Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 780845, 14 pages
doi:10.1155/2008/780845
Research Article
On a Li-Stević Integral-Type Operators between
Different Weighted Bloch-Type Spaces
Yanyan Yu1 and Yongmin Liu2
1
2
School of Mathematics and Physics Science, Xuzhou Institute of Technology, Xuzhou 221008, China
Department of Mathematics, Xuzhou Normal University, Xuzhou 221116, China
Correspondence should be addressed to Yongmin Liu, minliu@xznu.edu.cn
Received 25 December 2007; Revised 16 June 2008; Accepted 19 November 2008
Recommended by Ulrich Abel
Let ϕ be an analytic self-map of the unit disk D, let HD be the space of analytic functions
g
on D, and let g ∈ HD. Recently, Li and Stević defined the following operator: Cϕ fz z f ϕwgwdw, on HD. The boundedness and compactness of the operator between two
0
weighted Bloch-type spaces are investigated in this paper.
Copyright q 2008 Y. Yu and Y. Liu. 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
First, we introduce some basic notation which is used in this paper. Throughout the entire
paper, the unit disk in the finite complex plane C will be denoted by D. HD will denote
the space of all analytic functions on D. Every analytic self-map ϕ of the unit disk D induces
through composition a linear composition operator Cϕ from HD to itself. It is a well-known
consequence of Littlewood’s subordination principle 1 that the formula Cϕ f f ◦ ϕ
defines a bounded linear operator on the classical Hardy and Bergman spaces. That is,
Cϕ : H p → H p and Cϕ : Ap → Ap are bounded operators. A problem that has received
much attention recently is to relate function theoretic properties of ϕ to operator theoretic
properties of the restriction of Cϕ to various Banach spaces of analytic functions. Some
characterizations of the boundedness and compactness of the composition operator between
various Banach spaces of analytic functions can be found in 2–6. Recently, Yoneda in 7
gave some necessary and sufficient conditions for a composition operator Cϕ to be bounded
and compact on the logarithmic Bloch space defined as follows:
Blog f ∈ HD : f sup 1 − |z|2 log
z∈D
2
f z < ∞ .
1 − |z|2
1.1
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Journal of Inequalities and Applications
The space Blog is a Banach space under the norm fBlog |f0| f. Ye in 8 characterized
the boundedness and compactness of the weighted composition operator uCϕ between the
logarithmic Bloch space Blog and the α-Bloch space Bα on the unit disk as well as the
boundedness and compactness of the weighted composition operator uCϕ between the little
logarithmic Bloch space B0log and the little α-Bloch space Bα0 on the unit disk. A function
f ∈ HD is said to belong to the Bloch-type space or α-Bloch space, denoted by Bα Bα D,
if
α Bα f sup 1 − |z|2 f z < ∞.
1.2
z∈D
The space Bα becomes a Banach space with the norm fα |f0| Bα f. Let Bα0 denote the
subspace of Bα consisting of those f ∈ Bα such that
α lim 1 − |z|2 f z 0.
1.3
|z| → 1
This space is called the little Bloch-type space. For α 1, we obtain the well-known classical
Bloch space and the little Bloch space, simply denoted by B and B0 . Let B0log denote the
subspace of Blog consisting of those f ∈ Blog such that
lim 1 − |z|2 log
|z| → 1
2
f z 0.
2
1 − |z|
1.4
Ye in 9 proved that B0log is a closed subspace of Blog . Galanopoulos in 10 characterized
p
the boundedness and compactness of the composition operator Cϕ : Blog → Qlog and
the boundedness and compactness of the weighted composition operator uCϕ : Blog →
Blog . Some characterizations of the weighted composition operator between various Blochtype spaces can be found in 11–16. Li and Stević in 17 studied the boundedness and
compactness of the following two Volterra-type integral operators:
Jg fz Ig fz z
0
z
fξg ξdξ,
1.5
f ξgξdξ,
0
on the Zygmund space, for any g ∈ HD. Li and Stević in 18, for f, g ∈ HD, defined a
linear operator as follows:
g
Cϕ fz
z
f ϕw gwdw.
