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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(4),1764-1780
PublishedOnlineApril2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.114193
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¡HLS)Øªk'‘š‚5È©•§µ
f
q−1
(ξ) =
Z
Ω
G(ξ)f(η)G(η)
|η
−1
ξ|
Q−α
dη+λ
Z
Ω
f(η)
|η
−1
ξ|
Q−α−β
dη,ξ∈
¯
Ω,
Ù¥q>1,0 <α<Q§0 <β<Q−α§Q= 2n+2´H
n
àg‘ê§λ∈R,Ω ⊂H
n
´˜‡1w
k.•…G(ξ)´
¯
Ω¥šKëY¼ê"ùp§·‚ò?Øg.
2Q
Q+α
<q<2œ/ eT•§)
•35(J"
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°Ü+§HardyõLittlewoodõSobolevØª§Brezis-Nirenberg.¯K§È©•§§g
.œ/§•35
ExistenceofPositiveSolutionsto
NonlinearIntegralEquationswith
WeightsontheBoundedDomains
oftheHeisenbergGroupin
SubcriticalCase
∗Email:jnchen@zjnu.edu.cn
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2022,11(4):1764-1780.DOI:10.12677/aam.2022.114193
•ZV
JianiChen
∗
CollegeofMathematicsandComputerSciencesofZhejiangNormalUniversity,JinhuaZhejiang
Received:Mar.18
th
,2022;accepted:Apr.12
th
,2022;published:Apr.20
th
,2022
Abstract
Thispaperisdevotedtoakindofnonlinearintegralequationswithweightsrelated
tothesharpHardy-Littlewood-Sobolev(hereinafterreferredtoasHLS)inequalityon
theboundeddomainsoftheHeisenberggroupH
n
:
f
q−1
(ξ) =
Z
Ω
G(ξ)f(η)G(η)
|η
−1
ξ|
Q−α
dη+λ
Z
Ω
f(η)
|η
−1
ξ|
Q−α−β
dη,ξ∈
¯
Ω,
whereq>1,0<α<Q,0<β<Q−α,Q=2n+2isthehomogeneousdimensionof
H
n
,λ∈R,Ω ⊂H
n
isasmoothboundeddomainandG(ξ)isnonnegativecontinuousin
¯
Ω.Inthispaper,wewillstudytheexistenceresultsofthepositivesolutionsforthe
equationinsubcriticalcase
2Q
Q+α
<q<2.
Keywords
Heisenb ergGroup,Hardy-Littlewood-SobolevInequality,Brezis-Nirenb erg-Type
Problem,IntegralEquation,SubcriticalCase,Existence
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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|η
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|η
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DOI:10.12677/aam.2022.1141931766A^êÆ?Ð
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dη,f≥0, ξ∈
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DOI:10.12677/aam.2022.1141931767A^êÆ?Ð
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DOI:10.12677/aam.2022.1141931768A^êÆ?Ð
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DOI:10.12677/aam.2022.1141931769A^êÆ?Ð
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½Â1(òÈ[3])H
n
þü‡¼êf,gòÈ½Â•
f∗g(ξ) =
Z
f(η)g(η
−1
ξ)dη=
Z
f(ξη
−1
)g(η)dη.
ùp,·‚Ï~^ÎÒD
0
L«©Ù˜m.XJf∈C
∞
0
…G∈D
0
,@o·‚òrC
∞
¼êG∗fÚf∗
G½Â•
G∗f(ξ) = G(f(η
−1
ξ)),f∗G(ξ) = G(f(ξη
−1
)).
½Â2(K©Ù[3])·‚¡˜‡©ÙF´K,XJ•3˜‡3H
n
\{0}þ´C
∞
¼
êf¦é¤kg∈C
∞
0
(H
n
\{0})ÑkF(g) =
R
fgdξ¤á.
·K4(Ún8.7[3])XJF´˜‡λgKàg©Ù,Ù¥−Q<λ<0,@oé?¿
>0,Ng→g∗FÑŒ±*ÐlL
p
L
q
ÚlL
1
L
−
Q
λ
−
(loc)˜‡k.N,ùp
1
q
=
1
p
−
λ
Q
−1…1 <p<q<∞.
