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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(10),3446-3455
PublishedOnlineOctober2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1010363
lVBergman˜mA
p
ω
Bloch.˜m
Volterra.È©Žf
–––’’’¤¤¤,¯¯¯
∗
*H“‰ÆêƆÚOÆ§2ÀÖô
ÂvFϵ2021c921F¶¹^Fϵ2021c1018F¶uÙFϵ2021c1025F
Á‡
©ïÄlVBergman˜mA
p
ω
Bloch.˜mVolterra.È©EÜŽfT
ϕ
g
ÚS
ϕ
g
k
.5Ú;5¯K§‰ÑVolterra.È©EÜŽfk.5Ú;5•x"
'…c
VolterraÈ©EÜŽf§VBergman˜m§Bloch.˜m§k.5§;5
VolterraTypeOperatorsfrom
BergmanSpacesA
p
ω
withDoubling
WeightstotheBloch
TypeSpaces
YechengShi,ErminWang
∗
SchoolofMathematicsandStatistics,LingnanNormalUniversity,ZhanjiangGuangdong
Received:Sep.21
st
,2021;accepted:Oct.18
th
,2021;published:Oct.25
th
,2021
∗ÏÕŠö"
©ÙÚ^:–’¤,¯.lVBergman˜mA
p
ω
Bloch.˜mVolterra.È©Žf[J].A^êÆ?Ð,2021,
10(10):3446-3455.DOI:10.12677/aam.2021.1010363
–’¤,¯
Abstract
WeconsidertheboundednessandcompactnessofVolterratypeoperatorsfromthe
BergmanspacesA
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/
1.Úó
D´E²¡Cþü ,H(D)´Dþ)Û¼êN.ϕ´Dþ)ÛgN,½ÂÙ
3H(D)þpEÜŽfC
ϕ
•µ
(C
ϕ
f)(z) = f(ϕ(z)).
g∈H(D),Volterra.È©ŽfT
g
ÚS
g
½Â•
(T
g
f)(z) =
Z
z
0
f(ζ)g
0
(ζ)dζ,
Ú
(S
g
f)(z) =
Z
z
0
f
0
(ζ)g(ζ)dζ.
Pommerenke[1]31977cÄkïÄVolterraÈ©ŽfT
g
3H
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þk.5¯K,y²T