1.6
0
They called the operator the generalized composition operator and studied the boundedness
and compactness of the operator on the Zygmund space, the Bloch-type space Bα , and the
little Bloch-type space Bα0 . They also studied the weak compactness of the operator on the little
Y. Yu and Y. Liu
3
ϕ
Bloch-type space. When gw ϕ w, we see that the operator Cϕ -Cϕ is a point evaluation
g
operator. The operator Cϕ is closely related with some integral operators in papers 19–26.
Our goal here is to characterize the boundedness and compactness of the Li-Stević integraltype operator between different weighted Bloch-type spaces.
Throughout this paper, the letter C denotes a positive constant which may vary at each
occurrence but it is independent of the essential variables.
2. Preliminary material
In order to prove the main results, we need the following lemmas.
Lemma 2.1. There exist two functions f1 , f2 ∈ Blog such that
1−
|z|2
1
log 2/ 1 −
|z|2
≤ C f1 z f2 z ,
z ∈ D.
2.1
Proof. For each z ∈ D, it is easy to see that 1 ≤ 1 |z| < 2 and 0 < 1 − |z|2 ≤ 1. So
1 − |z| log
2
2
≤ 1 − |z|2 log
1 − |z|
1 − |z|
≤ 1 − |z|2 log
4
1 − |z|2
2.2
≤ 1 − |z|2 log 4
2 1 − |z|2 log
2
.
1 − |z|2
1 − |z|2
2
According to 10, Lemma 3.1, there exist two functions f1 , f2 ∈ Blog such that
1
≤ C f1 z f2 z ,
1 − |z| log 2/ 1 − |z|
z ∈ D.
2.3
From 2.2 and 2.3, the lemma follows.
Lemma 2.2 see 8. Let fz 1 − |z| log2/1 − |z|/|1 − z| log4/|1 − z|, z ∈ D, then
|fz| < 2.
Lemma 2.3 see 8. Let f ∈ Blog , then ft Blog ≤ CfBlog , where ft z ftz, 0 < t < 1.
Lemma 2.4 see 8. Suppose 0 ≤ t ≤ 1. Let fz, t 1−|z| log2/1−|z|/1−|tz| log2/1−
|tz|, z ∈ D, then |fz, t| < 2.
g
3. The boundedness of Cϕ : Blog or B0log → Bα or Bα0 g
In this section, we study the boundedness of Cϕ : Blog or B0log → Bα or Bα0 .
4
Journal of Inequalities and Applications
g
Theorem 3.1. Suppose 0 < α < ∞, ϕ is an analytic self-map of D, and g ∈ HD. Then Cϕ : Blog →
Bα is bounded if and only if
α 1 − |z|2 gz
sup 2 2 < ∞.
z∈D 1 − ϕz
log 2/ 1 − ϕz
3.1
Proof. We first prove that the condition is sufficient. Suppose
α 1 − |z|2 gz
M sup 2 2 < ∞.
z∈D 1 − ϕz
log 2/ 1 − ϕz
3.2
Then, for z ∈ D and f ∈ Blog , we have
1 − |z|2 α Cϕg f z 1 − |z|2 α f ϕz gz
α 1 − |z|2 gz
≤ Cf 2 2 1 − ϕz log 2/ 1 − ϕz
3.3
≤ CMfBlog < ∞,
thus
g Cϕ f Cϕg f0 Bα Cϕg f ≤ CMfB ,
log
α
3.4
g
so Cϕ : Blog → Bα is bounded.
Conversely, using Lemma 2.1, there exist two functions f1 , f2 ∈ Blog , satisfying
1−
|z|2
1
≤ C f1 z f2 z .
2
log 2/ 1 − |z|
3.5
Setting z ϕw in the above inequality, we obtain
1
2 2 ≤ C f1 ϕw f2 ϕw .
1 − ϕw log 2/ 1 − ϕw
3.6
Hence,
α 1 − |w|2 gw
2 α
2 2 ≤ C 1 − |w| gw f1 ϕw f2 ϕw 1 − ϕw log 2/ 1 − ϕw
3.7
g g ≤ C Cϕ f1 α Cϕ f2 α .