½n3(H
n
þHLSØªéó/ª[3])XJ©ÙF|ξ|
α−Q
,Ù¥0<α<Q,@oé
u1 <p<q<+∞Ú
1
q
=
1
p
−
α
Q
,þ¡(JÒC¤
kg∗|ξ|
α−Q
k
L
q
≤Ckgk
L
p
.(2.5)
•,·‚‰ÑH
n
þLipschitz˜mxΓ
α
½Â9Ù$^.
½Â3(Lipschitz˜m[3,17])(i)éu0 <α<1,
Γ
α
= {f∈L
∞
∪C: sup
ξ,η
|f(ξη)−f(ξ)|
|η|
α
<∞}.
(ii)éuα= 1,
Γ
1
= {f∈L
∞
∪C: sup
ξ,η
|f(ξη)+f(ξη
−1
)−2f(ξ)|
|η|
<∞}.
(iii)éuα= k+α
0
,ùpk´˜‡ê…0 <α
0
≤1,
Γ
α
= {f∈L
∞
∪C: f∈Γ
α
0
…é¤kD∈B
k
,ÑkDf∈Γ
α
0
},
Ù¥
B
k
= {L
a
1
L
a
2
···L
a
j
: 1 ≤a
i
≤2n, i= 1,2,···,j, j≤k},
L
j
= X
j
±9L
j+n
= Y
j
,j= 1,2,···,n.
·K5(½n20.1[3])éu0 <α<∞,Γ
α
⊂C
α
2
(loc).
DOI:10.12677/aam.2022.1141931770A^êÆ?Ð
•ZV
3.g.œ/eš‚5È©•§)•35
ù˜Ù,·‚òïÄg.q
α
<q<2œ¹e•§(1.1))•35.Äk,·‚y²e¡
;5Ún,dd·‚Œ±íäg.œ¹e•§(1.1))•35.
Ún1é?¿;«•ΩÚt<
2Q
Q−α
,ŽfI
G,α,Ω
:L
2Q
Q+α
(Ω)→L
t
(Ω)´;,Ù¥G(ξ)∈
C(
¯
Ω).•Ò´`, é?¿k.S{f
j
}
+∞
j=1
⊂L
2Q
Q+α
(Ω), •3˜‡¼êf∈L
2Q
Q+α
(Ω)Ú{I
G,α,Ω
f
j
(ξ)}
+∞
j=1
˜‡fS(E,^{I
G,α,Ω
f
j
(ξ)}
+∞
j=1
5L«),¦3L
t
(Ω)¥kI
G,α,Ω
f
j
(ξ)ÂñI
G,α,Ω
f.
y²Ï•{f
j
}
+∞
j=1
´L
2Q
Q+α
(Ω)¥k.S,L
2Q
Q+α
(Ω)´˜‡g‡Banach˜m…¼êG(ξ) ∈
C(
¯
Ω),¤±·‚k
kG(ξ)f
j
(η)G(η)k
L
2Q
Q+α
(Ω)
≤C,
…df;5Œ:•3{f
j
}˜‡fS(E,^{f
j
}L«)Ú˜‡¼êf∈L
2Q
Q+α
(Ω),¦j→
+∞ž,3L
2Q
Q+α
(Ω)¥kf
j
*f,ù¿›Xj→+∞ž,3L
2Q
Q+α
(Ω)¥k
G(ξ)f
j
(η)G(η) *G(ξ)f(η)G(η).(3.1)
©)|ξ|
α−Q
=|ξ|
α−Q
χ
{|ξ|>ρ}
+|ξ|
α−Q
χ
{|ξ|<ρ}
,Ù¥ρ>0ò3Yy²¥À.ùp·‚
½Â¼ê
I
G,α,Ω
f
j
(ξ) = I
1
G,α,Ω
f
j
(ξ)+I
2
G,α,Ω
f
j
(ξ)
:= G(ξ)f
j
(η)G(η)∗|ξ|
α−Q
χ
{|ξ|>ρ}
+G(ξ)f
j
(η)G(η)∗|ξ|
α−Q
χ
{|ξ|<ρ}
.
e¡,·‚ò©üÚ5?Ø{I
G,α,Ω
f
j
(ξ)}3L
t
(Ω)¥Âñ5.
