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PureMathematicsnØêÆ,2022,12(12),2133-2140
PublishedOnlineDecember2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.1212229
\Bergman˜mþ2ÂVolterra.
È©Žf
–––’’’¤¤¤
*H“‰Æ§2ÀÖô
ÂvFϵ2022c1124F¶¹^Fϵ2022c1222F¶uÙFϵ2022c1229F
Á‡
Cc5)Û¼ê˜mþ2ÂVolterra.È©Žfk.5Ú ;5Úå¯õÆö,",§
\Bergman˜mþ2ÂVolterra.ŽfïÄÿ™õ"Ø©?Ø\Bergman˜
mƒm2ÂVolterra.È©Žfk.5Ú;5¯K§|^BergmanCarlesonÿÝÚ
Littlewo od-Paleyúª‰Ñ \Bergman˜mƒm2ÂVolterra.È©Žfk.5Ú
;5•x§õ\Bergman˜mþ2ÂVolterra.Žf5Ÿ"
'…c
Bergman˜m§2ÂVolterra.È©Žf§k.5§;5
GeneralizedVolterraTypeIntegral
OperatorsbetweenWeighted
BergmanSpaces
YechengShi
SchoolofMathematicsandStatistics,LingnanNormalUniversity,ZhanjiangGuangdong
Received:Nov.24
th
,2022;accepted:Dec.22
nd
,2022;published:Dec.29
th
,2022
©ÙÚ^:–’¤.\Bergman˜mþ2ÂVolterra.È©Žf[J].nØêÆ,2022,12(12):2133-2140.
DOI:10.12677/pm.2022.1212229
–’¤
Abstract
Inrecentyears,theboundednessandcompactnessofgeneralizedVolterra-typeinte-
graloperatorsonanalyticfunctionspaceshaveattractedmanyscholars’interests,but
thestudyofthegeneralizedVolterra-typ eintegraloperatorsonweightedBergman
spaceisnotyetcomplete.Inthispaper,weconsidertheb oundednessandcom-
pactnessofGeneralizedVolterratypeintegraloperatorsbetweenweightedBergman
spaces.UsingtheBergmanCarlesonmeasureandLittlewood-Paleyformula,wechar-
acterizedtheboundednessandcompactnessofgeneralizedVolterra-typeoperators
weightedBergmanspace,andthepropertiesofgeneralizedVolterra-typeoperators
werefurtherimproved.
Keywords
BergmanSpaces,GeneralizedVolterraTypeIntegralOperators,Boundedness,
Compactness
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
D´E²¡Cþü ,=D:={z∈C:|z|<1}.H(D)´Dþ)Û¼êN.
0 <p<∞,−1 <α<∞ž,\Bergman˜mA
p
α
½Â•¤kL
p
(D,dA
α
)‰êk•)Û¼ê
N,=µ
A
p
α
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p
A
p
α
:=
Z
|f(z)|
p
dA
α
(z) <∞}.
ùpdA
α
(z) =
1
α+1
(1−|z|
2
)
α
dA(z),dA(z) =
1
π
dxdy´DþIOLebaneseÿÝ.
ϕ∈H(D) …ϕ•ü Dþ)ÛgN,EÜŽfC
ϕ
½Â•
(C
ϕ
f)(z) = f(ϕ(z)),z∈D,f∈H(D).
\EÜŽfuC
ϕ
½Â•
DOI:10.12677/pm.2022.12122292134nØêÆ
–’¤
uC
ϕ
(f)(z) = u(z)f(ϕ(z)), z∈D,f∈H(D).
ùpabL«•3~êC>0 ¦C
−1
b≤a≤Cb.
g∈H(), ½ÂVolterraÈ©ŽfT
g
•
T
g
f(z) =
Z
z
0
f(ζ)g
0
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g
•
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f)(z) =
Z
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0
f
0
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g
3Hardy˜mH
2
