Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 619525, 14 pages
doi:10.1155/2008/619525
Research Article
Weighted Composition Operators from
Generalized Weighted Bergman Spaces to
Weighted-Type Spaces
Dinggui Gu
Department of Mathematics, JiaYing University, Meizhou, GuangDong 514015, China
Correspondence should be addressed to Dinggui Gu, gudinggui@163.com
Received 3 November 2008; Revised 22 November 2008; Accepted 24 November 2008
Recommended by Kunquan Lan
Let ϕ be a holomorphic self-map and let ψ be a holomorphic function on the unit ball B. The
boundedness and compactness of the weighted composition operator ψCϕ from the generalized
weighted Bergman space into a class of weighted-type spaces are studied in this paper.
Copyright q 2008 Dinggui Gu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
Let B be the unit ball of Cn and let HB be the space of all holomorphic functions on B. For
f ∈ HB, let
Rfz n
∂f
zj
z
∂z
j
j1
1.1
represent the radial derivative of f ∈ HB. We write Rm f RRm−1 f.
For any p > 0 and α ∈ R, let N be the smallest nonnegative integer such that pN α >
p
−1. The generalized weighted Bergman space Aα is defined as follows:
p
Aα
f ∈ HB | fApα
f0 N
R fzp 1 − |z|2 pNα dvz
1/p
<∞ .
1.2
B
Here dv is the normalized Lebesgue measure of B i.e., vB 1. The generalized weighted
p
Bergman space Aα is introduced by Zhao and Zhu see, e.g., 1. This space covers the
2
Journal of Inequalities and Applications
p
classical weighted Bergman space α > −1, the Besov space A−n1 , and the Hardy space
H 2 . See 1, 2 for some basic facts on the weighted Bergman space.
Let μ be a positive continuous function on 0, 1. We say that μ is normal if there exist
positive numbers α and β, 0 < α < β, and δ ∈ 0, 1 such that see 3
μr
0;
1 − rα
μr
lim
∞.
r → 1 1 − rβ
μr
is decreasing on δ, 1,
1 − rα
μr
is increasing on δ, 1,
1 − rβ
lim
r →1
1.3
An f ∈ HB is said to belong to the weighted-type space, denoted by Hμ∞ Hμ∞ B,
if
fHμ∞ sup μ |z| fz < ∞,
1.4
z∈B
∞
where μ is normal on 0, 1. The little weighted-type space, denoted by Hμ,0
, is the subspace
∞
∞
of Hμ consisting of those f ∈ Hμ such that
lim μ |z| fz 0.
1.5
|z| → 1
See 4, 5 for more information on Hμ∞ .
Let ϕ be a holomorphic self-map of B. The composition operator Cϕ is defined as
follows:
Cϕ f z f ◦ ϕz,
f ∈ HB.
1.6
Let ψ ∈ HB. For f ∈ HB, the weighted composition operator ψCϕ is defined by
ψCϕ f z ψzf ϕz ,
z ∈ B.
1.7
The book 6 contains a plenty of information on the composition operator and the weighted
composition operator.
In the setting of the unit ball, Zhu studied the boundedness and compactness of
the weighted composition operator between Bergman-type spaces and H ∞ in 7. Some
extensions of these results can be found in 8. Some necessary and sufficient conditions
for the weighted composition operator to be bounded or compact between the Bloch space
and H ∞ are given in 9. In the setting of the unit polydisk, some necessary and sufficient
conditions for a weighted composition operator to be bounded and compact between the
Bloch space and H ∞ are given in 10, 11 see also 12 for the case of composition operators.
In 13, Zhu studied the boundedness and compactness of the Volterra composition operators
from generalized weighted Bergman space to μ-Bloch-type space. Other related results can
be found, for example, in 4, 5, 14–22.
Dinggui Gu
3
In this paper, we study the weighted composition operator ψCϕ from the generalized
∞
. Some necessary and sufficient
weighted Bergman space to the spaces Hμ∞ and Hμ,0
conditions for the weighted composition operator ψCϕ to be bounded and compact are
given.
