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AdvancesinAppliedMathematics
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,2021,10(10),3446-3455
PublishedOnlineOctober2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1010363
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VolterraTypeOperatorsfrom
BergmanSpaces
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TypeSpaces
YechengShi,ErminWang
∗
SchoolofMathematicsandStatistics,LingnanNormalUniversity,ZhanjiangGuangdong
Received:Sep.21
st
,2021;accepted:Oct.18
th
,2021;published:Oct.25
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Abstract
WeconsidertheboundednessandcompactnessofVolterratypeoperatorsfromthe
Bergmanspaces
A
p
ω
withexponentialweightstotheBlochSpace.Weobtainthe
characterizationsoftheboundednessandcompactnessofVolterratypeintegralcom-
positionoperators.
Keywords
VolterraTyp eIntegral-CompositionOperators,BergmanSpaceswithDoubling
Weights,BlcohTypeSpaces,Boundedness,Compactness
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
|
ω
(
S
(
ϕ
(
z
))
−
1
p
.
k
T
g
f
a,p
k
B
<
∞
.
y
.
.
í
Ø
3.1
b
g
∈
H
(
D
),
ω
∈
b
D
,
K
T
g
:
A
p
ω
→B
α
´
k
.
Ž
f
…
=
sup
z
∈
D
(1
−|
z
|
)
α
|
g
0
(
z
)
|
ω
(
S
(
z
)
−
1
p
<
∞
.
½
n
3.2
b
g
∈
H
(
D
),
ψ
∈W
0
,
K
T
ϕ
g
:
A
p
ω
→B
α
´
k
;
Ž
f
…
=
lim
|
ϕ
(
z
)
|→
1
(1
−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
|
ω
(
S
(
ϕ
(
z
))
−
1
p
= 0
.
y
²
¿
©
5
:
b
T
ϕ
g
:
A
p
ω
→B
α
´
;
Ž
f
.
P
f
z,p
(
w
)=
ω
(
S
(
z
))
−
1
p
1
−|
z
|
2
1
−
zw
λ
+1
p
,
K
k
f
z,p
k
A
p
ω
1,
¿
…
|
z
|→
1
ž
,
f
z,p
3
D
þ
;
f
8
˜
—
Â
ñ
u
0.
0 =lim
|
ϕ
(
z
)
|→
1
k
T
ϕ
g
f
ϕ
(
z
)
,p
k
B
α
&
lim
|
ϕ
(
z
)
|→
1
sup
w
∈
D
(1
−|
w
|
2
)
α
|
(
g
◦
ϕ
)
0
(
w
)
||
f
ϕ
(
z
)
,p
(
w
)
|
&
lim
|
ϕ
(
z
)
|→
1
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−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
||
f
ϕ
(
z
)
,p
(
ϕ
(
z
))
|
&
lim
|
ϕ
(
z
)
|→
1
(1
−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
|
ω
(
S
(
ϕ
(
z
))
−
1
p
.
7
‡
5
:
b
lim
|
ϕ
(
z
)
|→
1
(1
−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
|
ω
(
S
(
ϕ
(
z
))
−
1
p
,
K
é
?
¿
>
0,
•
3
r
0
∈
(0
,
1),
¦
|
ϕ
(
z
)
|
>r
0
ž
,
k
(1
−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
|
ω
(
S
(
ϕ
(
z
))
−
1
p
<.
q
Ï
•
|
ϕ
(
z
)
|≤
r
0
ž
,
w
,
k
(1
−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
|
ω
(
S
(
ϕ
(
z
))
−
1
p
<
∞
k
.
.
l
d
½
n
3.1,
T
ϕ
g
:
A
p
ω
→
B
α
´
.
Ž
f
.
5
¿
g
◦
ϕ
∈B
α
,
•
3
ê
M
,
¦
M
:=sup
|
ϕ
(
z
)
|≤
r
0
(1
−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
|
<
∞
.
DOI:10.12677/aam.2021.10103633451
A^
ê
Æ
?
Ð
–
’
¤
,
¯
{
f
n
}
•
A
p
ω
¥
k
.
S
,
…
3
D
;
f
8
þ
˜
—
Â
ñ
u
0.
k
T
ϕ
g
f
n
k
B
α
=sup
z
∈
D
(1
−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
||
f
n
(
ϕ
(
z
))
|
=sup
|
ϕ
(
z
)
|≤
r
0
(1
−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
||
f
n
(
ϕ
(
z
))
|
+sup
|
ϕ
(
z
))
|
>r
0
(1
−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
||
f
n
(
ϕ
(
z
))
|
.
