PRODUCTS OF INTEGRAL-TYPE AND COMPOSITION OPERATORS
BETWEEN GENERALLY WEIGHTED BLOCH SPACES
HAIYING LI and TIANSHUI MA
Abstract. Let ϕ be a holomorphic self-map of the open unit disk D on the complex plane and
0 < α, β < +∞. The boundedness and compactness of products of integral-type and composition
operators between generally weighted Bloch spaces are investigated.
1. Introduction and preliminaries
Let D be the unit disc on the complex plane and ϕ a holomorphic self-map of D. We denote by
H(D) the space of all holomorphic functions on D, denote by dm(z) the normalized Lebesgue area
measure and define the composition operator Cϕ on H(D) by Cϕ f = f ◦ ϕ.
The space of analytic functions on D such that
2
kf kBlog = |f (0)| + sup |f 0 (z)|(1 − |z|2 ) log
<∞
1
−
|z|2
z∈D
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is called weighted Bloch space Blog . Blog and BM OAlog first appeared in the study of boundedness
of the Hankel operators on the Bergman space
Z
1
A = {f ∈ H(D) :
|f (z)| dm(z) < ∞}
D
Received May 15, 2010; revised November 9, 2010.
2001 Mathematics Subject Classification. Primary 47B38.
Key words and phrases. holomorphic self-map; composition operator; generally weighted Bloch space.
and the Hardy space H 1 , respectively. BM OAlog also appeared in the study of a Volterra type
operator (see e.g. [1, 2, 3, 4, 9, 10]). In [11], Yoneda studied the composition operators from Blog
α
to BM OAlog . In [5, 6, 7], we introduced the space Blog
, α < 0, the space of analytic functions on
D such that
α
kf kBlog
= |f (0)| + sup |f 0 (z)|(1 − |z|2 )α log
z∈D
2
<∞
1 − |z|2
α
that is called generally weighted Bloch space Blog
.
Let g ∈ H(D), for f ∈ H(D) be the integral-type operator Ig and Jg respectively, defined by
Zz
Ig f (z) =
f 0 (ζ)g(ζ)dζ,
0
Zz
Jg f (z) =
f (ζ)g 0 (ζ)dζ,
z ∈ D.
0
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The importance of the operators Ig and Jg comes from the fact that
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Iφ f (z) + Jφ f (z) = Mφ f (z) − f (0)φ(0),
z ∈ D,
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where Mg is the multiplication operator
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(Mg f )(z) = g(z)f (z),
f ∈ H(D), z ∈ D.
The products of composition operators and integral-type operators are defined by
ϕ(z)
Z
Cϕ Jg f (z) =
0
f (ξ)g (ξ)dξ,
Zz
Jg Cϕ f (z) =
0
0
ϕ(z)
Z
Cϕ Iφ f (z) =
f (ϕ(ξ))g 0 (ξ)dξ,
0
f (ξ)φ(ξ)dξ,
Zz
Iφ Cϕ f (z) =
0
(f ◦ ϕ)0 (ξ)φ(ξ)dξ.
0
In this article, we consider the characterization of boundedness and compactness of products
of integral-type and composition operators between generally weighted Bloch spaces on the unit
disk. Throughout the remainder of this paper C will denote a positive constant, the exact value
of which will vary from one appearance to the next.
β
α
2. The boundedness and compactness of Cϕ Jg (Cϕ Ig ) : Blog
→ Blog
At the beginning, the following Lemma 2.1 can be seen in [5].
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α
Lemma 2.1. Let f ∈ Blog
and z ∈ D, then
1
α ;
(a) For 0 < α < 1, |f (z)| ≤ 1 +
kf kBlog
(1 − α) log 2
4
log 1−|z|
2
α ;
(b) For α = 1, |f (z)| ≤
kf kBlog
log 2
2α−1
1
α .
(c) For α > 1, |f (z)| ≤ 1 +
kf kBlog
(α − 1) log 2 (1 − |z|2 )α−1
Lemma 2.2. Assume that ϕ is a holomorphic self-map of D and α, β > 0. Then Cϕ Jg (or
β
α
α
Cϕ Ig ) : Blog
→ Blog
is compact if and only if for any bounded sequence (fj )j∈N in Blog
, when
fj → 0 uniformly on compact subsets of D, kCϕ Jg fj kB β → 0 or kCϕ Ig fj kB β ) → 0 as j → ∞.
log
log
The result follows from standard arguments similar to those in [4].
