# Norm and Essential Norm of Composition Followed by Differentiation from Logarithmic Bloch Spaces toH

Authors: Shanli Ye
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

In this note we express the norm of composition followed by differentiationD
C
φ from the logarithmic Bloch and the little logarithmic Bloch spaces to the weighted space H
μ
∞ on the unit disk and give an upper and a lower bound for the essential norm of this operator from the logarithmic Bloch space to H
μ
∞.


---

## Body

## 1. Introduction

Let𝔻
=
{
z
:
|
z
|
<
1
} be the open unit disk in the complex plane ℂ, H
(
𝔻
) be the space of all analytic functions on 𝔻, and H
∞ be the space of bounded analytic functions on 𝔻 with the norm ∥
f
∥
∞
=
sup
⁡
z
∈
𝔻
|
f
(
z
)
|.An analytic functionf
∈
H
(
𝔻
) is said to belong to the logarithmic Bloch space ℒ
ℬ if
(1)
∥
f
∥
ℒ
ℬ
=
sup
⁡
{
(
1
-
|
z
|
)
ln
⁡
(
2
e
1
-
|
z
|
)
|
f
′
(
z
)
|
:
z
∈
𝔻
}
<
∞
and to the little logarithmic Bloch space ℒ
ℬ
0 if
(2)
lim
⁡
|
z
|
→
1
-
(
1
-
|
z
|
)
ln
⁡
(
2
e
1
-
|
z
|
)
|
f
′
(
z
)
|
=
0
.
It can be easily proved that ℒ
ℬ is a Banach space, under the norm
(3)
∥
f
∥
ℒ
=
|
f
(
0
)
|
+
∥
f
∥
ℒ
ℬ
,
and that ℒ
ℬ
0 is a closed subspace of ℒ
ℬ. Some sources for results and references about the logarithmic Bloch functions are the papers of Yoneda [1], Stević [2], and the authors of [3–8].Letμ be a weight, that is, a positive continuous function on 𝔻. The weighted space H
μ
∞ consists of all f
∈
H
(
𝔻
) such that
(4)
∥
f
∥
H
μ
∞
=
sup
⁡
z
∈
𝔻
μ
(
z
)
|
f
(
z
)
|
<
∞
,
where μ is a weight.Letφ be a holomorphic self-map of 𝔻. The composition operator C
φ is defined by
(5)
C
φ
(
f
)
(
z
)
=
f
(
φ
(
z
)
)
,
f
∈
H
(
𝔻
)
.
Let D be the differentiation operator. The product D
C
φ is defined by
(6)
D
C
φ
(
f
)
=
(
f
∘
φ
)
′
=
f
′
(
φ
)
φ
′
,
f
∈
H
(
𝔻
)
.
The operator D
C
φ is probably studied for the first time by Hibschweiler and Portnoy in [9], where the boundedness and compactness of D
C
φ between Bergman and Hardy spaces are investigated. In [10], Stević calculated the norm of the operator D
C
φ from the classical Bloch space to H
μ
∞. Recently there has been some interest in calculating operator norms and essential norms of composition and related operators (see, e.g., [11–18] and the references therein). Motivated by the papers [10, 19], we continue here this line of research by calculating ∥
D
C
φ
∥
ℒ
ℬ
→
H
μ
∞.Suppose thatX
1 and X
2 are Banach spaces and L
:
X
1
→
X
2 is a bounded linear operator. The essential norm ∥
L
∥
e
,
X
1
→
X
2 of L is its distance to the compact operators. More precisely,
(7)
∥
L
∥
e
,
X
1
→
X
2
=
inf
⁡
{
∥
L
-
K
∥
X
1
→
X
2
:
0000
K
a
compact
operator
of
X
1
into
X
2
}
,
where ∥
·
∥
X
1
→
X
2 denotes the operator norm. If X
1
=
X
2, it is simply denoted by ∥
·
∥
e. Since the set of all compact operators is a closed subset of the set of bounded operators, it follows that an operator L is compact if and only if ∥
L
∥
e
,
X
1
→
X
2
=
0.Essential norm formulas for composition operators are known in various settings. WhenC
φ acts from the Hardy space H
2
(
𝔻
) to itself, Shapiro [20] gives a formula for ∥
C
φ
∥
e in terms of the Nevanlinna counting function for φ. In [21], Donaway gives upper and lower estimates for ∥
C
φ
∥
e when C
φ maps the Bloch, Dirichlet, or a Besov type space to itself. The essential norm of the D
C
φ operator from α-Bloch spaces to H
μ
∞ space was estimated recently by Stević in [10]. In this note we give upper and lower estimates for ∥
D
C
φ
∥
e
,
ℒ
ℬ
→
H
μ
∞.