Throughout the paper, constants are denoted by C, they are positive and may differ
from one occurrence to the other.
2. Main results and proofs
Before we formulate our main results, we state several auxiliary results which will be used in
the proofs. They are incorporated in the lemmas which follow.
Lemma 2.1 see 1. i Suppose that p > 0 and α n 1 > 0. Then there exists a constant C > 0
such that
fz ≤ CfApα
nα1/p
1 − |z|2
2.1
p
for all f ∈ Aα and z ∈ B.
ii Suppose that p > 0 and α n 1 < 0 or 0 < p ≤ 1 and α n 1 0. Then every function
p
in Aα is continuous on the closed unit ball. Moreover, there is a positive constant C such that
f∞ ≤ CfAp
−n1
2.2
,
p
for every f ∈ A−n1 .
iii Suppose that p > 1, 1/p 1/q 1, and α n 1 0. Then there exists a constant C > 0
such that
fz ≤ C ln
e
1 − |z|2
1/q
2.3
p
for all f ∈ Aα and z ∈ B.
The following criterion for compactness of weighted composition operators follows
from standard arguments similar to those outlined in 6, Proposition 3.11 see also 12, proof
of Lemma 2. We omit the details of the proof.
Lemma 2.2. Assume that ψ ∈ HB, ϕ is a holomorphic self-map of B, and μ is a normal function
p
p
on 0, 1. Then ψCϕ : Aα → Hμ∞ is compact if and only if ψCϕ : Aα → Hμ∞ is bounded and for
p
any bounded sequence fk k∈N in Aα which converges to zero uniformly on compact subsets of B as
k → ∞, one has ψCϕ fk Hμ∞ → 0 as k → ∞.
p
Note that when p > 0 and α n 1 < 0, the functions in Aα are Lipschitz continuous
see 1, Theorem 66. By Lemma 2.1 and Arzela-Ascoli theorem, similarly to 19, proof of
Lemma 3.6, we have the following result.
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p
Lemma 2.3. Let p > 0 and α n 1 < 0. Let fk be a bounded sequence in Aα which converges to 0
uniformly on compact subsets of B, then
lim supfk z 0.
k → ∞ z∈B
2.4
The following lemma is from 21 one-dimensional case is 20, Lemma 2.1.
∞
Lemma 2.4. Assume that μ is a normal function on 0, 1. A closed set K in Hμ,0
is compact if and
only if it is bounded and satisfies
lim sup μ |z| fz 0.
|z| → 1 f∈K
2.5
We will consider three cases: n 1 α > 0, n 1 α 0, and n 1 α < 0.
2.1. Case n 1 α > 0
Theorem 2.5. Assume that p > 0, α is a real number such that n α 1 > 0, ψ ∈ HB, ϕ is a
p
holomorphic self-map of B, and μ is a normal function on 0, 1. Then ψCϕ : Aα → Hμ∞ is bounded
if and only if
μ |z| ψz
M : sup n1α/p < ∞.
z∈B 1 − ϕz2
2.6
p
Proof. Assume that ψCϕ : Aα → Hμ∞ is bounded. Let
1
t > n max 1,
p
α1
.
p
2.7
For a ∈ B, set
t−n1α/p
1 − |a|2
fa z .
t
1 − z, a
2.8
p
It follows from 1, Theorem 32 that fa ∈ Aα and supa∈B fa Apα < ∞. Hence
CψCϕ Apα → Hμ∞ ≥ ψCϕ fϕb Hμ∞
sup μ |z| ψCϕ fϕb z
z∈B
μ |b| ψb
≥
2 n1α/p ,
1 − ϕb
from which we get 2.6.
2.9
Dinggui Gu
5
p
Conversely, suppose that 2.6 holds. Then for arbitrary z ∈ B and f ∈ Aα , by
Lemma 2.1 we have
μ |z| ψz
2 n1α/p .