It is easy to obtain the following result by a similar method in [8] for 0 < α < 1.
Lemma 2.3. Assume that ϕ is a holomorphic self-map of D and 0 < α < 1, β > 0. Then
β
α
α
Cϕ Jg : Blog
→ Blog
is compact if and only if for any bounded sequence (fj )j∈N in Blog
, when
fj → 0 uniformly on D, kCϕ Jg fj kB β → 0 as j → ∞.
log
α
Lemma 2.4. Assume that h ∈ H(D), f ∈ Blog
, α > 0 for a fixed z0 ∈ D. Then there exists a
positive constant C independent of f such that
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z
Z 0
f (ζ)h(ζ)dζ ≤ Ckf kB α max |h(ζ)|,
log
|ζ|≤|z0 |
0
z
Z 0
f 0 (ζ)h(ζ)dζ ≤ Ckf kB α max |h(ζ)|.
log
|ζ|≤|z0 |
0
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α
Proof. For h ∈ H(D), f ∈ Blog
, then
z
Z 0
f (ζ)h(ζ)dζ ≤ max |f (ζ)| max |h(ζ)|
|ζ|≤|z0 |
|ζ|≤|z0 |
0
0
≤ |f (0)| + |z0 | max |f (ζ)| max |h(ζ)|
|ζ|≤|z0 |
|ζ|≤|z0 |
)
(
|z0 |
α
kf kBlog
max |h(ζ)|.
≤ max 1,
2
|ζ|≤|z0 |
(1 − |z0 |2 )α log 1−|z
2
0|
Similarly, we have
z
Z 0
f 0 (ζ)h(ζ)dζ ≤ |z0 | max |f 0 | max |h(ζ)|
|ζ|≤|z0 |
|ζ|≤|z0 |
0
≤
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|z0 |
(1 − |z0 |2 )α log
2
1−|z0 |2
α
kf kBlog
max |h(ζ)|.
|ζ|≤|z0 |
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Theorem 2.5. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α ∈ (0, 1), β > 0,
β
α
then Cϕ Jg : Blog
→ Blog
is bounded if and only if
(2.1)
sup(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
z∈D
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2
< ∞.
1 − |z|2
β
α
Proof. Assume that Cϕ Jg : Blog
→ Blog
is bounded. Then by the definition of the operator
Cϕ Jg ,
(Cϕ Jg f )0 (z) = f (ϕ(z))g 0 (ϕ(z))ϕ0 (z).
(2.2)
α
Let f0 (z) = 1, then f0 ∈ Blog
. Then by the boundedness of Cϕ Jg
(2.3)
(1 − |z|2 )β |ϕ0 (z)|g 0 (ϕ(z))| log
2
α
≤ kCϕ Jg kkf0 kBlog
< ∞.
1 − |z|2
Then (2.1) holds by (2.3).
Conversely, assume that (2.1) holds. Then by Lemma 2.1 and (2.2)
(1 − |z|2 )β (Cϕ Jg f )0 (z) log
(2.4)
2
1 − |z|2
2 β 0
0
α (1 − |z| ) |ϕ (z)||g (ϕ(z))| log
≤ Ckf kBlog
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2
.
1 − |z|2
Then, by Lemma 2.4, with h = g 0 and z0 = ϕ(0),
Z
ϕ(0)
0
α
(2.5)
|(Cϕ Jg fj )(0)| = f (ζ)g (ζ)dζ ≤ Ckf kBlog
max |g 0 (ζ)|.
0
|ζ|≤|ϕ(0)|
By (2.4), we have
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2
kCϕ Jg f kB β ≤ C sup(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
log
1 − |z|2
z∈D
α .
+ max |g 0 (ζ)| kf kBlog
|ζ|≤|ϕ(0)|
By (2.1) and (2.5), the boundedness of Cϕ Jg is obtained.
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Theorem 2.6. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α ∈ (0, 1), β > 0,
β
α
then Cϕ Jg : Blog
→ Blog
is compact if and only if
sup(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
z∈D
2
< ∞.
1 − |z|2
β
α
Proof. Assume that Cϕ Jg : Blog
→ Blog
is compact, then it is bounded, hence (2.1) holds by
Theorem 2.5.