M
sup
|
w
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r
0
|
f
n
(
w
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|
+sup
|
ϕ
(
z
))
|
>r
0
(1
−|
z
|
2
)
α
|
(
g
◦
ϕ
)
0
(
z
)
|
S
(
ϕ
(
z
))
−
1
p
k
f
n
k
A
p
ω
.
M
sup
|
w
|≤
r
0
|
f
n
(
w
)
|
+
.
Ï
d
,
n
→∞
ž
,
k
T
g
f
n
k
B
α
→
0.
¤
±
T
ϕ
g
:
A
p
ω
→B
α
´
k
;
Ž
f
.
y
.
.
í
Ø
3.2
b
g
∈
H
(
D
),
ω
∈
b
D
,
K
T
g
:
A
p
ω
→B
α
´
;
Ž
f
…
=
lim
|
z
|→
1
(1
−|
z
|
2
)
α
|
g
0
(
z
)
|
ω
(
S
(
z
)
−
1
p
<
∞
.
½
n
3.3
0
<p<
∞
,0
<α<
∞
,
g
∈
H
(
D
)
…
ϕ
´
D
þ
)
Û
g
N
,
ω
∈
b
D
,
K
S
ϕ
g
:
A
p
ω
→B
α
´
k
.
Ž
f
…
=
sup
z
∈
D
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
|
g
(
ϕ
(
z
))
|
<
∞
.
y
²
¿
©
5
:
b
sup
z
∈
D
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
|
g
(
ϕ
(
z
))
|
<
∞
.
-
f
∈
A
p
ω
,
d
Ú
n
2.5,
k
(1
−|
z
|
2
)
α
|
g
(
ϕ
(
z
))
||
(
f
◦
ϕ
)
0
(
z
)
|
=(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
||
f
0
(
ϕ
(
z
))
|
.
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
k
f
k
A
p
ω
.
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
.
¤
±
k
S
ϕ
g
f
k
B
α
<
∞
.
S
ϕ
g
:
A
p
ω
→B
α
´
k
.
Ž
f
.
7
‡
5
:
b
S
ϕ
g
:
A
p
ω
→B
α
´
k
.
Ž
f
.
|
ϕ
(
z
)
|≥
1
2
ž
,
k
S
g
f
ϕ
(
z
)
,p
k
B
α
=sup
w
∈
D
(1
−|
w
|
2
)
α
|
g
(
ϕ
(
w
))
||
(
f
ϕ
(
z
)
,p
◦
ϕ
)
0
(
w
)
|
≥
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
f
0
ϕ
(
z
)
,p
(
ϕ
(
z
))
|
&
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
.
DOI:10.12677/aam.2021.10103633452
A^
ê
Æ
?
Ð
–
’
¤
,
¯
|
ϕ
(
z
)
|
<
1
2
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
.
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
.
k
S
ϕ
g
(
z
)
k
B
α
<
∞
.
y
.
.
í
Ø
3.3
0
<p<
∞
,0
<α<
∞
,
…
ϕ
´
D
þ
)
Û
g
N
,
ω
∈
b
D
,
K
C
ϕ
:
A
p
ω
→B
α
´
k
.
Ž
f
…
=
sup
z
∈
D
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
<
∞
.
í
Ø
3.4
0
<p<
∞
,0
<α<
∞
,
g
∈
H
(
D
),
ω
∈
b
D
,
K
S
g
:
A
p
ω
→B
α
´
k
.
Ž
f
…
=
sup
z
∈
D
(1
−|
z
|
2
)
α
−
1
ω
(
S
(
z
))
−
1
p
|
g
(
z
)
|
<
∞
.
½
n
3.4
0
<p<
∞
,0
<α<
∞
,
g
∈
H
(
D
)
…
ϕ
´
D
þ
)
Û
g
N
,
ω
∈
b
D
,
K
S
ϕ
g
:
A
p
ω
→B
α
´
;
Ž
f
…
=
lim
|
ϕ
(
z
)
|→
1
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
|
g
(
ϕ
(
z
))
|
= 0
.
y
²
¿
©
5
:
b
lim
|
ϕ
(
z
)
|→
1
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
|
g
(
ϕ
(
z
))
|
= 0
.
{
f
n
}
•
A
p
ω
¥
k
.
S
,
…
3
D
;
f
8
þ
˜
—
Â
ñ
u
0.
é
u
?