1 − ϕz
μ |z| ψCϕ f z μ |z| f ϕz ψz ≤ CfApα 2.10
p
In light of condition 2.6, the boundedness of the operator ψCϕ : Aα → Hμ∞ follows from
2.10 by taking the supremum over B. This proof is completed.
Theorem 2.6. Assume that p > 0, α is a real number such that n α 1 > 0, ψ ∈ HB, ϕ is a
p
holomorphic self-map of B, and μ is a normal function on 0, 1. Then ψCϕ : Aα → Hμ∞ is compact if
and only if ψ ∈ Hμ∞ and
μ |z| ψz
2 n1α/p 0.
|ϕz| → 1 1 − ϕz
2.11
lim
p
p
Proof. Assume that ψCϕ : Aα → Hμ∞ is compact, then ψCϕ : Aα → Hμ∞ is bounded. Taking
fz ≡ 1, we get that ψ ∈ Hμ∞ . Let zk k∈N be a sequence in B such that |ϕzk | → 1 as k → ∞
if such a sequence does not exist that condition 2.11 is vacuously satisfied. Set
2 t−nα1/p
1 − ϕ zk fk z ,
t
1 − z, ϕ zk
k ∈ N,
2.12
where t satisfies 2.7. From 1, Theorem 32, we see that fk k∈N is a bounded sequence in
p
Aα . Moreover, it is easy to see that fk converges to zero uniformly on compact subsects of B.
By Lemma 2.2, lim supk → ∞ ψCϕ fk Hμ∞ 0. On the other hand, we have
μ zk ψ zk ψCϕ fk ∞ sup μ |z| ψCϕ fk z| ≥
Hμ
2 n1α/p .
z∈B
1 − ϕ zk 2.13
μ zk ψ zk lim sup 2 n1α/p 0,
k→∞
1 − ϕ zk 2.14
Hence
from which 2.11 follows.
Conversely, assume that ψ ∈ Hμ∞ and 2.11 holds. Then, it is easy to check that 2.6
p
holds. Hence ψCϕ : Aα → Hμ∞ is bounded. According to 2.11, for given ε > 0, there is a
constant δ ∈ 0, 1 such that
μ |z| ψz
sup
n1α/p < ε.
{z∈B:δ<|ϕz|<1} 1 − ϕz2
2.15
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Journal of Inequalities and Applications
p
Let fk k∈N be a bounded sequence in Aα such that fk → 0 uniformly on compact subsets of
B as k → ∞. Let δD {w ∈ B : |w| ≤ δ}. From 2.15 and ψ ∈ Hμ∞ , we have
ψCϕ fk ∞ sup μ |z| fk ϕz ψz
Hμ
z∈B
sup
{z∈B: |ϕz|≤δ}
ψHμ∞
sup
{z∈B:δ<|ϕz|<1}
μ |z| ψzfk ϕz sup fk w Cfk Apα
w∈δD
≤ ψHμ∞ sup fk w Cε.
μ |z| ψz
sup
n1α/p
{z∈B:δ<|ϕz|<1} 1 − ϕz2
2.16
w∈δD
Since δD is a compact subset of B, we have limk → ∞ supw∈δD |fk w| 0. Using this fact and
letting k → ∞ in 2.16, we obtain
lim supψCϕ fk Hμ∞ ≤ Cε.
2.17
k→∞
Since ε is an arbitrary positive number, we obtain lim supk → ∞ ψCϕ fk Hμ∞ 0. By Lemma 2.2,
the implication follows.