β
α
Conversely, assume that (2.1) holds. Then by Theorem 2.5, Cϕ Jg : Blog
→ Blog
is bounded.
α
By Lemma 2.3 for any bounded sequence (fj )j∈N in Blog , when fj → 0 uniformly on D, we need
only to prove that kCϕ Jg fj kB β → 0 as j → ∞. Then
log
lim sup(1 − |z|2 )β (Cϕ Jg fj )0 (z) log
j→∞
z∈D
2
1 − |z|2
≤ sup(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
z∈D
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2
lim kfj k∞ = 0.
1 − |z|2 j→∞
Z
ϕ(0)
0
|(Cϕ Jg fj )(0)| = fj (ζ)g (ζ)dζ ≤ Ckfj k∞ max |g 0 (ζ)| → 0 as j → ∞.
0
|ζ|≤|ϕ(0)|
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Then the compactness of Cϕ Jg is completed.
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Theorem 2.7. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D),
β > 0.
(i) If
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(2.6)
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sup(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
z∈D
2
2
log
< ∞,
2
1 − |z|
1 − |ϕ(z)|2
β
then Cϕ Jg : Blog → Blog
is bounded.
β
(ii) If Cϕ Jg : Blog → Blog
is bounded, then
(2.7)
sup(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
z∈D
2
2
log log
< ∞.
2
1 − |z|
1 − |ϕ(z)|2
Proof. (i) For f ∈ Blog , by Lemma 2.1, it holds
2
(1 − |z|2 )β (Cϕ Jg f )0 (z) log
1 − |z|2
≤ Ckf kBlog (1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
2
2
log
.
2
1 − |z|
1 − |ϕ(z)|2
β
By (2.6), we have that Cϕ Jg : Blog → Blog
is bounded.
β
(ii) Assume that Cϕ Jg : Blog → Blog
is bounded. For w ∈ D, set
fw (z) = log log
Then
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fw0 (z) =
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1
log
2
1−wz
2
.
1 − wz
·
w
.
1 − wz
Then |fw (0)| = log log 2 and
(1 − |z|
2
)|fw0 (z)| log
2
(1 − |z|2 )|w| log 1−|z|
2
2
=
2
1 − |z|2
|1 − wz| log |1−wz|
≤
2
1−|z|2
2
|1−z|
(1 − |z|2 ) log
|1 − z| log
< ∞.
β
Thus fw ∈ Blog . Hence by the boundedness of Cϕ Jg : Blog → Blog
, we have
(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
2
2
log log
1 − |z|2
1 − |ϕ(z)|2
≤ CkCϕ Jg fϕ(z) kB β ≤ kCϕ Jg k · kfϕ(z) kBlog < ∞.
log
Theorem 2.8. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), β > 0.
(i) If
2
<∞
sup(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
1 − |z|2
z∈D
and
2
2
lim (1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
(2.8)
log
= 0,
1 − |z|2
1 − |ϕ(z)|2
|ϕ(z)|→1
β
then Cϕ Jg : Blog → Blog
is compact.
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β
(ii) If Cϕ Jg : Blog → Blog
is compact, then
(2.9)
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lim (1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
|ϕ(z)|→1
2
2
log log
= 0.
1 − |z|2
1 − |ϕ(z)|2
Proof. (i) By (2.8), we have that for any ε > 0 there exists an r0 ∈ (0, 1) such that
(2.10)
(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
for every |ϕ(z)| > r0 .
2
2
log
< ε,
1 − |z|2
1 − |ϕ(z)|2
Let (fj )j∈N be a norm bounded sequence in Blog such that fj → 0 uniformly on compact subsets
of D as j → ∞. By Lemma 2.1, (2.1) and (2.10), we have
(1−|z|2 )β (Cϕ Jg fj )0 (z) log
≤
sup
sup (1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
|fj (ϕ(z))|
|ϕ(z)|≤r0
+ Ckfj kBlog
2
1 − |z|2
|ϕ(z)|≤r0
sup (1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
|ϕ(z)|>r0
2
1 − |z|2
2
2
log
2
1 − |z|
1 − |ϕ(z)|2
≤ C sup |fj (ζ)| + Cεkfj kBlog .
|ζ|≤r0
ϕ(0)
Z
0
|(Cϕ Jg fj )(0)| = f (ζ)g (ζ)dζ 0
≤
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max
|ζ|≤|ϕ(0)|
|fj (ζ)|
max
|ζ|≤|ϕ(0)|
|g 0 (ζ)| → 0
(j → ∞).
