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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(7),4443-4450
PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.117471
5pBergman˜mþ
HankelŽf
¯¯¯§§§–––’’’¤¤¤
∗
*H“‰ÆêƆÚOÆ§2ÀÖô
ÂvFϵ2022c68F¶¹^Fϵ2022c75F¶uÙFϵ2022c712F
Á‡
©Ì‡•x1<q<p<∞ž§d÷v˜½^‡ÎÒ¼ê¤pHankelŽfH
f
,H
f
lA
p
ω
L
q
Ω
Óžk.½;A§Ù¥ω,Ω´5"
'…c
\Bergman˜m§k.5§HankelŽf
HankelOperatorsonBergmanSpaces
InducedbyRegularWeights
ErminWang,YechengShi
∗
SchoolofMathematicsandStatistics,LingnanNormalUniversity,ZhanjiangGuangdong
Received:Jun.8
th
,2021;accepted:Jul.5
th
,2022;published:Jul.12
th
,2022
Abstract
Givenω,Ω∈R,for1<q<p<∞,wecharacterizethosesymbolsfforwhichthe
∗ÏÕŠö"
©ÙÚ^:¯,–’¤.5pBergman˜mþHankelŽf[J].A^êÆ?Ð,2022,11(7):
4443-4450.DOI:10.12677/aam.2022.117471
¯§–’¤
inducedHankeloperatorsH
f
,H
f
arebothbounded(compact)fromweightedBergman
spaceA
p
ω
toLebesguespaceL
q
Ω
.
Keywords
WeightedBergmanSpaces,Boundedness,HankelOperators
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.có
D´E²¡þü .é1≤p<∞,‰½DþšKŒÿ¼êω,˜mL
p
ω
(½P•
L
p
(ωdA))´Dþ¤k÷v
kfk
L
p
ω
=

Z
D
|f(z)|
p
ω(z)dA(z)