Theorem 2.7. Assume that p > 0, α is a real number such that n α 1 > 0, ψ ∈ HB, ϕ is a
p
∞
is bounded
holomorphic self-map of B, and μ is a normal function on 0, 1. Then ψCϕ : Aα → Hμ,0
p
∞
if and only if ψCϕ : Aα → Hμ∞ is bounded and ψ ∈ Hμ,0
.
p
p
∞
Proof. Assume that ψCϕ : Aα → Hμ,0
is bounded. Then it is clear that ψCϕ : Aα → Hμ∞ is
p
∞
, we see that
bounded. Taking fz 1 and employing the boundedness of ψCϕ : Aα → Hμ,0
∞
ψ ∈ Hμ,0 .
p
p
∞
Conversely, assume that ψCϕ : Aα → Hμ∞ is bounded and ψ ∈ Hμ,0
. Suppose that
f ∈ Aα with fApα ≤ L, using polynomial approximations we obtain see, e.g., 1
n1α/p fz 0.
lim 1 − |z|2
|z| → 1
2.18
∞
From the above equality and ψ ∈ Hμ,0
, we have that for every ε > 0, there exists a δ ∈ 0, 1
such that when δ < |z| < 1,
n1α/p fz < ε ,
M
n1α/p
2
ε 1−δ
μ |z| ψz <
,
L
1 − |z|2
2.19
2.20
Dinggui Gu
7
where M is defined in 2.6. Therefore, if δ < |z| < 1 and δ < |ϕz| < 1, from 2.6 and 2.19
we have
2 n1α/p μ |z| ψz
f ϕz 2 n1α/p 1 − ϕz
1 − ϕz
2 n1α/p f ϕz < ε.
≤ M 1 − ϕz
μ |z| ψCϕ f z 2.21
If δ < |z| < 1 and |ϕz| ≤ δ, using Lemma 2.1 and 2.20 we have
2 n1α/p μ |z| ψz
f ϕz 2 n1α/p 1 − ϕz
1 − ϕz
μ |z| ψz
≤ CfApα 2 n1α/p
1 − ϕz
μ |z| ψCϕ f z 2.22
CfApα
μ |z| ψz < ε.
2
1 − δ n 1 α/p
≤
∞
Combining 2.21 and 2.22, we obtain that ψCϕ f ∈ Hμ,0
. Since f is an arbitrary element of
p
Aα we see that
p
∞
ψCϕ Aα ⊂ Hμ,0
,
2.23
p
which, along with the boundedness of ψCϕ : Aα → Hμ∞ , implies the result.
Theorem 2.8. Assume that p > 0, α is a real number such that n α 1 > 0, ψ ∈ HB, ϕ is a
p
∞
is compact
holomorphic self-map of B, and μ is a normal function on 0, 1. Then ψCϕ : Aα → Hμ,0
if and only if
μ |z| ψz
lim
2 n1α/p 0.
|z| → 1 1 − ϕz
2.24
p
Proof. Assume that 2.24 holds. For any f ∈ Aα with fApα ≤ 1, by 2.10 we have
μ |z| ψz
2 n1α/p .
1 − ϕz
μ |z| ψCϕ f z ≤ CfApα 2.25
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Journal of Inequalities and Applications
Using 2.24, we get
μ |z| ψz
2 n1α/p 0.
1 − ϕz
lim sup μ |z| ψCϕ f z ≤ C lim |z| → 1
|z| → 1
p
fA ≤1
α
2.26
p
∞
is compact.
From this and Lemma 2.4, we see that ψCϕ : Aα → Hμ,0
p
p
∞
∞
Conversely, assume that ψCϕ : Aα → Hμ,0
is compact. Then ψCϕ : Aα → Hμ,0
is
p
bounded and ψCϕ : Aα → Hμ∞ is compact. By Theorems 2.6 and 2.7, we obtain
μ |z| ψz
lim
2 n1α/p 0,
|ϕz| → 1 1 − ϕz
lim μ |z| ψz 0.
|z| → 1
2.27
2.28
If ϕ∞ < 1, it holds that
μ |z| ψz
1
lim
μ |z| ψz 0,
n1α/p |z|lim
2 n1α/p ≤ |z| → 1 →
1
2
1 − ϕ∞
1 − ϕz
2.29
from which the result follows in this case.
Hence, assume that ϕ∞ 1. In terms of 2.27, for every ε > 0, there exists a δ ∈ 0, 1,
such that when δ < |ϕz| < 1,
μ |z| ψz
2 n1α/p < ε.