k.5¯K.3d ƒ,N
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ϕ
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ϕ
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g
, T
g,ϕ
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x¾,ÿ™ïÄ.©·‚̇•Ä\Bergman˜mA
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p
α
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kfk
A
p
α
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0
k
A
p
α+p
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½Â2.2b0<p,q<∞,−1<α<∞,Pλ:=
p
q
.µ•ü DþBorel ÿÝ,ei
\N
I: A
p
α
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{f
n
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n
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n
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lim
n→∞
kf
n
k
L
q
(dµ)
= 0,
·‚K¡µ´ž«(λ,α)−BergmanCarleson ÿÝ.
z∈D, 0 <r<1, ^4(z,r) L«±z•%,r•Œ»[V-. ·‚I‡e¡¯¤±
•Bergman ˜mi\ŽfCarleson.ÿÝ•x,Œ„©z[11].
Ún2.3b−1 <α<∞,0 <p,q<∞,-µ•DþBorel ÿÝ,K
(1)0 <p≤q<∞ž,
DOI:10.12677/pm.2022.12122292135nØêÆ
–’¤
(a)I: A
p
α
→L
q
(dµ)´k.Žf…=
sup
z∈D
µ(4(z,r))
(1−|z|
2
)
q
p
(α+2)
<∞.
(b)I: A
p
α
→L
q
(dµ)´;Žf…=
lim
|z|→1
µ(4(z,r))
(1−|z|
2
)
q
p
(α+2)
= 0.
(2)0 <q<p<∞ž,
1
s
=
1
q
−
1
p
.I: A
p
α
→L
q
(dµ)´k.(;)Žf…=
µ(4(z,r))
(1−|z|
2
)
α+2
∈L
s
(dA
α
).
5µ½Â‡©ŽfD•Df=f
0
. dÚn2.1,‡©ŽfD: A
p
α
→L
q
(µ)k.(;)…=i\Ž
fI: A
p
α+p
→L
q
(µ)k.(;).
b−1 <α<∞, ϕ´ü Dþ)ÛgN,½ÂNevanlinna.Oê¼êN
ϕ,α+2
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P
z∈ϕ
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1
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),ùpw∈ϕ(D) …w6=ϕ(0).¿-N
ϕ,α+2
(w)=0,w/∈ϕ(D).dÚn2.1,·‚ê
þke¡Ún.
Ún2.4b−1 <α<∞, ϕ: D→D´)ÛN,g∈B,f∈A
2
α
.K
kT
ϕ
g
fk
2
A
2
α
|(T
g
f)(ϕ(0))|
2
+
Z
D
|f(w)|
2
|g
0
(w)|N
ϕ,α+2
(w)dA(w).
3.k.5Ú;5•x
|^±þÚn,·‚y3y²©̇½n.
½n3.1b0<p,q<∞, −1 <α,β<∞.bg,ϕ∈H(),ϕ(D)⊂D,½Âü Dþ
Borelÿݵ
β
g,ϕ
•
µ(E) := µ
q,β
g,ϕ
(E) =
Z
ϕ
−1
(E)
|(g◦ϕ)
0
(z)|
q
dA
β+q
(z),
ùpE•Dþ?˜Borel8.KT
ϕ
g
:A
p
α
→A
q
β
´k.(;)Žf…=µ´(ž«)(
p
q
,α)-
BergmanCarlesonÿÝ.=µ
(1)0 <p≤q<∞ž,
(a)T
ϕ
g
: A
p
α
→A
q
β
´k.Žf…=
sup
z∈
µ(4(z,r))
(1−|z|
2
)
α+2
<∞.
(b)T
ϕ
g
: A
p
α
→A
q
β
´;Žf…=
DOI:10.12677/pm.2022.12122292136nØêÆ
–’¤
lim
|z|→1
µ(4(z,r))
(1−|z|
2
)
α+2
<∞.
(2)0 <q<p<∞ž,Ps=
pq
p−q
.KT
ϕ
g
: A
p
α
→A
q
β
´k.(;)Žf…=
µ(4(z,r))
(1−|z|
2
)
α+2
∈L
s
(D,dA
α
)
y²:T
ϕ
g
: A
p
α
→A
q
β
´k.Žfdu
kT
ϕ
g
fk
A
q
β
≤Ckfk
A
p
α
.
Ø”ϕ(0) = 0, KdÚn2.1
kT
ϕ
g
fk
q
A
q
β
=