## 2. The Operator Norm ofD
C
φ
:
ℒ
ℬ
(or ℒ
ℬ
0
)
→
H
μ
∞

In this section we prove a nice formula. Namely, we calculate the norm of the operatorD
C
φ
:
ℒ
ℬ
(
or
ℒ
ℬ
0
)
→
H
μ
∞.Theorem 1.
Assumeμ is a weight on 𝔻. Then the following are equivalent:(a)
D
C
φ
:
ℒ
ℬ
→
H
μ
∞ is a bounded operator;
(b)
D
C
φ
:
ℒ
ℬ
0
→
H
μ
∞ is a bounded operator;
(c)
sup
⁡
z
∈
𝔻
(
μ
(
z
)
|
φ
′
(
z
)
|
)
/
(
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
)
<
∞
Moreover, one has(8)
∥
D
C
φ
∥
ℒ
ℬ
0
→
H
μ
∞
=
∥
D
C
φ
∥
ℒ
ℬ
→
H
μ
∞
=
sup
⁡
z
∈
𝔻
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
.Proof.
(
a
)
⇒
(
b
). By the fact ℒ
ℬ
0
⊂
ℒ
ℬ and the definition of operator norm, we easily obtain that D
C
φ
:
ℒ
ℬ
0
→
H
μ
∞ is a bounded operator and
(9)
∥
D
C
φ
∥
ℒ
ℬ
0
→
H
μ
∞
≤
∥
D
C
φ
∥
ℒ
ℬ
→
H
μ
∞
.
(
b
)
⇒
(
c
). Suppose that D
C
φ is a bounded operator from ℒ
ℬ
0 to H
μ
∞. Taking the test function f
(
z
)
=
z
∈
ℒ
ℬ
0, we easily have
(10)
μ
(
w
)
|
φ
′
(
w
)
|
≤
∥
φ
′
∥
H
μ
∞
=
∥
D
C
φ
(
z
)
∥
H
μ
∞
≤
∥
D
C
φ
∥
ℒ
ℬ
0
→
H
μ
∞
∥
z
∥
ℒ
=
∥
D
C
φ
∥
ℒ
ℬ
0
→
H
μ
∞
ln
⁡
2
e
,
for every w
∈
𝔻. It implies that (c) holds when φ
(
z
)
=
0.
Fixingw
∈
𝔻
∖
{
0
}, we consider the function
(11)
f
w
(
z
)
=
1
w
¯
ln
⁡
ln
⁡
(
2
e
1
-
w
¯
z
)
-
1
w
¯
ln
⁡
ln
⁡
2
e
.
Sincer
(
x
)
=
x
ln
⁡
(
2
e
/
x
) is increasing on (
0,2
] and f
w
(
0
)
=
0, we have
(12)
∥
f
w
∥
ℒ
=
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
ln
⁡
(
2
e
1
-
|
z
|
)
×
1
|
ln
⁡
(
2
e
/
(
1
-
w
¯
z
)
)
|
1
|
1
-
w
¯
z
|
≤
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
z
|
)
)
(
1
-
|
w
¯
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
w
¯
z
|
)
)
×
(
1
-
|
w
¯
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
w
¯
z
|
)
)
|
1
-
w
¯
z
|
ln
⁡
(
2
e
/
(
1
-
w
¯
z
)
)
≤
1
.