1˜Ú.Äk,·‚5©Û{I
1
G,α,Ω
f
j
(ξ)}Âñ5.
˜•¡,5¿|ξ|
α−Q
χ
{|ξ|>ρ}
∈L
2Q
Q−α
(Ω)Ú(3.1),dfÂñ½ÂŒI
1
G,α,Ω
f
j
(ξ)Å:Âñ
I
1
G,α,Ω
f(ξ),ddí|I
1
G,α,Ω
f
j
(ξ)|
t
Å:Âñ|I
1
G,α,Ω
f(ξ)|
t
.,˜•¡,Ï•
|I
1
G,α,Ω
f
j
(ξ)|≤kG(ξ)f
j
(η)G(η)k
L
2Q
Q+α
(Ω)
k|ξ|
α−Q
χ
{|ξ|>ρ}
k
L
2Q
Q−α
(Ω)
≤C(ρ),
ù`²|I
1
G,α,Ω
f
j
(ξ)|
t
≤C
t
(ρ),ùpC(ρ)†f
j
Ã',¤±dLebesgue››Âñ½n,Œ
Z
Ω
|I
1
G,α,Ω
f
j
(ξ)|
t
→
Z
Ω
|I
1
G,α,Ω
f(ξ)|
t
,
dd
kI
1
G,α,Ω
f
j
(ξ)k
L
t
(Ω)
→kI
1
G,α,Ω
f(ξ)k
L
t
(Ω)
.
DOI:10.12677/aam.2022.1141931771A^êÆ?Ð
•ZV
Ïd,·‚Œ±Ñ3L
t
(Ω)¥k
I
1
G,α,Ω
f
j
(ξ) →I
1
G,α,Ω
f(ξ).(3.2)
1Ú.e5,·‚5©Û{I
2
G,α,Ω
f
j
(ξ)}Âñ5.
•y²3L
t
(Ω)¥kI
2
G,α,Ω
f
j
(ξ) →I
2
G,α,Ω
f(ξ),·‚I‡y²j→+∞žk
kI
2
G,α,Ω
f
j
(ξ)−I
2
G,α,Ω
f(ξ)k
L
t
(Ω)
= kI
2
G,α,Ω
(f
j
(ξ)−f(ξ))k
L
t
(Ω)
→0.
ŠâYoungØª,·‚Œ±íÑ
kI
2
G,α,Ω
(f
j
(ξ)−f(ξ))k
L
t
(Ω)
≤CkG(ξ)G(η)(f
j
(η)−f(η))k
L
2Q
Q+α
(Ω)
k|ξ|
α−Q
χ
{|ξ|<ρ}
k
L
s
(Ω)
≤Ckf
j
(η)−f(η)k
L
2Q
Q+α
(Ω)
k|ξ|
α−Q
χ
{|ξ|<ρ}
k
L
s
(Ω)
,
(3.3)
ùp
1
t
+1 =
1
2Q
Q+α
+
1
s
, s= (
1
t
+
Q−α
2Q
)
−1
<
Q
Q−α
…k|ξ|
α−Q
χ
{|ξ|<ρ}
k
L
s
(Ω)
≤Cρ
β
, Ù¥β= Q(
1
s
−
Q−α
Q
).
K(3.3)=z¤
kI
2
G,α,Ω
(f
j
(ξ)−f(ξ))k
L
t
(Ω)
≤Cρ
β
.
dž,Àρ•vê,@oj→+∞k
kI
2
G,α,Ω
(f
j
(ξ)−f(ξ))k
L
t
(Ω)
→0,
=j→+∞ž,3L
t
(Ω)¥k
I
2
G,α,Ω
f
j
(ξ) →I
2
G,α,Ω
f(ξ).(3.4)
Ïd,d(3.2)Ú(3.4)Œj→+∞ž,3L
t
(Ω)¥k
I
G,α,Ω
f
j
(ξ) →I
G,α,Ω
f(ξ).