Z
ϕ(z)
0
f(ζ)g
0
(ζ)dζ





q
A
q
β
k(f◦ϕ))(g◦ϕ)
0
k
q
A
q
β+q

Z
D
|f(ϕ(z))|
q
|(g◦ϕ)
0
(z)|
q
dA
β+q
=
Z
D
|f(z)|
q
dµ
β
g,ϕ
(z).
·‚kT
ϕ
g
: A
p
α
→A
q
β
´k.(;)Žf…=i\ŽfI: A
p
α
→L
q
(dµ)´k.(;)Žf.Ï
d,dÚn2.3,½ny.
½n3.2b0<p,q<∞, −1 <α,β<∞.bg,ϕ∈H(),ϕ(D)⊂D,½Âü Dþ
Borelÿݵ•
µ(E) =
Z
ϕ
−1
(E)
|g
0
(z)|
q
dA
β+q
(z),
ùpE•Dþ?˜Borel 8.KT
g,ϕ
:A
p
α
→A
q
β
´k.(;)Žf…=µ´(ž«) (
p
q
,α)-
BergmanCarlesonÿÝ.
y²:T
g,ϕ
: A
p
α
→A
q
β
´k.Žfdu
kT
g,ϕ
fk
A
q
β
≤Ckfk
A
p
α
.
K
kT
g,ϕ
fk
q
A
q
β
=




Z
z
0
f(ϕ(ζ))g
0
(ζ)dζ




q
A
q
β
k(f◦ϕ))g
0
k
q
A
q
β+q

Z
D
|f(ϕ(z))|
q
|g
0
(z)|
q
dA
β+q
=
Z
D
|f(z)|
q
dµ(z).
DOI:10.12677/pm.2022.12122292137nØêÆ
–’¤
·‚kT
g,ϕ
:A
p
α
→A
q
β
´k.(;)Žf…=i\ŽfI:A
p
α
→L
q
(dµ)´k.(;)Žf.
Ïd,dÚn2.3,½ny.
½n3.3b0<p,q<∞, −1 <α,β<∞.bg,ϕ∈H(),ϕ(D)⊂D,½Âü Dþ
Borelÿݵ•
µ(E) =
Z
ϕ
−1
(E)
|g(ϕ(z))ϕ
0
(z)|
q
dA
β+q
(z),
ùpE•Dþ?˜Borel 8.KS
ϕ
g
:A
p
α
→A
q
β
´k.(;)Žf…=µ´(ž«) (
p
q
,α+p)-
BergmanCarlesonÿÝ.
y²:S
ϕ
g
: A
p
α
→A
q
β
´k.Žfdu
kS
ϕ
g
fk
A
q
β
≤Ckfk
A
p
α
.
Ø”ϕ(0) = 0, K
kS
ϕ
g
fk
q
A
q
β
=





Z
ϕ(z)
0
f
0
(ζ)g(ζ)dζ





q
A
q
β
k(f◦ϕ)
0
(g◦ϕ)k
q
A
q
β+q

Z
D
|(f◦ϕ)
0
(z))|
q
|g(ϕ(z))|
q
dA
β+q
=
Z
D
|f
0
(z)|
q
dµ(z).
·‚kS
ϕ
g
: A
p
α
→A
q
β
´k.(;)Žf…=‡©ŽfD: A
p
α
→L
q
(dµ)´k.(;)Žf.Ï
d,dÚn2.39Ù5, ½ny.
½n3.4b0<p,q<∞, −1 <α,β<∞.bg,ϕ∈H(),ϕ(D)⊂D,½Âü Dþ
Borelÿݵ•
µ(E) =
Z
ϕ
−1
(E)
|g(z)ϕ
0
(z)|
q
dA
β+q
(z),
ùpE•Dþ?˜Borel 8.KS
g,ϕ
: A
p
α
→A
q
β
´k.(;)Žf…=µ´(ž«) (
p
q
,α+p)-
BergmanCarlesonÿÝ.
y²:S
g,ϕ
: A
p
α
→A
q
β
´k.Žfdu
kS
g,ϕ
fk
A
q
β
≤Ckfk
A
p
α
.
Ø”ϕ(0) = 0, K
kS
g,ϕ
fk
q
A
q
β
=