Taking the supremum over z ∈ D and letting j → ∞, we have kCϕ Jg fj kB β → 0 as j → ∞. Thus
log
β
Cϕ Jg : Blog → Blog
is compact.
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β
(ii) Assume that Cϕ Jg : Blog → Blog
is compact and (zn )n∈N is a sequence in D such that
limn→∞ |ϕ(zn )| = 1. Let
!2
−1
2
2
log log
,
n ∈ N.
fn (z) = log log
1 − |ϕ(zn )|2
1 − ϕ(zn )z
Then fn is a uniformly bounded family on Blog that converges to 0 on compact subsets of D.
Then kCϕ Jg fn kB β → 0 as n → ∞.
log
2
1 − |z|2
z∈D
2
2
≥ 1 − |zn |2 )β |ϕ0 (zn )||g 0 (ϕ(zn ))| log
log log
.
1 − |zn |2
1 − |ϕ(zn )|2
kCϕ Jg fn kB β ≥ sup(1 − |z|2 )β (Cϕ Jg fn )0 (z) log
log
Hence
lim (1 − |zn |2 )β |ϕ0 (zn )||g 0 (ϕ(zn ))| log
n→∞
2
2
log log
= 0.
1 − |zn |2
1 − |ϕ(zn )|2
So (2.9) holds.
Theorem 2.9. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α > 1, β > 0. If
(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
(2.11)
sup
(1 − |ϕ(z)|2 )α−1
z∈D
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< ∞,
β
α
then Cϕ Jg : Blog
→ Blog
is bounded.
α
Proof. By Lemma 2.1 and (2.11), for f ∈ Blog
,
2
1 − |z|2
(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
(1 − |z|2 )β (Cϕ Jg f )0 (z) log
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α
≤ Ckf kBlog
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2
1−|z|2
(1 − |ϕ(z)|2 )α−1
2
1−|z|2
< ∞.
max |f (ζ)| max |g 0 (ζ)|
|ζ|≤|ϕ(0)|
)
(
|ϕ(z0 )|
α
kf kBlog
≤ max 1,
max |g 0 (ζ)|.
2
|ζ|≤|ϕ(0)|
(1 − |ϕ(z0 )|2 ) log 1−|z
2
0|
|(Cϕ Jg f )(0)| ≤
|ζ|≤|ϕ(0)|
Then the boundedness of Cϕ Jg is obtained.
Theorem 2.10. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α > 1, β > 0. If
sup(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
z∈D
2
<∞
1 − |z|2
and
(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
(2.12)
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lim
|ϕ(z)|→1
(1 − |ϕ(z)|2 )α−1
2
1−|z|2
= 0,
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β
α
then Cϕ Jg : Blog
→ Blog
is compact.
Proof. By (2.12), then for any ε > 0, there exists an r0 ∈ (0, 1) such that
(1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
(1 − |ϕ(z)|2 )α−1
2
1−|z|2
< ε,
for every |ϕ(z)| > r0 .
α
Let (fj )j∈N be a norm bounded sequence in Blog
such that fj → 0 uniformly on compact subsets
of D as j → ∞. By Lemma 2.1, we have
2
(1 − |z|2 )β (Cϕ Jg fj )0 (z) log
1 − |z|2
2
≤ sup |fj (ϕ(z))| sup (1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
1
−
|z|2
|ϕ(z)|≤r0
|ϕ(z)|≤r0
α
+ Ckfj kBlog
sup (1 − |z|2 )β |ϕ0 (z)||g 0 (ϕ(z))| log
|ϕ(z)|>r0
2
2
log
1 − |z|2
1 − |ϕ(z)|2
α .
≤ C sup |fj (ζ)| + Cεkfj kBlog
|ζ|≤r0
ϕ(0)
Z
α
|(Cϕ Jg fj )(0)| = f (ζ)g 0 (ζ)dζ ≤ Ckfj kBlog
max |g 0 (ζ)|.
|ζ|≤|ϕ(0)|
0
β
α
→ Blog
Taking the supremum over z ∈ D and letting j → ∞, kCϕ Jg fj kB β → 0. Thus Cϕ Jg : Blog
log
is compact.