Úny.2
3þãÚn1Ä:þ,·‚òy²e¡Ún,ddŒ±íѽn1•35(J.•{
üå„,·‚•y²λ= 0žœ/.Ty²•{†[16]¥Ún4.2y²•{ƒÓ.
Ún2éuq>q
α
,þ(.
D
G,α,q
(Ω) :=sup
f∈L
q
(Ω)\{0}
R
Ω
R
Ω
G(ξ)f(ξ)|η
−1
ξ|
α−Q
f(η)G(η)dηdξ
kfk
2
L
q
(Ω)
UL
q
(Ω)¥,˜šK¼êˆ,Ù¥0 <α<Q…¼êG(ξ) ∈C(
¯
Ω).
y²˜m©,·‚ò5y²D
G,α,q
(Ω) ≤C<+∞.
DOI:10.12677/aam.2022.1141931772A^êÆ?Ð
•ZV
Ï•G(ξ) ∈C(
¯
Ω),q>q
α
,f∈L
q
(Ω)…
e
f(ξ) :=
(
f(ξ)ξ∈Ω,
0ξ∈H
n
\Ω,
¤±|G(ξ)|≤C,f∈L
q
α
(Ω)…
e
f(ξ)∈L
q
α
(H
n
),ùpk
e
f(ξ)k
L
q
α
(H
n
)
=kf(ξ)k
L
q
α
(Ω)
.é?¿f∈
L
q
(Ω),Šâ°(HLSØª(2.1),·‚k
hI
G,α,Ω
f,fi=
Z
Ω
f(ξ)(
Z
Ω
G(ξ)f(η)G(η)
|η
−1
ξ|
Q−α
dη)dξ≤C
Z
Ω
Z
Ω
f(ξ)f(η)
|η
−1
ξ|
Q−α
dηdξ
= C
Z
H
n
Z
H
n
e
f(ξ)
e
f(η)
|η
−1
ξ|
Q−α
dηdξ≤C·D
n,α
k
e
fk
2
L
q
α
(H
n
)
= C·D
n,α
kfk
2
L
q
α
(Ω)
≤Ckfk
2
L
q
(Ω)
,
Ïd
D
G,α,q
(Ω) :=sup
f∈L
q
(Ω)\{0}
R
Ω
R
Ω
G(ξ)f(ξ)|η
−1
ξ|
α−Q
f(η)G(η)dηdξ
kfk
2
L
q
(Ω)
=sup
f∈L
q
(Ω)\{0}
R
Ω
f(ξ)(
R
Ω
G(ξ)f(η)G(η)|η
−1
ξ|
α−Q
dη)dξ
kfk
2
L
q
(Ω)
=sup
f∈L
q
(Ω)\{0}
hI
G,α,Ω
f,fi
kfk
2
L
q
(Ω)
≤sup
f∈L
q
(Ω)\{0}
Ckfk
2
L
q
(Ω)
kfk
2
L
q
(Ω)
= C<+∞,
=D
G,α,q
(Ω) ≤C<+∞.
e5,·‚‡y²þ(.D
G,α,q
(Ω)UL
q
(Ω)¥,˜šK¼êˆ.