Z
z
0
(f◦ϕ)
0
(ζ)g(ζ)dζ




q
A
q
β
k(f◦ϕ)
0
gk
q
A
q
β+q

Z
D
|(f◦ϕ)
0
(z)|
q
|g(z)|
q
dA
β+q
(z)
=
Z
D
|f
0
(z)|
q
dµ(z).
DOI:10.12677/pm.2022.12122292138nØêÆ
–’¤
¤±,·‚kS
g,ϕ
: A
p
α
→A
q
β
´k.(;)Žf…=‡©ŽfD: A
p
α
→L
q
(dµ)´k.(;)Žf.
Ïd,dÚn2.39Ù5, ½ny.
£Bloch˜m½Â.d¤k÷v
sup
z∈D
|f
0
(z)|(1−|z|
2
) <∞
¤k)Û¼ê¤˜m,¡•Bloch˜m,PŠB.d[12]·‚•µT
g
: A
p
α
→A
p
α
k.…=
g∈B.g∈Bž, ·‚¦^©z[13]•{, |^Nevanlinna .Oê¼ê•x2ÂVolterra.
È©ŽfT
ϕ
g
ÚT
g,ϕ
k.5Ú;5•x.•uŸÌ,·‚•‰ÑT
ϕ
g
;5•x.
½n3.5b0 <p<∞, −1 <α<∞.b ϕ∈H(),ϕ(D) ⊂D, g∈B.KT
ϕ
g
: A
p
α
→A
p
α
´;
Žf…=
|g
0
(z)|
2
N
ϕ,α
(z)dA(z)
´ž«(1,α)-BergmanCarleson ÿÝ.
y²:¿©5:Äk•Äp=2œ¹.b{f
n
}⊂A
2
α
,÷vkf
n
k
A
2
α
≤1…f
n
3D;f
8þ˜—Âñu0.@o,n→∞ž,
T
g
f
n
(ϕ(0)) =
Z
ϕ(0)
0
f
n
(w)g
0
(w)dw→0.
Ïd,e|g
0
|
2
N
ϕ,α
dA´ž«(1,α)−BergmanCarlesonÿÝ,Ki\NA
2
α
→L
2
(D,|g
0
|
2
N
ϕ,α
dA)
´;.u´,dÚn2.4,kT
ϕ
g
f
n
k
A
2
α
→0,n→∞.¤±,T
ϕ
g
: A
2
α
→A
2
α
´;Žf.
dug∈B,Œ•µé?¿0 <p<∞,T
ϕ
g
: A
p
α
→A
p
α
´k.Žf,ldKrasnoselskiiŠ
½n[14]µT
ϕ
g
: A
p
α
→A
p
α
´;Žf.
7‡5µbé,‡p∈(0,∞),kC
ϕ
T
g
:A
p
α
→A
p
α
´;Žf,@odŠnØ,T
ϕ
g
´A
2
α
þ;Žf.dÚn2.4,Œ•i\NA
2
α
→L
2
(D,|g
0
|
2
N
ϕ,α
dA)´;.|g
0
|
2
N
ϕ,α+2
dA´ž
«(1,α)-BergmanCarlesonÿÝ.
5µg∈H
∞
ž,ŽfS
ϕ
g
,S
g,ϕ
: A
p
α
→A
p
α
•kƒAÓa.(Ø.
Ä7‘8
Ø©É*H“‰Æ‰ï‘8(1170919634)]Ï.
ë•©z
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