1 − ϕz
2.30
According to 2.28, for the above ε, there exists an r ∈ 0, 1, such that when r < |z| < 1,
n1α/p
μ |z| ψz < ε 1 − δ 2
.
2.31
Therefore, when r < |z| < 1 and δ < |ϕz| < 1, we have that
μ |z| ψz
2 n1α/p < ε.
1 − ϕz
2.32
If r < |z| < 1 and |ϕz| ≤ δ, we obtain
μ |z| ψz
1
μ |z| ψz < ε.
2 n1α/p ≤ n1α/p
1 − δ2
1 − ϕz
Combining 2.32 with 2.33 we get 2.24, as desired.
2.33
Dinggui Gu
9
2.2. Case n 1 α 0
Theorem 2.9. Assume that p > 1, α is a real number such that n α 1 0, ψ ∈ HB, ϕ is a
p
holomorphic self-map of B, and μ is a normal function on 0, 1. Then ψCϕ : Aα → Hμ∞ is bounded
if and only if
M1 : sup μ |z| ψz
ln
z∈B
1−1/p
e
< ∞.
2
1 − ϕz
2.34
p
Proof. Assume that 2.34 holds. Then for arbitrary z ∈ B and f ∈ Aα , by Lemma 2.1 we have
μ |z| ψCϕ f z μ |z| f ϕz ψz
≤ Cf p μ |z| ψz ln
Aα
e
1−1/p
2
1 − ϕz
.
2.35
p
From 2.34 and 2.35, the boundedness of ψCϕ : Aα → Hμ∞ follows.
p
Now assume that ψCϕ : Aα → Hμ∞ is bounded. For a ∈ B, set
fa z ln
e
1 − |a|2
−1/p ln
e
.
1 − z, a
2.36
p
By using 2, Theorem 1.12, we easily check that fa ∈ A−n1 . Therefore,
CψCϕ Apα → Hμ∞ ≥ ψCϕ fϕb Hμ∞
sup μ |z| ψCϕ fϕb z
z∈B
≥ μ |b| ψb
ln
e
2
1 − ϕb
2.37
1−1/p
.
From the last inequality, we get the desired result.
Theorem 2.10. Assume that p > 1, α is a real number such that n α 1 0, ψ ∈ HB, ϕ is a
p
holomorphic self-map of B, and μ is a normal function on 0, 1. Then ψCϕ : Aα → Hμ∞ is compact if
and only if ψ ∈ Hμ∞ and
lim μ |z| ψz
|ϕz| → 1
ln
e
2
1 − ϕz
1−1/p
0.
2.38
Proof. First assume that 2.38 holds and ψ ∈ Hμ∞ . In this case, the proof of Theorem 2.6 still
works with minor changes, hence we omit the details.
10
Journal of Inequalities and Applications
p
p
Now we assume that ψCϕ : Aα → Hμ∞ is compact, then it is clear that ψCϕ : Aα →
is bounded. Similarly to the proof of Theorem 2.6, we see that ψ ∈ Hμ∞ . Let zk k∈N be
a sequence in B such that |ϕzk | → 1 as k → ∞ if such a sequence does not exist that
condition 2.38 is vacuously satisfied. Set
Hμ∞
fk z e
ln
2
1 − ϕ zk −1/p e
ln
1 − z, ϕ zk
,
k ∈ N.
2.39
p
From 2, Theorem 1.12, we see that fk k∈N is a bounded sequence in Aα . Moreover, fk → 0
uniformly on compact subsets of B as k → ∞. It follows from Lemma 2.2 that ψCϕ fk Hμ∞ →
0 as k → ∞. Because
ψCϕ fk ∞ sup μ |z| ψCϕ fk z
Hμ
z∈B
≥ μ zk ψ zk e
ln
2
1 − ϕ zk 1−1/p
2.40
,
we obtain
lim μ zk ψ zk k→∞
e
ln
2
1 − ϕ zk 1−1/p
0,
2.41
from which we get the desired result. The proof is completed.