3L
q
(Ω)¥À˜‡šK4ŒzS{f
j
}
+∞
j=1
,…?˜Ú¦ÙIOz¦kf
j
k
L
q
(Ω)
= 1,ùž·‚
Œ±
lim
j→+∞
Z
Ω
Z
Ω
G(ξ)f
j
(ξ)|η
−1
ξ|
α−Q
f
j
(η)G(η)dηdξ
=sup
f∈L
q
(Ω)\{0}
Z
Ω
Z
Ω
G(ξ)f(ξ)|η
−1
ξ|
α−Q
f(η)G(η)dηdξ
= D
G,α,q
(Ω)
…{f
j
}3L
q
(Ω)¥k..qÏ•g‡Banach˜mL
q
(Ω)¥k.S´fO;,…dÚn1Œ
•:ŽfI
G,α,Ω
äk;5,¤±·‚Œ±íäÑ•3{f
j
}˜‡fS(ùpE,^{f
j
}L«)Úf
∗
∈
L
q
(Ω),¦3L
q
(Ω)¥kf
j
*f
∗
±93L
q
0
(Ω)¥kI
G,α,Ω
f
j
→I
G,α,Ω
f
∗
,Ù¥1<q
0
<
2Q
Q−α
.Ï
dL
q
‰êfeŒëY5,Œ
kf
∗
k
L
q
(Ω)
≤liminf
j→+∞
kf
j
k
L
q
(Ω)
(3.5)
DOI:10.12677/aam.2022.1141931773A^êÆ?Ð
•ZV
Ú
lim
j→+∞
hI
G,α,Ω
f
j
,f
j
i= hI
G,α,Ω
f
∗
,f
∗
i.(3.6)
@o,Šâ(3.5)Ú(3.6),·‚k
D
G,α,q
(Ω) :=lim
j→+∞
R
Ω
R
Ω
G(ξ)f
j
(ξ)|η
−1
ξ|
α−Q
f
j
(η)G(η)dηdξ
kfk
2
L
q
(Ω)
=
lim
j→+∞
R
Ω
R
Ω
G(ξ)f
j
(ξ)|η
−1
ξ|
α−Q
f
j
(η)G(η)dηdξ
lim
j→+∞
kfk
2
L
q
(Ω)
=
lim
j→+∞
hI
G,α,Ω
f
j
,f
j
i
liminf
j→+∞
kf
j
k
2
L
q
(Ω)
≤
hI
G,α,Ω
f
∗
,f
∗
i
kf
∗
k
2
L
q
(Ω)
,
ù•Ò´`,f
∗
´˜‡4Œ.Úny.2
N´yUþD
G,α,q
(Ω)4Œf(ξ),3ƒ˜‡~ê¦fœ/e,÷ve•§:
f
q−1
(ξ) =
Z
Ω
G(ξ)f(η)G(η)
|η
−1
ξ|
Q−α
dη,ξ∈
¯
Ω.(3.7)
-g(ξ) = f
q−1
(ξ)…q
0
=
q
q−1
,Kf(ξ) = g
1
q−1
(ξ) = g
q
0
−1
(ξ),ùpf∈L
q
(Ω).Ïd,(3.7)=z¤
g(ξ) =
Z
Ω
G(ξ)g
q
0
−1
(η)G(η)
|η
−1
ξ|
Q−α
dη,ξ∈
¯
Ω,(3.8)
Ù¥2 <q
0
<p
α
…g∈L
q
0
(Ω).
•¤½n1•e(Øy²,·‚„I‡y²e¡K5Ún.
Ún3bg∈L
q
0
(Ω)´•§(3.8)˜‡)±9¼êG(ξ)∈C(
¯
Ω).XJq
0
<p
α
,@oé
u0 <α≤1,g∈Γ
α
(
¯
Ω) ⊂C
α
2
(
¯
Ω).
y²e¡,·‚ò©ü‡Ú½5y²g∈Γ
α
(
¯
Ω) ⊂C
α
2
(
¯
Ω).