Moreover, since
(13)
(
1
-
|
z
|
)
ln
⁡
2
e
1
-
|
z
|
|
f
w
′
(
z
)
|
≤
(
1
-
|
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
z
|
)
)
(
1
-
|
w
¯
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
w
¯
z
|
)
)
≤
(
1
-
|
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
z
|
)
)
(
1
-
|
w
|
)
ln
⁡
2
e
⟶
0
,
as |
z
|
→
1
-, it follows that f
w
∈
ℒ
ℬ
0 for every w
∈
𝔻
∖
{
0
}. Thus, for each t
∈
(
0,1
) we obtain that
(14)
∥
D
C
φ
∥
ℒ
ℬ
0
→
H
μ
∞
≥
∥
D
C
φ
(
f
t
(
φ
(
w
)
/
|
φ
(
w
)
|
)
)
∥
H
μ
∞
=
sup
⁡
z
∈
𝔻
μ
(
z
)
|
φ
′
(
z
)
f
t
(
φ
(
w
)
/
|
φ
(
w
)
|
)
′
(
φ
(
z
)
)
|
≥
μ
(
w
)
|
φ
′
(
w
)
|
(
1
-
t
|
φ
(
w
)
|
)
ln
⁡
(
2
e
/
(
1
-
t
|
φ
(
w
)
|
)
)
,
for every φ
(
w
)
≠
0. Letting t
→
1
-, we obtain that
(15)
∥
D
C
φ
∥
ℒ
ℬ
0
→
H
μ
∞
≥
μ
(
w
)
|
φ
′
(
w
)
|
(
1
-
|
φ
(
w
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
w
)
|
)
)
,
for every φ
(
w
)
≠
0. It implies that (c) also holds when φ
(
z
)
≠
0.
(
c
)
⇒
(
a
). For every f
∈
ℒ
ℬ, we easily obtain that
(16)
∥
D
C
φ
f
∥
H
μ
∞
≤
sup
⁡
z
∈
D
μ
(
z
)
|
(
D
C
φ
f
)
(
z
)
|
=
sup
⁡
z
∈
D
μ
(
z
)
|
φ
′
(
z
)
f
′
(
φ
(
z
)
)
|
≤
sup
⁡
z
∈
D
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
∥
f
∥
ℒ
.
Hence D
C
φ
:
ℒ
ℬ
→
H
μ
∞ is a bounded operator. Also, we obtain
(17)
∥
D
C
φ
∥
ℒ
ℬ
→
H
μ
∞
≤
sup
⁡
z
∈
𝔻
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
.
Moreover, from (9), (10), (15), and (17), we obtain
(18)
∥
D
C
φ
∥
ℒ
ℬ
0
→
H
μ
∞
=
∥
D
C
φ
∥
ℒ
ℬ
→
H
μ
∞
=
sup
⁡
z
∈
𝔻
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
.


## 3. Estimates of Essential Norm ofD
C
φ
:
ℒ
ℬ
(or ℒ
ℬ
0
)
→
H
μ
∞

In this section we will estimate the essential norm ofD
C
φ
:
ℒ
ℬ
(
or
ℒ
ℬ
0
)
→
H
μ
∞. For this purpose we need some lemmas.Lemma 2.
Iff
∈
ℒ
ℬ, then |
f
(
z
)
|
≤
(
1
/
2
+
ln
⁡
ln
⁡
(
e
/
(
1
-
|
z
|
)
)
)
∥
f
∥
ℒ.This can be done in exactly the same way as in the proof of [3, Lemma 2.1].Lemma 3.
Letφ be an analytic self-map of 𝔻 and μ be a weight on 𝔻. Assume that D
C
φ is a bounded operator from ℒ
ℬ
(
o
r
ℒ
ℬ
0
) to H
μ
∞; then D
C
φ is compact if and only if for any bounded sequence {
f
n
} in ℒ
ℬ
(
o
r
ℒ
ℬ
0
), which converges to 0 uniformly on compact subsets of 𝔻, one has ∥
D
C
φ
(
f
n
)
∥
H
μ
∞
→
0 as n
→
∞.Proof.