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ϕ
g
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DOI:10.12677/aam.2021.10103633447A^êÆ?Ð
–’¤,¯
mVolterra.È©EÜŽfk.5Ú;5¯K.
g∈H(D),ϕ´Dþ)ÛgN,ü«Volterra.È©EÜŽf½ÂXeµ
(T
ϕ
g
f)(z) =
Z
z
0
(f◦ϕ)(ζ)(g◦ϕ)
0
(ζ)dζ,
Ú
(S
ϕ
g
f)(z) =
Z
z
0
(f◦ϕ)
0
(ζ)(g◦ϕ)(ζ)dζ.
ϕ(z) = zž,T
ϕ
g
= T
g
,S
ϕ
g
= S
g
;g= 1ž,S
ϕ
g
= C
ϕ
−C
ψ
,ùpψ(z) = ϕ(0).
·‚¡ω´¼ê,XJω´DþšKŒÈ¼ê.0 <p<∞ž,\Bergman˜mA
p
ω
½
奵
A
p
ω
= {f∈H(D) : kfk
p
A
p
ω
:=
Z
D
|f(z)|
p
ω(z)dA(z) <∞}.
·‚¡A
p
ω
´VBergman˜m,´•é»•ω,÷vVØªˆω(r)≤Cˆω(
1+r
2
),ùp
ˆω(r) =
R
1
r
ω(s)ds.dž·‚Pω∈
b
D.VBergman˜mƒ'•£Œ±„[3,4].
éu0 <α<∞,Bloch.˜mB
α
½Â•dDþ÷v^‡
||f||
B
α
= |f(0)|+sup
z∈D
(1−|z|
2
)
α
|f
0
(z)|<∞.
)Û¼êf|¤8Ü.
©Ì‡ïÄlVBergman˜mA
p
ω
Bloch.˜mB
α
Volterra.È©EÜŽfT
ϕ
g
ÚS
ϕ
g
k.5Ú;5•x.©¥,ÎÒCL«†?ØþÃ'~ê,zgÑy™7ƒ Ó.éu‰
½ü‡þAÚB,XJ•3~êC,¦
1
C
B≤A≤CB,·‚¡A†Bd,PŠAB.
2.ý•£
Äk,·‚‰Ñe¡(Ø.
Ún2.10 <p<∞,ω∈ω∈
b
D.ef∈A
p
ω
,K
|f(z)|≤Cω(S(z))
−
1
p
kfk
A
p
ω
.
y²f∈A
p
ω
,d|f(z)|
p
|gNÚ5Ÿ,·‚k
|f(z)|
p
≤
R
1
1−|z|
2
1
2π
R
2π
0
|f(z+se
iθ
)|
p
dθω(s)sds
R
1
1−|z|
2
ω(s)sds
≤
1
R
1
1−|z|
2
ω(s)sds
kfk
p
A
p
ω
DOI:10.12677/aam.2021.10103633448A^êÆ?Ð
–’¤,¯
,˜•¡,
Z
1
1−|z|
2
ω(s)sds≥
1−|z|
2
Z
1
1−|z|
2
ω(s)ds
≥
1−|z|
2
Z
1
1+|z|
2
ω(s)ds
&
1−|z|
2
Z
1
|z|
ω(s)ds
&ω(S(z))
y..
e¡Ún•©z[3]Ún2.4.
Ún2.20<p<∞,ω∈ω∈
b
D,K•3λ
0
(ω)>0¦λ≥λ
0
ž,é?¿a∈D,)Û¼
êF
a,p
(z) =