Y. Yu and Y. Liu
5
g
g
g
Since f1 , f2 ∈ Blog , and Cϕ : Blog → Bα is bounded, then Cϕ f1 and Cϕ f2 are in Bα . So the
supremum over w ∈ D in 3.7 is finite, which implies that
α 1 − |z|2 gz
sup 2 2 < ∞,
z∈D 1 − ϕz log 2/ 1 − ϕz
3.8
as desired.
g
Theorem 3.2. Suppose 0 < α < ∞, ϕ is an analytic self-map of D, and g ∈ HD. Then Cϕ : B0log →
Bα is bounded if and only if 3.1 holds.
g
Proof. If 3.1 holds, from Theorem 3.1, Cϕ : Blog → Bα is bounded, which along with the fact
g
B0log ⊂ Blog implies Cϕ : B0log → Bα is bounded.
g
Conversely, assume that Cϕ : B0log → Bα is bounded. For w ∈ D, put
fw z z
1 − wζ
−1
log
0
4
1 − wζ
−1
dζ.
3.9
By the inequality 1/1 |z| < 2/1 − |z|, z ∈ D, Lemmas 2.2 and 2.4, we get
2
fw z
2
1 − |z|
1 − |z|2 log 2/ 1 − |z|2
sup
z∈D |1 − wz| log 4/1 − wz
1 − |z|2 log 2/ 1 − |z|2
1 − |wz| log 2/ 1 − |wz|
≤ sup × sup
z∈D 1 − |wz| log 2/ 1 − |wz|
z∈D |1 − wz| log 4/1 − wz
1 − |z| log 2/ 1 − |z|
1 − |u| log 2/ 1 − |u|
≤ 4 sup × sup
|1 − u| log 4/|1 − u|
z∈D 1 − |wz| log 2/ 1 − |wz|
u∈D
sup 1 − |z|2 log
z∈D
3.10
≤ 16,
so fw Blog ≤ 16. Since
lim 1 − |z|
|z| → 1
2
1 − |z|2 log 2/ 1 − |z|2
2
fw z lim
log
|z| → 1 |1 − wz| log 4/1 − wz 1 − |z|2
1 − |z|2 log 2/ 1 − |z|2
≤ lim
|z| → 1
1 − |w| log 2
0,
3.11
6
Journal of Inequalities and Applications
we see that fw ∈ B0log . Thus, for w ∈ D,
α α 1 − |w|2 gw
1 − |w|2 gw
2 2 ≤ C 2 2 1 − ϕw log 2/ 1 − ϕw
1 − ϕw log 4/ 1 − ϕw
α C 1 − |w|2 fϕw
ϕw gw
g
g ≤ CCϕ fϕw α ≤ CCϕ fϕw Blog
g
≤ CCϕ < ∞,
3.12
which gives
α g
1 − |w|2 gw
sup 2 2 ≤ CCϕ < ∞,
w∈D 1 − ϕw
log 2/ 1 − ϕw
3.13
finishing the proof of the theorem.
g
Theorem 3.3. Suppose 0 < α < ∞, ϕ is an analytic self-map of D, and g ∈ HD. Then Cϕ : Blog →
Bα0 is bounded if and only if
α 1 − |z|2 gz
lim 0.
|z| → 1 1 − ϕz2 log 2/ 1 − ϕz2
3.14
g
Proof (Necessity). If Cϕ : Blog → Bα0 is bounded, we use the fact that for each function f ∈ Blog ,
g
the analytic function Cϕ f ∈ Bα0 . Then using the functions of Lemma 2.1, we get the following:
α 1 − |w|2 gw
2 α
2 2 ≤ C 1 − |w| gw f1 ϕw f2 ϕw 1 − ϕw log 2/ 1 − ϕw
g α g ≤ C 1 − |w|2 Cϕ f1 w Cϕ f2 w
−→ 0 as |w| −→ 1 ,
3.15
hence, 3.14 holds.