Necessity. Suppose that D
C
φ
:
ℒ
ℬ
(
or
ℒ
ℬ
0
)
→
H
μ
∞ is compact. Let {
f
n
} be a bounded sequence in ℒ
ℬ
(
or
ℒ
ℬ
0
) with f
n
→
0 uniformly on compact subsets of D. Assume that there is a subsequence {
f
n
k
} and an ϵ
0
>
0 such that ∥
D
C
φ
f
n
k
∥
≥
ϵ
0 for all k
=
1,2
,
3
,
…. Since D
C
φ is compact, we can find a further subsequence {
f
n
k
j
} and a function f
∈
H
μ
∞ such that lim
⁡
j
→
∞
∥
D
C
φ
f
n
k
j
-
f
∥
H
μ
∞
=
0. Then we obtain that, for z
∈
D,
(19)
|
(
D
C
φ
f
n
k
j
-
f
)
(
z
)
|
≤
∥
D
C
φ
f
n
k
j
-
f
∥
H
μ
∞
μ
(
z
)
.
Hence D
C
φ
f
n
k
j
-
f
→
0 uniformly on compact subsets of D. Also, since f
n
k
j
→
0 uniformly on compact subsets of D, D
C
φ
f
n
k
j
→
0 uniformly on compact subsets of D. It follows that f
=
0 and hence lim
⁡
j
→
∞
∥
D
C
φ
f
n
k
j
∥
H
μ
∞
=
0, contradicting the fact that ∥
D
C
φ
f
n
k
∥
≥
ϵ
0 for all k
=
1,2
,
3
,
…. Therefore we must have that lim
⁡
n
→
∞
∥
D
C
φ
(
f
n
)
∥
H
μ
∞
=
0.
Sufficiency. Let {
f
n
} be a bounded sequence in ℒ
ℬ
(
or
ℒ
ℬ
0
). Then Lemma 2 and Montel’s Theorem tell us that {
f
n
} forms a normal family, and hence there exists a subsequence {
f
n
k
} converging uniformly on compact sets to some function f. It is easy to see that f must be in ℒ
ℬ
(
ℒ
ℬ
0
). Then {
f
n
k
-
f
} is a bounded sequence in ℒ
ℬ
(
or
ℒ
ℬ
0
) converging to 0 uniformly on compact subsets of 𝔻 and by the hypothesis guarantees that D
C
φ
f
n
k
→
D
C
φ
f in H
μ
∞. Thus D
C
φ is compact.Lemma 4.
Letμ be a weight on 𝔻 and φ be an analytic self-map of 𝔻 with ∥
φ
∥
∞
<
1. Suppose that D
C
φ
:
ℒ
ℬ
(
o
r
ℒ
ℬ
0
)
→
H
μ
∞ is bounded. Then D
C
φ
:
ℒ
ℬ
(
o
r
ℒ
ℬ
0
)
→
H
μ
∞ is compact.Proof.
Suppose that{
f
n
} is a bounded sequence in ℒ
ℬ
(
or
ℒ
ℬ
0
) which converges to 0 uniformly on compact subsets of 𝔻. By Cauchy’s inequality we easily obtain that {
f
n
′
} also converges to 0 uniformly on compact subsets of 𝔻. Since D
C
φ is bounded, one can take the test function f
(
z
)
=
z to see that φ
′
∈
H
μ
∞. Then we obtain that
(20)
∥
D
C
φ
f
n
∥
H
μ
∞
≤
∥
φ
′
∥
H
μ
∞
sup
⁡
w
∈
φ
(
𝔻
)
|
f
n
′
(
w
)
|
⟶
0
,
as n
→
∞, since φ
(
𝔻
) is contained in the disk |
w
|
≤
∥
φ
∥
∞
<
1, which is a compact subset of 𝔻. Hence, by Lemma 3, the operator D
C
φ
:
ℒ
ℬ
(
or
ℒ
ℬ
0
)
→
H
μ
∞ is compact.Lemma 5.