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Theorem 2.11. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α ∈ (0, 1), β > 0,
β
β
α
α
then Jg Cϕ : Blog
→ Blog
is bounded if and only if Jg Cϕ : Blog
→ Blog
is compact if and only if
β
g ∈ Blog
.
Theorem 2.12. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), β > 0,. If
2
2
log
< ∞,
sup(1 − |z|2 )β |g 0 (z)| log
2
2
1
−
|z|
1
−
|ϕ(z)|
z∈D
β
then Jg Cϕ : Blog → Blog
is bounded.
Theorem 2.13. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), β > 0, if
2
sup(1 − |z|2 )β |g 0 (z)| log
<∞
1 − |z|2
z∈D
and
lim (1 − |z|2 )β |g 0 (z)| log
|ϕ(z)|→1
2
2
log
= 0,
2
1 − |z|
1 − |ϕ(z)|2
β
then Jg Cϕ : Blog → Blog
is compact.
Theorem 2.14. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α > 1, β > 0. If
(1 − |z|2 )β |g 0 (z)| log
sup
z∈D
(1 −
2
1−|z|2
|ϕ(z)|2 )α−1
< ∞,
β
α
then Jg Cϕ : Blog
→ Blog
is bounded.
Theorem 2.15. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α > 1, β > 0. If
2
sup(1 − |z|2 )β |g 0 (z)| log
<∞
1
−
|z|2
z∈D
and
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(1 − |z|2 )β |g 0 (z)| log
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lim
|ϕ(z)|→1
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= 0,
β
α
then Jg Cϕ : Blog
→ Blog
is compact.
Theorem 2.16. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α > 0, β > 0,. If
(1 − |z|2 )β |ϕ0 (z)||g(ϕ(z))| log
(2.13)
sup
z∈D
Quit
(1 −
2
1−|z|2
|ϕ(z)|2 )α−1
(1 − |ϕ(z)|2 )α log
β
α
then Cϕ Ig : Blog
→ Blog
is bounded.
2
1−|z|2
2
1−|ϕ(z)|2
< ∞,
α
Proof. By the definition of Cϕ Ig , (Cϕ Ig f )0 (z) = ϕ0 (z)g(ϕ(z))f 0 (ϕ(z)). For f ∈ Blog
, we have
2
1 − |z|2
(1 − |z|2 )β |ϕ0 (z)||g(ϕ(z))| log
(1 − |z|2 )β (Cϕ Ig f )0 (z) log
≤
2
1−|z|2
α .
kf kBlog
2
(1 − |ϕ(z)|2 )α log 1−|ϕ(z)|
2
ϕ(0)
Z
α
|(Cϕ Ig f )(0)| = f 0 (ζ)g(ζ)dζ ≤ Ckf kBlog
max |g(ζ)|.
|ζ|≤|ϕ(0)|
0
β
α
By (2.13), we have Cϕ Ig : Blog
→ Blog
is bounded.
Theorem 2.17. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α > 0, β > 0,. If
2
(2.14)
<∞
sup (1 − |z|2 )β |ϕ0 (z)||g(ϕ(z))| log
1 − |z|2
z∈D
and
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(1 − |z|2 )β |ϕ0 (z)||g(ϕ(z))| log
(2.15)
|ϕ(z)|→1
(1 − |ϕ(z)|2 )α log
2
1−|z|2
2
1−|ϕ(z)|2
= 0,
β
α
then Cϕ Ig : Blog
→ Blog
is compact.
Proof. By (2.15), for any ε > 0, there exists an r ∈ (0, 1) such that
(1 − |z|2 )β |ϕ0 (z)||g(ϕ(z))| log
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(2.16)
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lim
(1 − |ϕ(z)|2 )α log
2
1−|z|2
2
1−|ϕ(z)|2
<ε
for every r < |ϕ(z)| < 1.