1˜Ú.Äk,·‚5y²g∈L
∞
(
¯
Ω)∪C(
¯
Ω).
dH¨olderØª,Œ
g(ξ) =
Z
Ω
G(ξ)g
q
0
−1
(η)G(η)
|η
−1
ξ|
Q−α
dη≤C
Z
Ω
g
q
0
−1
(η)
|η
−1
ξ|
Q−α
dη
≤Ckg
q
0
−1
k
L
m
(Ω)
·k|η
−1
ξ|
α−Q
k
L
m
0
(Ω)
= Ckg
q
0
−1
k
L
s
∗
q
0
−1
(Ω)
·k|η
−1
ξ|
α−Q
k
L
(
s
∗
q
0
−1
)
0
(Ω)
= Ckgk
q
0
−1
L
s
∗
(Ω)
·(
Z
Ω
|η
−1
ξ|
(α−Q)·(
s
∗
q
0
−1
)
0
dη)
1
(
s
∗
q
0
−1
)
0
≤Ckgk
q
0
−1
L
s
∗
(Ω)
,
DOI:10.12677/aam.2022.1141931774A^êÆ?Ð
•ZV
Ù¥m=
s
∗
q
0
−1
†m
0
= (
s
∗
q
0
−1
)
0
p•Ýê…
s
∗
q
0
−1
>
Q
α
>1.@o,XJ·‚Ž‡yg∈L
∞
(
¯
Ω),·‚
•I‡Øy•3,˜~ês
∗
>0¦g∈L
s
∗
(Ω)…
s
∗
q
0
−1
>
Q
α
=Œ.˜g∈L
∞
(
¯
Ω)y²,·‚Ò
Œ±Šâ››Âñ½nêþíÑg∈C(
¯
Ω).e5,·‚ò©n«œ/5(½s
∗
Š.
œ/I.eq
0
<
Q
Q−α
,·‚Œs
∗
= q
0
,Ks
∗
w,÷vg∈L
s
∗
(Ω) = L
q
0
(Ω)…
s
∗
q
0
−1
=
q
0
q
0
−1
>
Q
α
.
œ/II.eq
0
=
Q
Q−α
,-k=[
Q
Q−α
] + 1…q
1
=(1 −
1
k
)
Q
Q−α
<
Q
Q−α
=q
0
,Kkg∈L
q
1
(Ω).Š
â(3.8)ÚHLSØªéó/ª(2.5),·‚
kg(ξ)k
L
s
∗
(Ω)
= k
Z
Ω
G(ξ)g
q
0
−1
(η)G(η)
|η
−1
ξ|
Q−α
dηk
L
s
∗
(Ω)
≤kC
Z
Ω
g
q
0
−1
(η)
|η
−1
ξ|
Q−α
dηk
L
s
∗
(Ω)
= Ckg
q
0
−1
∗|ξ|
α−Q
k
L
s
∗
(Ω)
≤Ckg
q
0
−1
k
L
ν
(Ω)
,
ùp
1
s
∗
=
1
ν
−
α
Q
,Ù¥1 <ν<s
∗
<+∞.Ø”-ν=
q
1
q
0
−1
>1,K
1
s
∗
=
q
0
−1
q
1
−
α
Q
=
α
(k−1)Q
.Ïd,·
‚Œs
∗
=
(k−1)Q
α
,Ks
∗
w,÷vg∈L
s
∗
(Ω)…
s
∗
q
0
−1
>
Q
α
.
œ/III.e
Q
Q−α
<q
0
<
2Q
Q−α
,d?·‚ò|^S“•{é~ês
∗
.Šâ(3.8)ÚHLSØª
éó/ª(2.5),·‚Œ±íÑ
kg(ξ)k
L
s
(Ω)
= k
Z
Ω
G(ξ)g
q
0
−1
(η)G(η)
|η
−1
ξ|
Q−α
dηk
L
s
(Ω)
≤kC
Z
Ω
g
q
0
−1
(η)
|η
−1
ξ|
Q−α
dηk
L
s
(Ω)
= Ckg
q
0
−1
∗|ξ|
α−Q
k
L
s
(Ω)
≤Ckg
q
0
−1
k
L
µ
(Ω)
,
(3.9)
ùp
1
s
=
1
µ
−
α
Q
,Ù¥1<µ<s<+∞.;X,·‚ò$^þãØª(3.9)5ŠS“.˜
m©,-µ=
q
0
q
0
−1
:=
µ
1
q
0
−1
±9µ
2
=s.쥵
2
=s=(
1
µ
−
α
Q
)
−1
…
2Q
Q+α
<µ=
1
1−
1
q
0
<
Q
α
,¤
±µ
2
= s>
2Q
Q−α
>q
0
= µ
1
.
