we have that 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ is bounded. In order to prove 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ
,
0 is bounded, it is enough to show that, for any f
∈
ℬ
log
⁡, 𝒟
φ
,
u
n
f
∈
𝒵
μ
,
0. Using (71), (72), and (73) we have that, for any ɛ
>
0, there is a constant 0
<
η
<
1, such that η
<
|
z
|
<
1 implies
(74)
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
ɛ
,
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
ɛ
,
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
ɛ
.
Then, for any f
∈
ℬ
log
⁡, from Lemma 1 we obtain that
(75)
μ
(
z
)
|
(
𝒟
φ
,
u
n
f
)
′′
(
z
)
|
=
μ
(
z
)
|
(
φ
′
(
z
)
)
2
u
′′
(
z
)
f
(
n
)
(
φ
(
z
)
)
=
μ
(
z
)
+
(
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
)
f
(
n
+
1
)
(
φ
(
z
)
)
=
μ
(
z
)
+
u
(
z
)
(
φ
′
(
z
)
)
2
f
(
n
+
2
)
(
φ
(
z
)
)
|
≤
μ
(
z
)
|
u
′′
(
z
)
|
|
f
(
n
)
(
φ
(
z
)
)
|
+
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
|
f
(
n
+
1
)
(
φ
(
z
)
)
|
+
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
|
f
(
n
+
2
)
(
φ
(
z
)
)
|
≤
C
∥
f
∥
ℬ
log
⁡
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
+
C
∥
f
∥
ℬ
log
⁡
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
1
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
+
C
∥
f
∥
ℬ
log
⁡
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
(
1
-
|
φ
(
z
)
|
2
)
n
+
2
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
3
C
∥
f
∥
ℬ
log
⁡
ɛ
,
when η
<
|
z
|
<
1. Hence 𝒟
φ
,
u
n
f
∈
𝒵
μ
,
0 for all f
∈
ℬ
log
⁡, completing the proof of the theorem.Theorem 8.
Letu
∈
H
(
𝔻
), and let n be a nonnegative integer, φ a holomorphic self-map of 𝔻, and μ a weight. If 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ
,
0 is a bounded operator, then (17), (18), and (19) hold and the following conditions are satisfied:
(76)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
′′
(
z
)
|
=
0
,
(77)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
=
0
,
(78)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
=
0
.Proof.
Assume that𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ
,
0 is bounded; it is clear that 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ is bounded. By Theorem 5 we have that (17), (18), and (19) hold. On the other hand, for all f
∈
ℬ
log
⁡, 𝒟
φ
,
u
n
f
∈
𝒵
μ
,
0. Take f
(
z
)
=
z
n
/
n
!
∈
ℬ
log
⁡; we have that
(79)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
′′
(
z
)
|
=
0
;
then (76) holds. Let f
(
z
)
=
z
n
+
1
/
(
n
+
1
)
!
∈
ℬ
log
⁡; we have that
(80)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
′′
(
z
)
φ
(
z
)
+
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
=
0
.
By (80), (76), and the boundedness of the function φ
(
z
), we get
(81)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
=
0
.
Hence, (77) holds. In the same way, let f
(
z
)
=
z
n
+
2
/
(
n
+
2
)
!
∈
ℬ
log
⁡; we have that
(82)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
′′
(
z
)
(
φ
(
z
)
)
2
+
2
(
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
)
φ
(
z
)
+
2
u
(
z
)
(
φ
′
(
z
)
)
2
|
=
0
.
By (76), (77), (82), and the boundedness of the function φ
(
z
), we have that
(83)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
=
0
.
That is, (78) holds. The proof is completed.Theorem 9.
Letu
∈
H
(
𝔻
), and let n be a nonnegative integer, φ a holomorphic self-map of 𝔻, and μ a weight. Then 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ
,
0 is a bounded operator if and only if 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ is a bounded operator and (76), (77), and (78) hold.Proof.
Assume that𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ
,
0 is a bounded operator; it is clear that 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ is a bounded operator. On the other hand, for all f
∈
ℬ
log
⁡
,
0, 𝒟
φ
,
u
n
f
∈
𝒵
μ
,
0. Taking f
(
z
)
=
z
n
/
n
!
∈
ℬ
log
⁡
,
0, we have that
(84)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
′′
(
z
)
|
=
0
;
then (76) holds. Let f
(
z
)
=
z
n
+
1
/
(
n
+
1
)
!
∈
ℬ
log
⁡
,
0; we have that
(85)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
′′
(
z
)
φ
(
z
)
+
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
=
0
.
By (85), (76), and the boundedness of the function φ
(
z
), we get
(86)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
=
0
.
Hence, (77) holds. In the same way, take f
(
z
)
=
z
n
+
2
/
(
n
+
2
)
!
∈
ℬ
log
⁡
,
0; we have that
(87)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
′′
(
z
)
(
φ
(
z
)
)
2
+
2
(
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
)
φ
(
z
)
+
2
u
(
z
)
(
φ
′
(
z
)
)
2
|
=
0
.
By (76), (77), (87), and the boundedness of the function φ
(
z
), we have that
(88)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
=
0
.
That is, (78) holds.
Conversely, suppose that𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ is a bounded operator and (76), (77), and (78) hold. For each polynomial p
(
z
) we get that
(89)
μ
(
z
)
|
(
𝒟
φ
,
u
n
p
)
′′
(
z
)
|
=
μ
(
z
)
|
(
φ
′
(
z
)
)
2
u
′′
(
z
)
p
(
n
)
(
φ
(
z
)
)
=
μ
(
z
)
+
(
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
)
p
(
n
+
1
)
(
φ
(
z
)
)
=
μ
(
z
)