Theorem 2.11. Assume that p > 1, α is a real number such that n α 1 0, ψ ∈ HB, ϕ is a
p
∞
is bounded
holomorphic self-map of B, and μ is a normal function on 0, 1. Then ψCϕ : Aα → Hμ,0
p
∞
if and only if ψCϕ : Aα → Hμ∞ is bounded and ψ ∈ Hμ,0
.
p
p
∞
Proof. First assume that ψCϕ : Aα → Hμ,0
is bounded. Then clearly ψCϕ : Aα → Hμ∞ is
p
∞
bounded. Taking fz 1, then employing the boundedness of ψCϕ : Aα → Hμ,0
, we have
∞
that ψ ∈ Hμ,0 , as desired.
p
∞
Conversely, assume that ψCϕ : Aα → Hμ∞ is bounded and ψ ∈ Hμ,0
. For each
polynomial p, we have
μ |z| ψCϕ p z μ |z| p ϕz ψz ≤ p∞ μ |z| ψz,
∞
from which we have that ψCϕ p ∈ Hμ,0
.
p
2.42
p
Since the set of all polynomials is dense in Aα see 2, for every f ∈ Aα there is a
sequence of polynomials pk k∈N such that
pk − f p −→ 0
Aα
as k −→ ∞.
2.43
Dinggui Gu
11
p
From the boundedness of ψCϕ : Aα → Hμ∞ , we have that
ψCϕ pk − ψCϕ f Hμ∞
≤ ψCϕ pk − f Apα −→ 0
as k −→ ∞.
2.44
∞
Since Hμ,0
is a closed subset of Hμ∞ , we obtain
∞
.
ψCϕ f lim ψCϕ pk ∈ Hμ,0
2.45
k→∞
p
∞
Therefore, ψCϕ : Aα → Hμ,0
is bounded.
Using Theorems 2.10 and 2.11, similarly to the proof of Theorem 2.8 we obtain the
following result. We omit the proof.
Theorem 2.12. Assume that p > 1, α is a real number such that n α 1 0, ψ ∈ HB, ϕ is a
p
∞
is compact
holomorphic self-map of B, and μ is a normal function on 0, 1. Then ψCϕ : Aα → Hμ,0
if and only if
lim μ |z| ψz
ln
|z| → 1
e
1−1/p
2
1 − ϕz
0.
2.46
Theorem 2.13. Assume that 0 < p ≤ 1, α is a real number such that n α 1 0, ψ ∈ HB, ϕ is a
p
holomorphic self-map of B, and μ is a normal function on 0, 1. Then ψCϕ : Aα → Hμ∞ is bounded
∞
if and only if ψ ∈ Hμ .
p
Proof. Assume that ψ ∈ Hμ∞ . For any f ∈ Aα , by Lemma 2.1 we have
sup μ |z| ψCϕ f z ≤ CfApα sup μ |z| ψz.
z∈B
2.47
z∈B
p
From the above inequality, we obtain that ψCϕ : Aα → Hμ∞ is bounded.
p
Conversely, assume that ψCϕ : Aα → Hμ∞ is bounded. Taking fz 1 and using the
p
boundedness of ψCϕ : Aα → Hμ∞ , we get ψ ∈ Hμ∞ , as desired.
Theorem 2.14. Assume that 0 < p ≤ 1, α is a real number such that n α 1 0, ψ ∈ HB, ϕ is
p
a holomorphic self-map of B, and μ is a normal function on 0, 1. Then ψCϕ : Aα → Hμ∞ is compact
∞
if and only if ψ ∈ Hμ and
lim μ |z| ψz 0.
|ϕz| → 1
2.48
Proof. First assume that ψ ∈ Hμ∞ and 2.48 holds. In this case, the proof is similar to the
corresponding part of the proof of Theorem 2.6 and hence will be omitted.