1
p
<∞,
LebesgueŒÿ¼êN,Ù¥dA´DþIO¡ÈÿÝ.·‚^ÎÒL
p
L«²;pg
Lebesgue˜m,Ù‰ê•k·k
L
p
=

R
D
|·|
p
dA

1
p
.·‚^H(D)L«DþX¼êN.\
Bergman˜m½Â•A
p
ω
= L
p
ω
∩H(D).w,A
p
ω
´L
p
ω
4f˜m,A
2
ω
´˜‡Hilbert˜m.
e˜‡»•¼êω÷ven‡^‡µ (1)ω∈L
1
[0,1);(2)¼êˆω(z)=
R
1
|z|
ω(s)ds÷v
V^‡,=é¤k0≤r<1,kˆω(r)≤Kˆω(
1+r
2
),Ù¥K´†rÃ'~ê;(3)é¤k
0 ≤r<1,k
ω(r) '
R
1
r
ω(s)ds
1−r
.
K¡ω´˜‡5,P•ω∈R.„©z[1].‰½˜‡»•ω,·‚Œ±Ï L-ω(z)=ω(|z|)
ù«•ªòω½ÂDþ.d5pBergman˜méõÆö\±ïÄ,„©z[1–4].
‰½ω∈R,éz˜‡z∈D,Nf7→f(z)´A
2
ω
þëY‚5•¼.dRieszL«½n Œ
•,•3•˜¼êB
z
∈A
2
ω
,¦阃f∈A
2
ω
,Ñkf(z) = hf,B
z
i
ω
,Ù¥
hf,gi
ω
=
Z
D
f(z)g(z)ω(z)dA(z),f,g∈A
2
ω
.
DOI:10.12677/aam.2022.1174714444A^êÆ?Ð
¯§–’¤
Ù¥B
z
¡•A
2
ω
BergmanØ.lL
2
ω
A
2
ω
ÝKP
ω
ŒL«•
P
ω
(g)(z) =
Z
D
g(ζ)B
z
(ζ)ω(ζ)dA(ζ).
é?¿1 ≤p<∞,P
ω
•´L
p
ω
A
p
ω
k.‚5Žf.e¡·‚Œ±½ÂA
p
ω
þHankelŽf.
dÎÒfpHankelŽf½Â•
H
ω
f
(g) = (Id−P
ω
)(fg),
Ù¥Id´ðŽf.
éHankelŽfïÄ©u[5],T©ÙïÄdÝ)Û¼êpHankel Žfk.5Ú;
5A.éu •˜„¼ê,[6]•k‰ÑHankel ŽfÚBergmanÝþeÎÒ¼ê²þƒm
éX.Ùù˜gŽ$^uk.é¡•Úr[à•þHankelŽfïÄ¥,„[7,8].3n
‘E ˜m¥ü ¥þ,•›1 <p≤q<∞,Pau<[9]ïÄH
f
ÚH
f
Óžl\Bergman˜m
A
p
(1−|·|
2
)
α
ƒALebesgue˜mL
q
(1−|·|
2
)
β
•k.½;Žf¿‡^‡.é1<q<p<∞œ/e,
ƒA(JÙ•®²Ñ,„[10].5,é¤k1<p,q<∞,Hu</•xHankel
Žf´d5ωpBergman˜mA
p
ω
L
q
ω
k.½;Žf•x.•C,þã(J
í2–A
p
ω
L
q
Ω
œ/,Ù¥1 <p,q<∞,¿…1 <p≤q<∞ž,H
f
,H
f
: A
p
ω
→L
q
Ω
Óžk.
½;A•Óž•x,„[11].
©̇8´•x1<q<p<∞ž,d÷v˜½^‡ÎÒ¼ê¤p HankelŽ
fH
f
,H
f
lA
p
ω
L
q
Ω
Óžk.½;A .°(/`,‰½ω,Ω∈R,·‚•Äd¼êF∈L
1
Ω
pH
f
,H
f
:A
p
ω
→L
q
Ω
Óžk.½;¿‡^‡.©ò[10]¥̇(Øí2–5p
Bergman˜mœ/.
3©¥,·‚o´^CL«†¤‡•Ä¼êÃ'~ê,3ØÓªf¥z?C¤“L
~êŒU؃.ü‡þA,BXJ÷vA≤CB,KP•A.B.XJA,B÷vA.B.A,·
‚Ò`A,Bd,P•A'B.
2.ý•£
3ùÜ©,·‚‰Ñ˜ý•£.-
β(z,ξ) =
1
2
log
1+|ϕ
z
(ξ)|
1−|ϕ
z
(ξ)|
L«DþBergman ål,Ù¥ϕ
z
(ξ) =
ξ−z
1−zξ
.z∈D,r>0, ^D(z,r) = {w∈D: β(z,w) <r}
L«±z•¥:!r•Œ»Bergman.‰½ω∈R,P
˘ω(z) = (1−|z|)
2
ω(z).
-r>0,•3D¥ê{z
j
}
∞
j=1
¦
D= ∪
∞
j=1
D(z
j
,r),D

z
j
,
r
4

∩D

z
k
,
r
4

= ∅,j6= k.
DOI:10.12677/aam.2022.1174714445A^êÆ?Ð
¯§–’¤
þãê¡•D˜‡r-‚.eE´DþLebesgueŒÿ8,·‚^χ
E
L«EA¼ê,P
|E|=
R
D
χ
E
dA.‰½r-‚{z
j
}
∞
j=1
ÚR>0,•3~êN¦
∞
X
j=1
χ
D(z
j
,R)
≤N.
r>0,L
1
loc
þÛܲþ¼êM
r
½Â•
M
r
(f)(z) =
1
|D(z,r)|
Z
D(z,r)
f(ξ)dA(ξ).
?‰½r>0,1 ≤p≤∞ž,M
r
´L
p
ω
þk.‚5Žf.f∈L
p
loc
,½ÂG
p,r
(f)•
G
p,r
(f)(z) = inf
(

