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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(11),3835-3840
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1011407
\Bergman˜mBloch˜mVolterra
È©Žf
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\Bergman˜m§Bloch˜m§VolterraÈ©Žf
VolterraTypeOperatorsfromWeighted
BergmanSpacetotheBlochSpace
ErminWang,JiajiaXu
∗
SchoolofMathematicsandStatistics,LingnanNormalUniversity,ZhanjiangGuangdong
Received:Oct.15
th
,2021;accepted:Nov.5
th
,2021;published:Nov.17
th
,2021
Abstract
WeconsidertheboundednessandcompactnessofVolterratypeoperatorsfromthe
∗ÏÕŠö"
©ÙÚ^:¯,N[[.\Bergman˜mBloch˜mVolterraÈ©Žf[J].A^êÆ?Ð,2021,10(11):
3835-3840.DOI:10.12677/aam.2021.1011407
¯,N[[
weightedBergmanspacesinducedbyB´ekoll´efunctionstotheBlochSpace.
Keywords
WeightedBergmanSpaces,BlochSpace,VolterraTypeOperators
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
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p
L
p
ω
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Z
D
|f(z)|
p
ω(z)dA(z) <∞}.
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ž,eéu?¿Carleson•¬
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þ•3~êk>0,¦
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DOI:10.12677/aam.2021.10114073836A^êÆ?Ð
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g
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lim
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DOI:10.12677/aam.2021.10114073837A^êÆ?Ð
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0
>1,α∈(0,1),η>−1,g∈H(D),
ω
(1−|z|
2
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∈B
p
0
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g
:A
p
ω
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k.Žf…=
sup
z∈D
(1−|z|
2
)|g
0
(z)|
1

R
D
z,α
ωdA

1/p
<∞.
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g
: A
p
ω
→B´k.Žf.-
k
p,z
(w) =
K
γ+1
z
(w)
kK
γ+1
z
k
L
p
ω
,z∈D,
Ù¥γ≥(η+2)p
0
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k
L
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k
p,z
k
B
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k
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A
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(1−|z|
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0
(z)|
1
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z,α
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(1−|z|
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(z)|f(z)|.(1−|z|
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z,α
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ω
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g
: A
p
ω
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lim
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(1−|z|)|g
0
(z)|
1

R
D
z,α
ωdA

1/p
= 0.
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g
:A
p
ω
→B´;Žf.d u|z|→1ž, k
p,z
3D?¿;f8þ˜—
DOI:10.12677/aam.2021.10114073838A^êÆ?Ð
¯,N[[
Âñu0.l
0 =lim
|z|→1
kT
g
k
p,z
k
B
&lim
|z|→1
sup
w∈D
(1−|z|
2
)|g
0
(w)|k
p,z
(w)|
&lim
|z|→1
(1−|z|
2
)|g
0
(z)|k
p,z
(z)|
&lim
|z|→1
(1−|z|
2
)|g
0
(z)|
1

R
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z,α
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1/p
.
l,
lim
|z|→1
(1−|z|)|g
0
(z)|
1

R
D
z,α
ωdA

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= 0.
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(1−|z|
2
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0
(z)|
1

R
D
z,α
ωdA
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0
ž,k
(1−|z|
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(z)|
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0
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(z)|
1
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R
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z,α
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ld½n3.1•,T
g
: A
p
ω
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M:=sup
|w|≤r
0
(1−|w|
2
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0
(w)|<∞.
{f
n
}•A
p
ω
¥k.S,…3D;f8þ˜—Âñu0.K
kT
g
f
n
k
B
=sup
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(1−|w|
2
)|g
0
(w)|f
n
(w)|
=sup
|w|≤r
0
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n
(w)|+sup
|w|>r
0
(1−|w|
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n
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n
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0
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1
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R
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w,α
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kf
n
k
L
p
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n
(w)|+
Ïd,n→∞ž, kT
g
f
n
k
B
→0.¤±T
g
: A
p
ω
→B´k;Žf.
Ä7‘8
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DOI:10.12677/aam.2021.10114073839A^êÆ?Ð
¯,N[[
ë•©z
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deC
n
.StudiaMathematica,71,305-323.https://doi.org/10.4064/sm-71-3-305-323
[2]Constantin,O.(2007)DiscretizationsofIntegralOperatorsandAtomicDecompositionsin
Vector-ValuedWeightedBergmanSpaces.IntegralEquationsandOperatorTheory,59,523-
554.https://doi.org/10.1007/s00020-007-1536-7
[3]Constantin,O.(2010)CarlesonEmbeddingsandSomeClassesofOperatorsonWeighted
BergmanSpaces.JournalofMathematicalAnalysisandApplications,365,668-682.
https://doi.org/10.1016/j.jmaa.2009.11.035
[4]Pommerenke,C.(1977)SchlichteFunktionenundanalytischeFunktionenvonbeschr¨ankter
mittlererOszillation.CommentariiMathematiciHelvetici,52,591-602.
https://doi.org/10.1007/BF02567392
[5]Li,S.(2008)VolterraCompositionOperatorsbetweenWeightedBergmanSpacesandBloch
TypeSpaces.JournaloftheKoreanMathematicalSociety,45,229-248.
https://doi.org/10.4134/JKMS.2008.45.1.229
[6]–’¤,¯.l•êBergman˜mA
p
ψ
Bloch˜mVolterraÈ©Žf[J].nØêÆ,
2021(11):1643-1648.https://doi.org/10.12677/PM.2021.119182
DOI:10.12677/aam.2021.10114073840A^êÆ?Ð

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