Sufficiency. For f ∈ Blog , we have
1 − |z|2 α Cϕg f z 1 − |z|2 α f ϕz gz
α
1 − |z|2 |gz|
≤ CfBlog 2 2 1 − ϕz log 2/ 1 − ϕz
−→ 0 as |z| −→ 1 ,
3.16
Y. Yu and Y. Liu
7
g
g
thus, Cϕ f ∈ Bα0 . Since 3.14 implies 3.1, by Theorem 3.1, Cϕ : Blog → Bα is bounded. Using
g
these two facts, we obtain that Cϕ : Blog → Bα0 is bounded. The proof is complete.
g
Theorem 3.4. Let 0 < α < ∞, g ∈ HD, and ϕ be an analytic self-map of D. Then Cϕ : B0log → Bα0
g
α
is bounded if and only if Cϕ : Blog → Bα is bounded and lim|z| → 1 1 − |z|2 |gz| 0.
g
Proof. If Cϕ : B0log → Bα0 is bounded, then for f ∈ Blog , ft ∈ B0log 0 < t < 1. From Lemma 2.3,
we have
g
g Cϕ ft ≤ Cϕg ft ≤ CCϕ fBlog ,
α
Blog
3.17
from this, it follows that
g
α 1 − |z|2 tf tϕz gz ≤ CCϕ fBlog ,
z ∈ D.
3.18
Letting t → 1 and taking the supremum in the above inequality over z ∈ D, we obtain that
g Cϕ f ≤ CCϕg fB ,
log
α
g
3.19
g
thus Cϕ : Blog → Bα is bounded. Since fz z is in B0log , the boundedness of Cϕ : B0log → Bα0
implies that
α α g lim 1 − |z|2 gz lim 1 − |z|2 Cϕ f z 0.
|z| → 1
|z| → 1
3.20
For the converse, by Theorem 3.1,
α 1 − |z|2 gz
M1 sup 2 2 < ∞.
z∈D 1 − ϕz
log 2/ 1 − ϕz
3.21
For any f ∈ B0log , we have
lim 1 − |z|2 log
|z| → 1
2
f z 0,
2
1 − |z|
3.22
so that for any > 0, there exists a δ1 ∈ 0, 1, such that when δ1 < |z| < 1,
1 − |z|
2
2
f z < .
log
2M1
1 − |z|2
3.23
8
Journal of Inequalities and Applications
Hence, when |ϕz| > δ1 , we get
1 − |z|2 α Cϕg f z 1 − |z|2 α f ϕz gz
α 1 − |z|2 gz
<
2M1 1 − ϕz2 log 2/1 − ϕz2 <
3.24
.
2
We know that there exists a constant M2 such that |f ϕz| ≤ M2 , for any z belonging to the
α
set {z ∈ D : |ϕz| ≤ δ1 }. Since lim|z| → 1 1 − |z|2 |gz| 0, there exists a δ ∈ 0, 1, such that
2 α
when δ < |z| < 1, 1 − |z| |gz| < /2M2 , then
1 − |z|2 α Cϕg f z 1 − |z|2 α f ϕz gz
α ≤ M2 1 − |z|2 gz
3.25
< ,
2
thus, we get that for z ∈ D1 {z ∈ D : δ < |z| < 1},
α g sup 1 − |z|2 Cϕ f z z∈D1
{z∈D1 :|ϕz|>δ1 }
<
g
α g 1 − |z|2 Cϕ f z
sup
sup
{z∈D1 :|ϕz|≤δ1 }
α g 1 − |z|2 Cϕ f z
3.26
,
2 2
g
so that Cϕ f ∈ Bα0 , that is, Cϕ : B0log → Bα0 is bounded. The proof of Theorem 3.4 is complete.
g
4. The compactness of Cϕ : Blog or B0log → Bα or Bα0 g
In this section, we characterize the compactness of Cϕ : Blog or B0log → Bα or Bα0 . For this
purpose, we start this section by stating some useful lemmas. By standard arguments see,
e.g., 2, the following lemmas follow.
Lemma 4.1. Suppose 0 < α < ∞, ϕ is an analytic self-map of D, and g ∈ HD. Let X Blog or
g
g
B0log , Y Bα or Bα0 . Then Cϕ : X → Y is compact if and only if Cϕ : X → Y is bounded and for any
bounded sequence {fn } in X which converges to zero uniformly on compact subsets of D as n → ∞,
g
one has Cϕ fn Y → 0 as n → ∞.
Lemma 4.2. Let 0 < α < ∞. A closed set K in Bα0 is compact if and only if K is bounded and satisfies
α lim sup 1 − |z|2 f z 0.
|z| → 1 f∈K
4.1
Y. Yu and Y. Liu
9
The proof of Lemma 4.2 is similar to 27, Lemma 2.1 see, also 28 and is omitted.