1
|D(z,r)|
Z
D(z,r)
|f−h|
p
dA

1
p
: h∈H(D(z,r))
)
.
duω∈R,k
G
p,r
(f)(z) 'inf
(

1
ω(D(z,r))
Z
D(z,r)
|f−h|
p
ωdA

1
p
: h∈H(D(z,r))
)
.
-f∈L
p
loc
,r>0,P
MO
p,r
(f)(z) =

1
|D(z,r)|
Z
D(z,r)
|f−f
D(z,r)
|
q
dA

1
p
Ù¥f
D(z,r)
=
1
|D(z,r)|
R
D(z,r)
fdA.±9
Oss
r
(f)(z) =sup
ξ∈D(z,r)
|f(ξ)−f(z)|.
3[11]¥,·‚y²Xe½nµ
½nA:ω,Ω ∈R,1 <q<p<∞.Kéf∈L
1
Ω
,eØãd:
(A)H
Ω
f
: A
p
ω
→L
q
Ω
k.;
(B)H
Ω
f
: A
p
ω
→L
q
Ω
;;
(C)•3(du:?¿)0 <r≤α/2,kΩ
1
q
ω
−
1
p
G
q,r
(f) ∈L
pq
p−q
;
(D)fäkXe©)f= f
1
+f
2
,Ù¥f
1
÷v:
f
1
∈C
1
(D),Ω
1
q
ω
−
1
p
(1−|·|)|∂f
1
|∈L
pq
p−q
;
f
2
÷v:•3(du:?¿)r>0,k
Ω
1
q
ω
−
1
p
M
r
(|f
2
|
q
)
1
q
∈L
pq
p−q
.
DOI:10.12677/aam.2022.1174714446A^êÆ?Ð
¯§–’¤
?k
kH
Ω
f
k
A
p
ω
→L
q
Ω
'



Ω
1
q
ω
−
1
p
G
q,r
(f)



L
pq
p−q
.(2.1)
3.̇(Ø
3ùÜ©,·‚‰Ñ©̇½n¿\±y².
½n1:ω,Ω ∈R,1 <q<p<∞.Kéf∈L
1
Ω
,eØãd:
(A)H
Ω
f
,H
Ω
f
: A
p
ω
→L
q
Ω
•k.Žf;
(B)H
Ω
f
,H
Ω
f
: A
p
ω
→L
q
Ω
•;Žf;
(C)•3(du:?¿)r>0,kΩ
1
q
ω
−
1
p
MO
q,r
(f) ∈L
pq
p−q
;
(D)fäkXe©)f= f
1
+f
2
,Ù¥f
1
÷v:
f
1
∈C
1
(D),Ω
1
q
ω
−
1
p
Oss
r
(f
1
) ∈L
pq
p−q
;
f
2
÷v:•3(du:?¿)r>0,k
Ω
1
q
ω
−
1
p
M
r
(|f
2
|
q
)
1
q
∈L
pq
p−q
.
¿…k


H
Ω
f


A
p
ω
→L
q
Ω
+



H
Ω
f



A
p
ω
→L
q
Ω
'



˘
Ω
1
q
˘ω
−
1
p
MO
q,r
(f)