òþãL§?1S“:-µ=
µ
i
q
0
−1
,Kµ
i+1
=s,Ù¥i=1,2,···.5¿q
0
−2 <
2α
Q−α
…
1
t
i+1
<
Q−α
2Q
.Ïd,²L˜•©Û,ØJÑt
i+1
>0ž,kt
i+1
>t
i
.¤±,3S“êõgƒ,•Ò´
`k
0
gž,·‚¬ü«œ¹:
t
k
0
q
0
−1
<
Q
α
Ú
t
k
0
+1
q
0
−1
≥
Q
α
.
XJ
t
k
0
+1
q
0
−1
>
Q
α
,·‚Œs
∗
= t
k
0
+1
,@os
∗
w,÷vg∈L
s
∗
(Ω)…
s
∗
q
0
−1
>
Q
α
.
XJ
t
k
0
+1
q
0
−1
=
Q
α
, @og∈L
t
k
0
+1
(Ω) = L
Q
α
(q
0
−1)
(Ω).-k= [
2Q
Q−α
]+1±9q
2
= (1−
1
k
)(q
0
−1)
Q
α
<
t
k
0
+1
=
Q
α
(q
0
−1),dd·‚Œ±íäg∈L
q
2
(Ω).Šâ(3.8)ÚHLSØªéó/ª(2.5),·‚
˜‡aq(3.9)Øª:
kg(ξ)k
L
s
∗
(Ω)
≤Ckg
q
0
−1
k
L
ω
(Ω)
,
ùp
1
s
∗
=
1
ω
−
α
Q
,Ù¥1 <ω<s
∗
<+∞.Ø”-ω=
q
2
q
0
−1
>1,Kk
1
s
∗
=
q
0
−1
q
2
−
α
Q
=
α
(k−1)Q
.¤±,
·‚Œs
∗
=
(k−1)Q
α
,w,s
∗
÷vg∈L
s
∗
(Ω)…
s
∗
q
0
−1
>
Q
α
.
Ïd,g∈L
∞
(
¯
Ω)∪C(
¯
Ω).
1Ú.Ùg,·‚‡y²g∈Γ
α
(
¯
Ω),Ù¥0<α≤1.ùp,·‚ò|^Lipschitz˜mΓ
α
½
©±eü«œ/5?Ø.
œ/i.éu0 <α<1,·‚I‡Øysup
ξ,γ
|g(ξγ)−g(ξ)|
|γ|
α
<∞.
DOI:10.12677/aam.2022.1141931775A^êÆ?Ð
•ZV
dg∈L
∞
(Ω)Ú(3.8),Œ
|g(ξγ)−g(ξ)|= |
Z
Ω
G(ξγ)g
q
0
−1
(η)G(η)
|η
−1
ξγ|
Q−α
dη−
Z
Ω
G(ξ)g
q
0
−1
(η)G(η)
|η
−1
ξ|
Q−α
dη|
≤Ckgk
q
0
−1
L
∞
(Ω)
|
Z
Ω
(G(ξγ)|ηγ|
α−Q
−G(ξ)|η|
α−Q
)dη|
≤Ckgk
q
0
−1
L
∞
(Ω)
Z
Ω
|G(ξγ)|ηγ|
α−Q
−G(ξ)|η|
α−Q
|dη
≤Ckgk
q
0
−1
L
∞
(Ω)
(
Z
Ω
G(ξγ)|ηγ|
α−Q
dη+
Z
Ω
G(ξ)|η|
α−Q
dη)
≤Ckgk
q
0
−1
L
∞
(Ω)
(
Z
Ω
|ηγ|
α−Q
dη+
Z
Ω
|η|
α−Q
dη)
= Ckgk
q
0
−1
L
∞
(Ω)
[
Z
Ω
(|ηγ|
α−Q
−|η|
α−Q
)dη+2
Z
Ω
|η|
α−Q
dη]
≤Ckgk
q
0
−1
L
∞
(Ω)
[
Z
Ω
||ηγ|
α−Q
−|η|
α−Q
|dη+2
Z
Ω
|η|
α−Q
dη].