1−|a|
2
1−az

λ+1
p
÷v
|F
a,p
(z)|1,z∈S(a),a∈D.
…
kF
a,p
(z)k
p
A
p
ω
ω(S(a)),a∈D.
dÚn2.2,·‚kXeÚn.
Ún2.30 <p<∞,ω∈ω∈
b
D,Pf
a,p
(z) = ω(S(a))
−
1
p

1−|a|
2
1−az

λ+1
p
,K
|f
a,p
(z)|ω(S(a))
−
1
p
, z∈S(a),a∈D.
…
kf
a,p
k
A
p
ω
1,
…|z|→1ž,f
a,p
3Dþ;f8˜—Âñu0.
Ún2.4T
ϕ
g
:A
p
ω
→B
α
´k.Žf,KT
ϕ
g
:A
p
ω
→B
α
´;Žfdué?¿k.S{f
n
}⊂
A
p
ω
…3Dþ?¿;f8˜—Âñu0,K
lim
n→∞
kT
ϕ
g
f
n
k
B
α
= 0.
y²¿©5.{f
n
}•A
p
ω
¥?¿k.S,dÚn2.1,k
|f
n
(z)|.ω(S(z))
−
1
p
kf
n
k
A
p
ω
.
Ïd,{f
n
}3D?¿;f8þ˜—Âñ.dMontel½n,•3{f
n
}fS,Ø”EP•{f
n
},9)
DOI:10.12677/aam.2021.10103633449A^êÆ?Ð
–’¤,¯
Û¼êf,¦{f
n
}3D;f8˜—Âñuf.ŠâFatouÚn,´f∈A
p
ω
.l
kT
ϕ
g
f
n
−fk
B
α
→0(n→∞).
T
ϕ
g
: A
p
ω
→B
α
´;Žf.
7‡5.k.S{f
n
}⊂A
p
ω
÷v3Dþ?¿;f8˜—Âñu0.duT
ϕ
g
: A
p
ω
→B
α
´;
Žf,•3{f
n
}fS,EP•{f
n
}9f∈B
α
,¦
lim
n→∞
kT
ϕ
g
f
n
−fk
B
α
= 0.
¤±,f(0) = 0…
sup
z∈D
(1−|z|
2
)
α
|f
n
(ϕ(z))(g◦ϕ)
0
(z)−f(z)|→0(n→∞).
Ï•f
n
(ϕ(z))(g◦ϕ)
0
(z)3Dþ˜—Âñuf
0
(z).{f
n
}3D˜—Âñu0,g∈H(D),¤±kf
0
(z)=
0.f(z) ≡0.Ïd,·‚y²
lim
n→∞
kT
g
f
n
k
B
α
= 0.
Ún2.50 <p<∞,ω∈ω∈
b
D.ef∈A
p
ω
,K
|f
0
(z)|.
kfk
A
p
w
ω(S(z))
1
p
(1−|z|
2
)
.
y²d…ÜÈ©úª
f
0
(z) =
1
2πi
Z
|ξ−z|=
1−|z|
2
f(ξ)dξ
(ξ−z)
2
.
|^Ún2.1,¿(Üω(S(ξ)) ≥ω(S(
1+|z|
2
)) ω(S(z)),=
|f
0
(z)|.
kfk
A
p
w
ω(S(z))
1
p
(1−|z|
2
)
.
3.lA
p
ω
˜mBloch.˜mVolterra.È©EÜŽf
(ܱþÚn©̇½n
½n3.1bg∈H(D),ω∈
b
D,KT
ϕ
g
: A
p
ω
→B
α
´k.Žf…=
sup
z∈D
(1−|z|)
α
|(g◦ϕ)
0
(z)|ω(S(ϕ(z))
−
1
p
<∞.
y²¿©5:b
sup
z∈D
(1−|z|)
α
|(g◦ϕ)
0
(z)|ω(S(ϕ(z))
−
1
p
<∞.
DOI:10.12677/aam.2021.10103633450A^êÆ?Ð
–’¤,¯
-f∈A
p
ω
,dÚn2.1,k
(1−|z|
2
)
α
|(g◦ϕ)
0
(z)|f(ϕ(z))|.(1−|z|
2
)
α
|(g◦ϕ)
0
(z)|ω(S(ϕ(z))
−
1
p
kfk
A
p
ω
.(1−|z|
2
)
α
|(g◦ϕ)
0
(z)|ω(S(ϕ(z))
−
1
p
.
¤±kT
α
g
fk
B
α
<∞.T
ϕ
g
: A
p
ψ
→B
α
´k.Žf.
7‡5:bT
ϕ
g
: A
p
ω
→B
α
´k.Žf.dÚn2.4ÚÚn2.3,·‚k
(1−|z|
2
)
α
|(g◦ϕ)
0
(z)|ω(S(ϕ(z))
−
1
p
.kT
g
f
a,p
k
B
<∞.
y..
íØ3.1bg∈H(D),ω∈
b
D,KT
g
: A
p
ω
→B
α
´k.Žf…=
sup
z∈D
(1−|z|)
α
|g
0
(z)|ω(S(z)
−
1
p
<∞.
½n3.2bg∈H(D),ψ∈W
0
,KT
ϕ
g
: A
p
ω
→B
α
´k;Žf…=
lim
|ϕ(z)|→1
(1−|z|
2
)
α
|(g◦ϕ)
0
(z)|ω(S(ϕ(z))
−
1
p
= 0.
y²¿ ©5:bT
ϕ
g
:A
p
ω
→B
α
´;Žf.Pf
z,p
(w)=ω(S(z))
−
1
p