+
u
(
z
)
(
φ
′
(
z
)
)
2
p
(
n
+
2
)
(
φ
(
z
)
)
|
≤
μ
(
z
)
|
u
′′
(
z
)
|
|
p
(
n
)
(
φ
(
z
)
)
|
+
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
|
p
(
n
+
1
)
(
φ
(
z
)
)
|
+
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
|
p
(
n
+
2
)
(
φ
(
z
)
)
|
≤
μ
(
z
)
|
u
′′
(
z
)
|
∥
p
(
n
)
∥
∞
+
μ
(
z
)
|
2
u
′
(
z
)
φ
′
(
z
)
+
u
(
z
)
φ
′′
(
z
)
|
∥
p
(
n
+
1
)
∥
∞
+
μ
(
z
)
|
u
(
z
)
(
φ
′
(
z
)
)
2
|
∥
p
(
n
+
2
)
∥
∞
⟶
0
SSSSSSS
(
as
|
z
|
⟶
1
)
.
Hence, 𝒟
φ
,
u
n
p
∈
𝒵
μ
,
0. On the other hand, since polynomials are dense in ℬ
log
⁡
,
0, thus, for each f
∈
ℬ
log
⁡
,
0, there is a sequence of polynomials {
p
k
}
k
∈
ℕ such that
(90)
lim
⁡
k
→
∞
∥
p
k
-
f
∥
ℬ
log
⁡
=
0
.
Since 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ is a bounded operator, by Theorem 5 we have 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ is a bounded operator. Since
(91)
∥
𝒟
φ
,
u
n
p
k
-
𝒟
φ
,
u
n
f
∥
𝒵
μ
≤
∥
𝒟
φ
,
u
n
∥
∥
p
k
-
f
∥
ℬ
log
⁡
and 𝒵
μ
,
0 is the closed subset of 𝒵
μ, we see that 𝒟
φ
,
u
n
f
∈
𝒵
μ
,
0; thus 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ
,
0 is a bounded operator. The proof is completed.Theorem 10.
Letu
∈
H
(
𝔻
), and let n be a nonnegative integer, φ a holomorphic self-map of 𝔻, and μ a weight. Then the following statements are equivalent:(1)
𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ
,
0 is compact;
(2)
𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ
,
0 is compact;
(3)
(71), (72), and (73) hold.Proof.
(
3
) ⇒ (
1
). Suppose that (71) (72), and (73) hold. By Theorem 7 we know that 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ
,
0 is bounded. Taking the supremum in inequality (20) over all f
∈
ℬ
log
⁡ such that ∥
f
∥
ℬ
log
⁡
≤
1 and letting |
z
|
→
1 yields
(92)
lim
⁡
|
z
|
→
1
sup
⁡
∥
f
∥
ℬ
log
⁡
≤
1
μ
(
z
)
|
(
𝒟
φ
,
u
n
f
)
′
′
(
z
)
|
=
0
.
Hence, by Lemma 4, we see that the operator 𝒟
φ
,
u
n
:
ℬ
log
⁡
→
𝒵
μ
,
0 is compact.
(
1
) ⇒ (
2
). This implication is clear.
(
2
) ⇒ (
3
). Assume that 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ
,
0 is compact. Firstly, it is obvious 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ
,
0 is bounded. By Theorem 9 we have that (76), (77), and (78) hold. On the other hand, we have that 𝒟
φ
,
u
n
:
ℬ
log
⁡
,
0
→
𝒵
μ is compact. By Theorem 6 we have that (48), (49), and (50) hold. We prove that (76) and (48) imply (71). The proof of (72) and (73) is similar; hence, it will be omitted. From (48), it follows that, for every ɛ
>
0, there exists δ
∈
(
0,1
) such that
(93)
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
ɛ
,
when δ
<
|
φ
(
z
)
|
<
1. Using (76) we see that there exists τ
∈
(
0,1
) such that
(94)
μ
(
z
)
|
u
′′
(
z
)
|
≤
ɛ
inf
⁡
t
∈
[
0
,
δ
]
(
1
-
t
2
)
n
log
⁡
2
,
when τ
<
|
z
|
<
1. Therefore when τ
<
|
z
|
<
1 and δ
<
|
φ
(
z
)
|
<
1, by (93), we have
(95)
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
<
ɛ
.
On the other hand, when τ
<
|
z
|
<
1 and |
φ
(
z
)
|
≤
δ, by (94), we obtain
(96)
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
≤
μ
(
z
)
|
u
′′
(
z
)
|
inf
⁡
t
∈
[
0
,
δ
]
⁡
(
1
-
t
2
)
n
log
⁡
2
<
ɛ
.
From (95) and (96) we have
(97)
lim
⁡
|
z
|
→
1
μ
(
z
)
|
u
′′
(
z
)
|
(
1
-
|
φ
(
z
)
|
2
)
n
log
⁡
(
2
/
(
1
-
|
φ
(
z
)
|
)
)
=
0
;
we obtain that (71) holds, as desired. The proof is completed.



---

*Source: 832713-2014-04-17.xml*