12
Journal of Inequalities and Applications
p
Now we suppose that ψCϕ : Aα → Hμ∞ is compact. It follows from Theorem 2.13 and
p
the boundedness of ψCϕ : Aα → Hμ∞ that ψ ∈ Hμ∞ . Let zk k∈N be a sequence in B such that
|ϕzk | → 1 as k → ∞. Set
2
1 − ϕ zk fk z ,
1 − z, ϕ zk
k ∈ N.
2.49
p
From 2, Theorem 6.6, we see that fk k∈N is a bounded sequence in Aα . Moreover, fk
converges to zero uniformly on compact subsets of B. Hence by Lemma 2.2 it follows that
lim supψCϕ fk Hμ∞ 0.
2.50
ψCϕ fk ∞ sup μ |z| ψCϕ fk z ≥ μ zk ψ zk .
Hμ
2.51
k→∞
On the other hand, we obtain
z∈B
Combining 2.50 with 2.51 we obtain that 2.48 holds.
Theorem 2.15. Assume that 0 < p ≤ 1, α is a real number such that n α 1 0, ψ ∈ HB, ϕ is
a holomorphic self-map of B, and μ is a normal function on 0, 1. Then the following statements are
equivalent:
p
∞
is bounded;
i ψCϕ : Aα → Hμ,0
p
∞
is compact;
ii ψCϕ : Aα → Hμ,0
∞
iii ψ ∈ Hμ,0
.
Proof. ii⇒i. This implication is clear.
p
∞
, we
i⇒iii. Taking fz 1 and employing the boundedness of ψCϕ : Aα → Hμ,0
∞
obtain that ψ ∈ Hμ,0 .
p
iii⇒ii. For any f ∈ Aα with fApα ≤ 1, we have
μ |z| ψCϕ f z ≤ CfApα μ |z| ψz ≤ Cμ |z| ψz,
2.52
from which we obtain
lim sup μ |z| ψCϕ f z ≤ C lim μ |z| ψz 0.
|z| → 1 f
|z| → 1
p ≤1
Aα
2.53
p
∞
Using Lemma 2.4, we obtain that ψCϕ : Aα → Hμ,0
is compact.
2.3. Case n 1 α < 0
Theorem 2.16. Assume that p > 0, α is a real number such that n α 1 < 0, ψ ∈ HB, ϕ is
a holomorphic self-map of B, and μ is a normal function on 0, 1. Then the following statements are
equivalent:
Dinggui Gu
13
p
i ψCϕ : Aα → Hμ∞ is bounded;
p
ii ψCϕ : Aα → Hμ∞ is compact;
iii ψ ∈ Hμ∞ .
Proof. ii⇒i. This implication is obvious.
p
i⇒iii. Taking fz 1, then using the boundedness of ψCϕ : Aα → Hμ∞ , we obtain
that ψ ∈ Hμ∞ .
p
iii⇒ii. If f ∈ Aα , by Lemma 2.1 we obtain
μ |z| ψCϕ f z ≤ CfApα μ |z| ψz,
2.54
p
from which it follows that ψCϕ : Aα → Hμ∞ is bounded. Let fk k∈N be any bounded sequence
p
in Aα and fk → 0 uniformly on B as k → ∞. By Lemma 2.3, we have
ψCϕ fk ∞ sup μ |z| fk ϕz ψz ≤ ψH ∞ sup fk ϕz −→ 0,
μ
Hμ
z∈B
2.55
z∈B
as k → ∞. The result follows from Lemma 2.2.
Similarly to the proof of Theorem 2.15, we have the following result. We omit the proof
here.
Theorem 2.17. Assume that p > 0, α is a real number such that n α 1 < 0, ψ ∈ HB, ϕ is
a holomorphic self-map of B, and μ is a normal function on 0, 1. Then the following statements are
equivalent:
p
∞
is bounded;
i ψCϕ : Aα → Hμ,0
p
∞
is compact;
ii ψCϕ : Aα → Hμ,0
∞
iii ψ ∈ Hμ,0
.
Acknowledgment
The author is supported partly by the NSF of Guangdong Province of China no. 07006700.
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