1−|z|
2
1−zw

λ+1
p
,Kkf
z,p
k
A
p
ω
1,
¿…|z|→1ž,f
z,p
3Dþ;f8˜—Âñu0.
0 =lim
|ϕ(z)|→1
kT
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
(1−|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
,Ké?¿>0,•3r
0
∈(0,1),¦
|ϕ(z)|>r
0
ž,k
(1−|z|
2
)
α
|(g◦ϕ)
0
(z)|ω(S(ϕ(z))
−
1
p
<.
qÏ•|ϕ(z)|≤r
0
ž,w,k(1−|z|
2
)
α
|(g◦ϕ)
0
(z)|ω(S(ϕ(z))
−
1
p
<∞k..ld½n3.1,T
ϕ
g
: A
p
ω
→
B
α
´.Žf.5¿g◦ϕ∈B
α
,•3êM,¦
M:=sup
|ϕ(z)|≤r
0
(1−|z|
2
)
α
|(g◦ϕ)
0
(z)|<∞.
DOI:10.12677/aam.2021.10103633451A^êÆ?Ð
–’¤,¯
{f
n
}•A
p
ω
¥k.S,…3D;f8þ˜—Âñu0.
kT
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))|
.Msup
|w|≤r
0
|f
n
(w)|+sup
|ϕ(z))|>r
0
(1−|z|
2
)
α
|(g◦ϕ)
0
(z)|S(ϕ(z))
−
1
p
kf
n
k
A
p
ω
.Msup
|w|≤r
0
|f
n
(w)|+.
Ïd,n→∞ž,kT
g
f
n
k
B
α
→0.¤±T
ϕ
g
: A
p
ω
→B
α
´k;Žf.
y..
íØ3.2bg∈H(D),ω∈
b
D,KT
g
: A
p
ω
→B
α
´;Žf…=
lim
|z|→1
(1−|z|
2
)
α
|g
0
(z)|ω(S(z)
−
1
p
<∞.
½n3.30<p<∞,0<α<∞,g∈H(D)…ϕ´Dþ)ÛgN,ω∈
b
D,KS
ϕ
g
:A
p
ω
→B
α
´k.Žf…=
sup
z∈D
(1−|z|
2
)
α
|ϕ
0
(z)|
(1−|ϕ(z)|
2
)ω(S(ϕ(z)))
1
p
|g(ϕ(z))|<∞.
y²¿©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
,dÚn2.5,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
kfk
A
p
ω
.
(1−|z|
2
)
α
|ϕ
0
(z)||g(ϕ(z))|
(1−|ϕ(z)|
2
)ω(S(ϕ(z)))
1
p
.
¤±kS
ϕ
g
fk
B
α
<∞.S
ϕ
g
: A
p
ω
→B
α
´k.Žf.
7‡5:bS
ϕ
g
: A
p
ω
→B
α
´k.Žf.
|ϕ(z)|≥
1
2
ž,
kS
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.2021.10103633452A^êÆ?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))|
.kS
ϕ
g
(z)k
B
α
<∞.
y..
íØ3.30 <p<∞,0 <α<∞,…ϕ´Dþ)ÛgN,ω∈
b
D,KC
ϕ
: A
p
ω
→B
α
´k.Žf
…=
sup
z∈D
(1−|z|
2
)
α
|ϕ
0
(z)|
(1−|ϕ(z)|
2
)ω(S(ϕ(z)))
1
p
<∞.
íØ3.40 <p<∞,0 <α<∞,g∈H(D),ω∈
b
D,KS
g
: A
p
ω
→B
α
´k.Žf…=
sup
z∈D
(1−|z|
2
)
α−1
ω(S(z))
−
1
p
|g(z)|<∞.
½n3.40<p<∞,0<α<∞,g∈H(D)…ϕ´Dþ)ÛgN,ω∈
b
D,KS
ϕ
g
:A
p
ω
→B
α
´;Žf…=
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.
y²¿©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,…3D;f8þ˜—Âñu0.éu?¿½0<r<1.dÚn2.4Ú
Ún2.5,k
kS
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
kf
n
k
A
p
ω
.
Ï•|ϕ(z)|≤rž,(1−|z|
2
)
α
|ϕ
0
(z)||g(ϕ(z))|.kS
ϕ
g
(z)k
B
α
<∞…
lim
n→∞
sup
|ϕ(z)|≤r
|f
n
(ϕ(z))|= 0.
DOI:10.12677/aam.2021.10103633453A^êÆ?Ð
–’¤,¯
¤±-n→∞,r→1,k
lim
n→∞
kS
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
´;Žf.
7‡5:bS
ϕ
g
: A
p
ω
→B
α
´;Žf.dÚn2.4,Œ
0=lim
|ϕ(z)|→1
kS
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..
íØ3.50 <p<∞,0 <α<∞,…ϕ´Dþ)ÛgN,ω∈
b
D,KC
ϕ
: A
p
ω
→B
α
´;Žf
…=
lim
|ϕ(z)|→1
(1−|z|
2
)
α
|ϕ
0
(z)|
(1−|ϕ(z)|
2
)ω(S(ϕ(z)))
1
p
= 0.
íØ3.60 <p<∞,0 <α<∞,g∈H(D),ω∈
b
D,KS
g
: A
p
ω
→B
α
´;Žf…=
lim
|z|→1
(1−|z|
2
)
α−1
ω(S(z))
−
1
p
|g(z)|= 0.
Ä7‘8
Ø©ÉI[g,‰ÆÄ7‘8(11901271)Ú*H“‰Æ‰ï‘8(1170919634)]Ï.
ë•©z
[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.2021.10103633454A^êÆ?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.2021.10103633455A^êÆ?Ð

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