We begin with the following necessary and sufficient condition for the compactness of
g
Cϕ : B0log → Bα0 .
Theorem 4.3. Suppose 0 < α < ∞, ϕ is an analytic self-map of D, and g ∈ HD. Then the following
statements are equivalent:
g
1 Cϕ : Blog → Bα0 is compact;
g
2 Cϕ : B0log → Bα0 is compact;
3
α 1 − |z|2 gz
lim 0.
|z| → 1 1 − ϕz2 log 2/ 1 − ϕz2
4.2
Proof. 1 ⇒ 2 is obvious.
g
2 ⇒ 3 Since Cϕ : B0log → Bα0 is compact, we obtain by Lemma 4.2
lim
sup
|z| → 1 f
α g 1 − |z|2 Cϕ f z 0.
Blog ≤1
4.3
Thus, for any > 0, there exists a δ ∈ 0, 1, such that when δ < |z| < 1,
sup
α g 1 − |z|2 Cϕ f z < .
C
fBlog ≤1
4.4
Let fw be defined by 3.9. It is easy to see that
1
1
≤ .
C fw B
log
4.5
Set hw fw /fw , then for δ < |z| < 1, w ϕz,
α 1 − |z|2 gz
2 α 2 2 ≤ C 1 − |z| hw ϕz gz
1 − ϕz log 2/ 1 − ϕz
α g C 1 − |z|2 Cϕ hw z
α g ≤ C sup 1 − |z|2 Cϕ f z < ,
4.6
fBlog ≤1
which gives that
α 1 − |z|2 gz
lim 0.
|z| → 1 1 − ϕz2 log 2/ 1 − ϕz2
4.7
10
Journal of Inequalities and Applications
3 ⇒ 1 For any bounded sequence {fn } in Blog with fn → 0 uniformly on compact
subsets of D, we must prove that by Lemma 4.1
g Cϕ fn −→ 0
α
as n −→ ∞.
4.8
We assume that fn Blog ≤ 1. From 4.2, given > 0, there exists a δ ∈ 0, 1, when δ < |z| < 1,
α 1 − |z|2 gz
2 2 < ,
2
1 − ϕz log 2/ 1 − ϕz
4.9
then using 4.9, we get for δ < |z| < 1,
1 − |z|2 α Cϕg fn z 1 − |z|2 α fn ϕz gz < .
2
4.10
Since {fn } converges uniformly to 0 on a compact subset {ϕz : |z| ≤ δ} of D and there exists
α
a constant M3 such that sup|z|≤δ 1 − |z|2 |gz| ≤ M3 , we see that there exists an N > 0, such
that for all n ≥ N,
supfn ϕz <
|z|≤δ
.
2M3
4.11
Therefore, for all n ≥ N, |z| ≤ δ,
1 − |z|2 α Cϕg fn z 1 − |z|2 α fn ϕz gz < ε .
2
4.12
g
Note that |Cϕ fn 0| 0. Combining 4.10 and 4.12, we obtain
g Cϕ fn −→ 0
α
as n −→ ∞.
4.13
The proof is complete.
By Theorems 3.3 and 4.3, we get the following corollary.
g
Corollary 4.4. Suppose 0 < α < ∞, ϕ is an analytic self-map of D, and g ∈ HD. Then Cϕ : Blog →
g
Bα0 is bounded if and only if Cϕ : Blog → Bα0 is compact.
g
Theorem 4.5. Suppose 0 < α < ∞, ϕ is an analytic self-map of D, and g ∈ HD. Then Cϕ : Blog →
g
Bα is compact if and only if Cϕ : Blog → Bα is bounded and
α 1 − |z|2 gz
lim 0.
|ϕz| → 1 1 − ϕz2 log 2/ 1 − ϕz2
4.14
Y. Yu and Y. Liu
11
Proof. Suppose that 4.14 is true. For any sequence {fn } in Blog such that fn Blog ≤ 1 and
fn → 0 uniformly on compact subsets of D, it is required to show that by Lemma 4.1,
g Cϕ fn −→ 0
α
as n −→ ∞.