Letf
∈
ℒ
ℬ. Then ∥
f
t
∥
ℒ
≤
∥
f
∥
ℒ
,
<
t
<
1, where f
t
(
z
)
=
f
(
t
z
).Sincer
(
x
)
=
(
1
-
x
)
ln
⁡
(
2
e
/
(
1
-
x
)
) is decreasing on [
0,1
), one may easily prove the result.Theorem 6.
Letμ be a weight on 𝔻 and φ be an analytic self-map of 𝔻. Suppose that D
C
φ
:
ℒ
ℬ
(
or
ℒ
ℬ
0
)
→
H
μ
∞ is bounded. Then
(21)
1
2
limsup
⁡
|
φ
(
z
)
|
→
1
-
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
≤
∥
D
C
φ
∥
e
,
ℒ
ℬ
0
→
H
μ
∞
≤
∥
D
C
φ
∥
e
,
ℒ
ℬ
→
H
μ
∞
≤
2
limsup
⁡
|
φ
(
z
)
|
→
1
-
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
.Proof.
If∥
φ
∥
∞
<
1, by Lemma 4, it follows that D
C
φ
:
ℒ
ℬ
(
or
ℒ
ℬ
0
)
→
H
μ
∞ is compact which is equivalent to ∥
D
C
φ
∥
e
,
ℒ
ℬ
0
→
H
μ
∞
=
∥
D
C
φ
∥
e
,
ℒ
ℬ
→
H
μ
∞
=
0. On the other hand, it is clear that in this case the condition |
φ
(
z
)
|
→
1 is vacuous, so that it is understood that
(22)
limsup
⁡
|
φ
(
z
)
|
→
1
-
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
=
0
.
Now suppose that∥
φ
∥
∞
=
1. Assume that {
z
n
} is a sequence in 𝔻 such that |
φ
(
z
n
)
|
→
1 as n
→
∞. Let
(23)
f
n
(
z
)
=
1
2
φ
(
z
n
)
¯
a
n
(
ln
⁡
ln
⁡
2
e
1
-
φ
(
z
n
)
¯
z
)
2
-
1
2
φ
(
z
n
)
¯
a
n
(
ln
⁡
ln
⁡
2
e
)
2
,
where a
n
=
ln
⁡
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
n
)
|
2
)
). Then we have f
n
(
0
)
=
0,
(24)
f
n
′
(
φ
(
z
n
)
)
=
1
(
1
-
|
φ
(
z
n
)
|
2
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
n
)
|
2
)
)
.
Clearly f
n
(
z
)
→
0 uniformly on compact subsets of 𝔻 as n
→
∞. It follows that
(25)
∥
f
n
∥
ℒ
=
sup
⁡
z
∈
𝔻
(
1
-
|
z
|
)
ln
⁡
2
e
1
-
|
z
|
1
a
n
|
ln
⁡
ln
⁡
2
e
1
-
φ
(
z
n
)
¯
z
|
×
1
|
ln
⁡
(
2
e
/
(
1
-
φ
(
z
n
)
¯
z
)
)
|
1
|
1
-
φ
(
z
n
)
¯
z
|
≤
sup
⁡
z
∈
𝔻
2
π
+
ln
⁡
(
2
π
+
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
n
)
|
)
)
)
ln
⁡
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
n
)
|
2
)
)
×
(
1
-
|
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
z
|
)
)
(
1
-
|
φ
(
z
n
)
¯
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
n
)
¯
z
|
)
)
×
(
1
-
|
φ
(
z
n
)
¯
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
n
)
¯
z
|
)
)
|
1
-
φ
(
z
n
)
¯
z
|
ln
⁡
(
2
e
/
|
1
-
φ
(
z
n
)
¯
z
|
)
≤
2
π
+
ln
⁡
(
2
π
+
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
n
)
|
)
)
)
ln
⁡
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
n
)
|
2
)
)
.