α
Let (fj )j∈N be a norm bounded sequence in Blog
such that fj → 0 uniformly on compact subsets
of D as j → ∞. Then
kCϕ Ig fj kB β ≤
log
sup (1 − |z|2 )β |ϕ0 (z)||g(ϕ(z))||fj0 (ϕ(z))| log
|ϕ(z)|≤r
2
1 − |z|2
+ sup (1 − |z|2 )β |ϕ0 (z)||g(ϕ(z))||fj0 (ϕ(z))| log
|ϕ(z)|>r
+
(2.17)
max
|ζ|≤|ϕ(0)|
|fj0 (ζ)|
max
|ζ|≤|ϕ(0)|
|g(ζ)|
≤ sup(1 − |z|2 )β |ϕ0 (z)||g(ϕ(z))| log
z∈D
2
sup |f 0 (ζ)|
1 − |z|2 |ζ|≤r j
(1 − |z|2 )β |ϕ0 (z)||g(ϕ(z))| log
+ sup
|ϕ(z)|>r
+
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max
|ζ|≤|ϕ(0)|
(1 − |ϕ(z)|2 )α log
|fj0 (ζ)|
max
|ζ|≤|ϕ(0)|
2
1 − |z|2
2
1−|z|2
2
1−|ϕ(z)|2
α
kfj kBlog
|g(ζ)|.
Since fj → 0 uniformly on compact subsets of D as j → ∞, by Cauchy’s estimate, fj0 → 0
uniformly on compact subsets of D as j → ∞. Hence by (2.14), (2.16) and (2.17), we have
β
α
kCϕ Ig fj kB β → 0 as j → ∞. Hence Cϕ Ig : Blog
→ Blog
is compact.
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Theorem 2.18. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α > 0, β > 0,. If
(1 − |z|2 )β |ϕ0 (z)||g(z)| log
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sup
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z∈D
(1 − |ϕ(z)|2 )α log
2
1−|z|2
2
1−|ϕ(z)|2
< ∞,
β
α
then Ig Cϕ : Blog
→ Blog
is bounded.
Theorem 2.19. Assume that ϕ is a holomorphic self-map of D, g ∈ H(D), α > 0, β > 0. If
sup(1 − |z|2 )β |ϕ0 (z)||g(z)| log
z∈D
2
<∞
1 − |z|2
and
(1 − |z|2 )β |ϕ0 (z)||g(z)| log
lim
|ϕ(z)|→1
(1 − |ϕ(z)|2 )α log
2
1−|z|2
2
1−|ϕ(z)|2
= 0,
β
α
then Ig Cϕ : Blog
→ Blog
is compact.
Acknowledgment. This work is supported by the Natural Science Foundation of Henan (No.
2008B110006; 2010A110009; 102300410012) and the Foster Foundation of Henan Normal University (No. 2010PL01).
JJ J
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1. Attele K. R., M. Toeplitz and Hankel on Bergman one space, Hokaido Math J., 21 (1992), 279–293.
2. Benke G., and Chang D. C., A note on weighted Bergman spaces and the Cesàro operator, Nagoya Mathematical
J., 159 (2000), 25–43.
3. Cima J., Stegenda D., Hankel operators on H p , in: Berkson E. R., Peck N. T.and Ulh J. (Eds.), Analysis at
Urbana 1, London Math. Soc. Lecture Note ser., Cambridge Univ. Press, Cambridge, 137 (1989), 133–150.
4. Cowen C. C. and MacCluer B. D., Composition operators on spaces of analytic functions, Boca Roton: CRC
Press, 1995.
5. Li H., Liu P. and Wang M., Composition operators between generally weighted Bloch spaces of polydisk, J.
Inequal Pure Appl. Math, 8(3) (2007), 1–8.
6. Li H. and Liu P., Composition operators between generally weighted Bloch space and Qqlog space. Banach Journal
of Mathematical Analysis, 3(1) (2009), 99–110.
7. Li H. and Yang X., Products of integral-type and composition operators from generally weighted Bloch space to
F (p, q, s) space. Filomat, 23(3) (2009), 231–241.
8. Ohno S., Stroethoff K. and Zhao R., Weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math., 33(1) (2003), 191–215.
9. Ramey W. and Ulrich D., Bounded mean oscillation of Bloch pull-backs, Math. Ann., 291 (1991), 591–606.
10. Siskakis A. and Zhao R. A Volterra type operator on spaces of analytic functions, Contemp. Math, 232 (1999),
299–311.
11. Yoneda R., The composition operators on the weighted Bloch space, Arch. Math., 78 (2002), 310–317.
Haiying Li, College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China,
e-mail: tslhy2001@yahoo.com.cn
Tianshui Ma, College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China,
e-mail: matianshui@yahoo.com
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