4.15
From 4.14, we have that for every > 0, there exists a δ ∈ 0, 1, such that δ < |ϕz| < 1
implies
α 1 − |z|2 gz
2 2 < ,
2
1 − ϕz log 2/ 1 − ϕz
4.16
then using 4.16, we get for δ < |ϕz| < 1,
1 − |z|2 α Cϕg fn z 1 − |z|2 α fn ϕz gz
α 1 − |z|2 gz
≤
2 2 1 − ϕz log 2/ 1 − ϕz
<
4.17
.
2
g
α
Since Cϕ : Blog → Bα is bounded, taking fz z, we see that L supz∈D 1 − |z|2 |gz| < ∞.
Let U {w ∈ D : |w| ≤ δ}, since {fn } converges uniformly to 0 on a compact subset U of D,
then there exists an N > 0, such that for all n ≥ N,
.
supfn w <
2L
w∈U
4.18
Therefore, for all n ≥ N,
sup
{|ϕz|≤δ}
1 − |z|2
g
Cϕ fn z α sup
α 1 − |z|2 fn ϕz gz
{|ϕz|≤δ}
≤ L supfn w
w∈U
<
4.19
.
2
g
Note that |Cϕ fn 0| 0. Combining 4.17 and 4.19, we obtain
g Cϕ fn −→ 0
α
as n −→ ∞.
4.20
12
Journal of Inequalities and Applications
g
g
Conversely, suppose that Cϕ : Blog → Bα is compact, then Cϕ : Blog → Bα is bounded.
Hence, we only need to prove that 4.14 holds. Assume to the contrary that there is a positive
number 0 and a sequence {zn } in D such that limn → ∞ |ϕzn | 1, and
2 α 1 − zn g zn 2 2 ≥ 0 ,
1 − ϕ zn log 2/ 1 − ϕ zn 4.21
for all n. For each n, writing ϕzn rn eiθn , we choose the test functions fn defined by
fn z z
rn
1 − e−iθn rn w
0
−
rn2
log
1 − e−iθn rn2 w
−1
4
dw,
1 − e−iθn rn2 w
4.22
then
fn z rn2
rn
−
1 − e−iθn rn z 1 − e−iθn rn2 z
log
4
1 − e−iθn rn2 z
−1
,
4.23
thus, |fn z| ≤ 1 − rn /1 − |z|2 log4/1 − |z|−1 , we see that fn converges to zero
uniformly on compact subsets of D as n → ∞. Using Lemmas 2.2 and 2.4, we have
fn Blog ≤ C. In view of Lemma 4.1, it follows that
g Cϕ fn −→ 0
α
as n −→ ∞.
4.24
Since
g α Cϕ fn ≥ 1 − zn 2 fn ϕ zn g zn α
−1
2 α rn2
rn
4
−
1 − zn g zn log
1 − rn2 1 − rn3
1 − rn3
2 α 1 − zn ϕ zn g zn ≥ C
2 3 1 − ϕ zn log 4/ 1 − ϕ zn 4.25
2 α 1 − zn ϕ zn g zn ≥ C
2 2 ,
1 − ϕ zn log 2/ 1 − ϕ zn α
and |ϕzn | → 1 as n → ∞, we obtain limn → ∞ 1 − |zn |2 |gzn |/1 − |ϕzn |2 log2/1 −
|ϕzn |2 0, which is a contradiction with 4.21. Hence, we are done.
Similarly, we can obtain the following result. The proof of the following theorem will
be omitted.
g
Theorem 4.6. Suppose 0 < α < ∞, ϕ is an analytic self-map of D, g ∈ HD, and Cϕ : Blog → Bα
g
is bounded. Then Cϕ : B0log → Bα is compact if and only if 4.14 holds.
Y. Yu and Y. Liu
13
Acknowledgments
This project is supported by the National Natural Science Foundation of China 10471039
and the Grant of Higher Schools’ Natural Science Basic Research of Jiangsu Province of
China 06KJD110175, 07KJB110115. The second author thanks Professors Shaozong Yan
and Xiaoman Chen for their encouragement and help while visiting Fudan University. The
authors also wish to thank the referees for carefully reading the manuscript and making
several useful suggestions which greatly improved the final version of this paper.
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p
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