Thus, limsup
⁡
n
→
∞
∥
f
n
∥
ℒ
≤
1. Let g
n
=
f
n
/
∥
f
n
∥
ℒ. Then ∥
g
n
∥
ℒ
=
1 and g
n
→
0 uniformly on compact subsets of 𝔻 as n
→
∞. Since g
n
∈
ℒ
ℬ
0, then it follows that g
n converges to 0 weakly in ℒ
ℬ
0. Thus, for any compact operator K
:
ℒ
ℬ
0
→
H
μ
∞, lim
⁡
n
→
∞
∥
K
g
n
∥
H
μ
∞
=
0. Therefore
(26)
∥
D
C
φ
-
K
∥
ℒ
ℬ
0
→
H
μ
∞
=
sup
⁡
∥
f
∥
ℒ
≤
1
∥
(
D
C
φ
-
K
)
f
∥
H
μ
∞
≥
limsup
⁡
n
→
∞
∥
(
D
C
φ
-
K
)
g
n
∥
H
μ
∞
≥
limsup
⁡
n
→
∞
∥
D
C
φ
g
n
∥
H
μ
∞
.
Hence
(27)
∥
D
C
φ
∥
e
,
ℒ
ℬ
0
→
H
μ
∞
≥
limsup
⁡
n
→
∞
∥
D
C
φ
g
n
∥
H
μ
∞
=
limsup
⁡
n
→
∞
sup
⁡
z
∈
𝔻
|
μ
(
z
)
g
n
′
(
φ
(
z
)
)
φ
(
z
)
|
≥
limsup
⁡
n
→
∞
1
∥
f
n
∥
ℒ
|
μ
(
z
n
)
f
n
′
(
φ
(
z
n
)
)
φ
(
z
n
)
|
≥
limsup
⁡
n
→
∞
μ
(
z
n
)
|
φ
(
z
n
)
|
(
1
-
|
φ
(
z
n
)
|
2
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
n
)
|
2
)
)
=
1
2
limsup
⁡
n
→
∞
μ
(
z
n
)
|
φ
(
z
n
)
|
(
1
-
|
φ
(
z
n
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
n
)
|
)
)
.
Thus the first inequality in (21) follows. The second inequality in (21) is obvious. Now we prove the third one.
Lets
∈
(
0,1
) be fixed and ρ
n
=
1
-
1
/
(
n
+
1
)
,
n
=
1,2
,
…. By Lemma 4 we obtain that the operator D
C
ρ
n
φ
:
ℒ
ℬ
→
H
μ
∞ is compact for every n. It follows that
(28)
∥
D
C
φ
∥
e
,
ℒ
ℬ
→
H
μ
∞
≤
∥
D
C
φ
-
D
C
ρ
n
φ
∥
ℒ
ℬ
→
H
μ
∞
=
sup
⁡
∥
f
∥
ℒ
≤
1
∥
(
D
C
φ
-
D
C
ρ
n
φ
)
(
f
)
∥
H
μ
∞
=
sup
⁡
∥
f
∥
ℒ
≤
1
sup
⁡
|
φ
(
z
)
|
≤
s
μ
(
z
)
|
φ
′
(
z
)
|
×
|
f
′
(
φ
(
z
)
)
-
ρ
n
f
′
(
ρ
n
φ
(
z
)
)
|
+
sup
⁡
∥
f
∥
ℒ
≤
1
sup
⁡
|
φ
(
z
)
|
>
s
μ
(
z
)
|
φ
′
(
z
)
|
×
|
f
′
(
φ
(
z
)
)
-
ρ
n
f
′
(
ρ
n
φ
(
z
)
)
|
≜
I
1
+
I
2
.