¿
½
0
<r<
1
.
d
Ú
n
2.4
Ú
Ú
n
2.5,
k
k
S
ϕ
g
f
n
k
B
α
=sup
z
∈
D
(1
−|
z
|
2
)
α
|
g
(
ϕ
(
z
))
||
(
f
n
◦
ϕ
)
0
(
z
)
|
=sup
|
ϕ
(
z
)
|≤
r
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
||
f
0
n
(
ϕ
(
z
))
|
+sup
|
ϕ
(
z
)
|
>r
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
||
f
0
n
(
ϕ
(
z
))
|
.
sup
|
ϕ
(
z
)
|≤
r
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
||
f
0
n
(
ϕ
(
z
))
|
+sup
|
ϕ
(
z
)
|
>r
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
k
f
n
k
A
p
ω
.
Ï
•
|
ϕ
(
z
)
|≤
r
ž
,(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
.
k
S
ϕ
g
(
z
)
k
B
α
<
∞
…
lim
n
→∞
sup
|
ϕ
(
z
)
|≤
r
|
f
n
(
ϕ
(
z
))
|
= 0
.
DOI:10.12677/aam.2021.10103633453
A^
ê
Æ
?
Ð
–
’
¤
,
¯
¤
±
-
n
→∞
,
r
→
1,
k
lim
n
→∞
k
S
ϕ
g
f
n
k
B
α
.
lim
|
ϕ
(
z
)
|→
1
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
= 0
.
S
ϕ
g
:
A
p
ω
→B
α
´
;
Ž
f
.
7
‡
5
:
b
S
ϕ
g
:
A
p
ω
→B
α
´
;
Ž
f
.
d
Ú
n
2.4,
Œ
0=lim
|
ϕ
(
z
)
|→
1
k
S
g
f
ϕ
(
z
)
,p
k
B
α
=lim
|
ϕ
(
z
)
|→
1
sup
w
∈
D
(1
−|
w
|
2
)
α
|
g
(
ϕ
(
w
))
||
(
f
ϕ
(
z
)
,p
◦
ϕ
)
0
(
w
)
|
≥
lim
|
ϕ
(
z
)
|→
1
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
f
0
ϕ
(
z
)
,p
(
ϕ
(
z
))
|
&
lim
|
ϕ
(
z
)
|→
1
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
||
g
(
ϕ
(
z
))
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
.
y
.
.
í
Ø
3.5
0
<p<
∞
,0
<α<
∞
,
…
ϕ
´
D
þ
)
Û
g
N
,
ω
∈
b
D
,
K
C
ϕ
:
A
p
ω
→B
α
´
;
Ž
f
…
=
lim
|
ϕ
(
z
)
|→
1
(1
−|
z
|
2
)
α
|
ϕ
0
(
z
)
|
(1
−|
ϕ
(
z
)
|
2
)
ω
(
S
(
ϕ
(
z
)))
1
p
= 0
.
í
Ø
3.6
0
<p<
∞
,0
<α<
∞
,
g
∈
H
(
D
),
ω
∈
b
D
,
K
S
g
:
A
p
ω
→B
α
´
;
Ž
f
…
=
lim
|
z
|→
1
(1
−|
z
|
2
)
α
−
1
ω
(
S
(
z
))
−
1
p
|
g
(
z
)
|
= 0
.
Ä
7
‘
8
Ø
©
É
I
[
g
,
‰
Æ
Ä
7
‘
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[1]Pommerenke,C.(1977)SchlichteFunktionenundanalytischeFunktionenvonbeschr¨ankter
mittlererOszillation.
CommentariiMathematiciHelvetici
,
52
,591-602.
https://doi.org/10.1007/BF02567392
[2]Li,S.(2008)VolterraCompositionOperatorsbetweenWeightedBergmanSpacesandBloch
TypeSpaces.
JournaloftheKoreanMathematicalSociety
,
45
,229-248.
DOI:10.12677/aam.2021.10103633454
A^
ê
Æ
?
Ð
–
’
¤
,
¯
[3]Pel´aez,J.andR¨atty¨a,J.(2014)WeightedBergmanSpacesInducedbyRapidlyIncreasing
Weights.AmericanMathematicalSociety,Providence.
[4]Pel´aez,J.andR¨atty¨a,J.(2015)EmbeddingTheoremsforBergmanSpacesviaHarmonic
Analysis.
MathematischeAnnalen
,
362
,205-239.
DOI:10.12677/aam.2021.10103633455
A^
ê
Æ
?
Ð