L
pq
p−q
.(3.1)
y²:(B) ⇒(A)w,¤á.ey(A) ⇔(C) ⇔(D)±9(C) ⇒(B).
(C)⇔(D).b•3r>0¦f÷v(C).-f
1
=M
r
2
(f),f
2
=1 −f
1
.Kβ(z,ξ)≤
r
2
ž,·‚k
|f
1
(z)−f
1
(ξ)|≤|f
1
(z)−M
r
(f)(z)|+|M
r
(f)(z)−f
1
(ξ)|
≤
1
|D(z,r/2)|
Z
D(z,r/2)
|f
1
(·)−M
r
(f)(z)|dA
+
1
|D(ξ,r/2)|
Z
D(ξ,r/2)
|f
1
(·)−M
r
(f)(z)|dA
≤CMO
q,r
(f)(z).
¤±
˘
Ω(z)
1
q
˘ω(z)
−
1
p
Oss
r
(f)(z) .
˘
Ω(z)
1
q
˘ω(z)
−
1
p
MO
q,r
(f)(z).(3.2)
DOI:10.12677/aam.2022.1174714447A^êÆ?Ð
¯§–’¤
éf
2
,·‚k
M
r
2
(|f
2
|
q
)(z)
1
q
≤M
r
2
(|f−f
1
(z)|
q
)(z)
1
q
+Oss
r
2
(f
1
)(z)
≤CMO
q,
r
2
(f)(z)+Oss
r
2
(f
1
)(z).
ddÚ(3.2)Œ±íÑ
˘
Ω(z)
1
q
˘ω(z)
−
1
p
M
r
2
(|f
2
|
q
)(z)
1
q
.
˘
Ω(z)
1
q
˘ω(z)
−
1
p
MO
q,r
(f)(z).
5¿(D)¥^‡†rÃ',·‚Ñ(C) ⇒(D).
‡ƒ,bfäkX(D)¥©): f=f
1
+f
2
.dMO
q,r
(f)(z).Oss
r
(f)(z),Υf
1
÷v
(C).¿…
MO
q,r
(f
2
)(z)≤M
r
(|f
2
|
q
)(z)
1
q
+M
r
(|f|)
≤2M
r
(|f
2
|
q
)(z)
1
q
L²f
2
•÷v(C).(D)ŒíÑ(C).
du(D)¥^‡†rÃ',·‚Œ±Ω
1
q
ω
−
1
p
MO
q,r
(f) ∈L
pq
p−q
†rÃ'.
(A) ⇔(C).Ø”0 <r<α.eH
Ω
f
ÚH
Ω
f
Ñk.,d(2.1)Œ•
˘
Ω(z)
1
q
˘ω(z)
−
1
p
G
q,r
(f)(z) .


H
Ω
f


A
p
ω
→L
q
Ω
±9
˘
Ω(z)
1
q
˘ω(z)
−
1
p
G
q,r
(f)(z) .



H
Ω
f



A
p
ω
→L
q
Ω
.
aqu©z[12]¥·K2.5y²Œ•
˘
Ω(z)
1
q
˘ω(z)
−
1
p
MO
q,r
(f)(z) .


H
Ω
f


A
p
ω
→L
q
Ω
+



H
Ω
f



A
p
ω
→L
q
Ω
.(3.3)
‡ƒ,dG
q,r
(f)½Â´•
G
q,r
(f)(z) ≤MO
q,r
(f)(z),G
q,r
(f)(z) ≤MO
q,r
(f)(z).(3.4)
d½nAŒ•


H
Ω
f


A
p
ω
→L
q
Ω
+



H
Ω
f



A
p
ω
→L
q
Ω
.



˘
Ω(z)
1
q
˘ω(z)
−
1
p
MO
q,r
(f)



L
∞
.(3.5)
(3.1)ªd(3.3)9(3.5)=ŒÑ.
(C) ⇒(B).d(C)9(3.4)Œ•:Ω
1
q
ω
−
1
p
G
q,r
(f) ∈L
pq
p−q
.2d½nAŒ•H
Ω
f
: A
p
ω
→L
q
Ω
•
;Žf;Ón,H
Ω
f
: A
p
ω
→L
q
Ω
•;Žf.½ny.
DOI:10.12677/aam.2022.1174714448A^êÆ?Ð
¯§–’¤
—
a"v<@ý".
Ä7‘8
Ø©ÉI[g,‰ÆÄ7‘8(12001258)Ú*H“‰Æ‰ï‘8(ZL1925)]Ï.
ë•©z
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[2]Pavlovic,M.andPelaez,J.A.(2008)AnEquivalenceforWeightedIntegralsofanAnalytic
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https://doi.org/10.1002/mana.200510701
[3]Pelaez,J.A.andRattya,J.(2013)WeightedBergmanSpacesInducedbyRapidlyIncreasing
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