By Cauchy’s inequality, we obtain that
(29)
I
1
≤
sup
⁡
∥
f
∥
ℒ
≤
1
sup
⁡
|
φ
(
z
)
|
≤
s
⁡
∥
φ
′
∥
H
μ
∞
|
f
′
(
φ
(
z
)
)
-
f
′
(
ρ
n
φ
(
z
)
)
|
+
sup
⁡
∥
f
∥
ℒ
≤
1
sup
⁡
|
φ
(
z
)
|
≤
s
∥
φ
′
∥
H
μ
∞
(
1
-
ρ
n
)
|
f
′
(
ρ
n
φ
(
z
)
)
|
≤
(
1
-
ρ
n
)
∥
φ
′
∥
H
μ
∞
sup
⁡
∥
f
∥
ℒ
≤
1
sup
⁡
|
w
|
≤
s
|
f
′′
(
w
)
|
+
(
1
-
ρ
n
)
∥
φ
′
∥
H
μ
∞
sup
⁡
∥
f
∥
ℒ
≤
1
sup
⁡
|
w
|
≤
s
|
f
′
(
w
)
|
≤
(
1
-
ρ
n
)
∥
φ
′
∥
H
μ
∞
sup
⁡
∥
f
∥
ℒ
≤
1
2
1
-
s
max
⁡
|
z
|
≤
(
1
+
s
)
/
2
|
f
′
(
z
)
|
+
(
1
-
ρ
n
)
∥
φ
′
∥
H
μ
∞
sup
⁡
∥
f
∥
ℒ
≤
1
sup
⁡
|
w
|
≤
s
|
f
′
(
w
)
|
≤
(
1
-
ρ
n
)
∥
φ
′
∥
H
μ
∞
sup
⁡
∥
f
∥
ℒ
≤
1
(
1
+
2
1
-
s
)
×
max
⁡
|
z
|
≤
(
1
+
s
)
/
2
(
1
-
|
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
z
|
)
)
|
f
′
(
z
)
|
(
1
-
|
z
|
)
ln
⁡
(
2
e
/
(
1
-
|
z
|
)
)
≤
1
n
+
1
∥
φ
′
∥
H
μ
∞
(
1
+
2
1
-
s
)
2
(
1
-
s
)
ln
⁡
(
4
e
/
(
1
-
s
)
)
.
On the other hand, by Lemma 5, we obtain that
(30)
I
2
≤
sup
⁡
∥
f
∥
ℒ
≤
1
sup
⁡
|
φ
(
z
)
|
>
s
μ
(
z
)
|
φ
′
(
z
)
|
∥
f
∥
ℒ
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
+
sup
⁡
∥
f
∥
ℒ
≤
1
sup
⁡
|
φ
(
z
)
|
>
s
μ
(
z
)
|
φ
′
(
z
)
|
∥
f
ρ
n
∥
ℒ
(
1
-
ρ
n
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
ρ
n
|
φ
(
z
)
|
)
)
≤
2
sup
⁡
∥
f
∥
ℒ
≤
1
sup
⁡
|
φ
(
z
)
|
>
s
μ
(
z
)
|
φ
′
(
z
)
|
∥
f
∥
ℒ
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
≤
2
sup
⁡
|
φ
(
z
)
|
>
s
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
,
where f
ρ
n
(
z
)
=
f
(
ρ
n
z
). Hence, for for all s
∈
(
0,1
) and all n, we have
(31)
∥
D
C
φ
∥
e
,
ℒ
ℬ
→
H
μ
∞
≤
1
n
+
1
∥
φ
′
∥
H
μ
∞
(
1
+
2
1
-
s
)
2
(
1
-
s
)
ln
⁡
(
4
e
/
(
1
-
s
)
)
+
2
sup
⁡
|
φ
(
z
)
|
>
s
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
.
Letting n
→
∞ and then letting s
→
1
-, we obtain that
(32)
∥
D
C
φ
∥
e
,
ℒ
ℬ
→
H
μ
∞
≤
2
limsup
⁡
|
φ
(
z
)
|
→
1
-
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
.
The proof of the theorem is finished.Corollary 7.
Letφ be an analytic self-map of 𝔻, μ be a weight on 𝔻, and D
C
φ be a bounded operator from ℒ
ℬ
(
o
r
ℒ
ℬ
0
) to H
μ
∞. Then D
C
φ is a compact operator from ℒ
ℬ
(
o
r
ℒ
ℬ
0
) to H
μ
∞ if and only if
(33)
limsup
⁡
|
φ
(
z
)
|
→
1
-
μ
(
z
)
|
φ
′
(
z
)
|
(
1
-
|
φ
(
z
)
|
)
ln
⁡
(
2
e
/
(
1
-
|
φ
(
z
)
|
)
)
=
0
.



---

*Source: 725145-2014-05-14.xml*