(3.10)
©)
Z
Ω
||ηγ|
α−Q
−|η|
α−Q
|dη=
Z
Ω∩{|η|≥2|γ|}
||ηγ|
α−Q
−|η|
α−Q
|dη+
Z
Ω∩{|η|≤2|γ|}
||ηγ|
α−Q
−|η|
α−Q
|dη
Ú
Z
Ω
|η|
α−Q
dη=
Z
Ω∩{|η|≥2|γ|}
|η|
α−Q
dη+
Z
Ω∩{|η|≤2|γ|}
|η|
α−Q
dη.
e¡,·‚ò駂?1ŘO.˜•¡,Šâ·K3¥ª(2.3),·‚UO
Z
Ω∩{|η|≥2|γ|}
||ηγ|
α−Q
−|η|
α−Q
|dη≤C
Z
Ω∩{|η|≥2|γ|}
|γ||η|
α−Q−1
dη≤C|γ|
α
.(3.11)
Ï•|η|≤2|γ|,¤±d·K2¥ª(2.2)Œ:•3˜‡~êC≥1,¦|ηγ|≤C(|η|+ |γ|)≤
C(2|γ|+|γ|) = 3C|γ|.Ï,·‚Œ±O
Z
Ω∩{|η|≤2|γ|}
||ηγ|
α−Q
−|η|
α−Q
|dη≤
Z
Ω∩{|η|≤2|γ|}
|ηγ|
α−Q
dη+
Z
Ω∩{|η|≤2|γ|}
|η|
α−Q
dη
≤
Z
Ω∩{|ηγ|≤3C|γ|}
|ηγ|
α−Q
d(ηγ)+
Z
Ω∩{|η|≤2|γ|}
|η|
α−Q
dη
≤C|γ|
α
.
(3.12)
Ïd,d(3.11)Ú(3.12)ŒíÑ
Z
Ω
||ηγ|
α−Q
−|η|
α−Q
|dη≤C|γ|
α
.(3.13)
,˜•¡,·‚Œ±O
Z
Ω∩{|η|≥2|γ|}
|η|
α−Q
dη≤C|γ|
α
(3.14)
DOI:10.12677/aam.2022.1141931776A^êÆ?Ð
•ZV
Ú
Z
Ω∩{|η|≤2|γ|}
|η|
α−Q
dη≤C|γ|
α
.(3.15)
¤±,Šâ(3.14)Ú(3.15),·‚k
Z
Ω
|η|
α−Q
dη≤C|γ|
α
.(3.16)
ò(3.13)Ú(3.16)“\(3.10)¥,
|g(ξγ)−g(ξ)|≤Ckgk
q
0
−1
L
∞
(Ω)
|γ|
α
,
†é{`,sup
ξ,γ
|g(ξγ)−g(ξ)|
|γ|
α
<∞,ùL²0 <α<1ž,g∈Γ
α
(Ω).
œ/ii.éuα= 1,·‚I‡ysup
ξ,γ
|g(ξγ)+g(ξγ
−1
)−2g(ξ)|
|γ|
<∞.
dg∈L
∞
(Ω)Ú(3.8),Œ
|g(ξγ)+g(ξγ
−1
)−2g(ξ)|
= |
Z
Ω
G(ξγ)g
q
0
−1
(η)G(η)
|η
−1
ξγ|
Q−α
dη+
Z
Ω
G(ξγ
−1
)g
q
0
−1
(η)G(η)
|η
−1
ξγ
−1
|
Q−α
dη−2
Z
Ω
G(ξ)g
q
0
−1
(η)G(η)
|η
−1
ξ|
Q−α
dη|
≤Ckgk
q
0
−1
L
∞
(Ω)
|
Z
Ω
(G(ξγ)|η
−1
ξγ|
α−Q
+G(ξγ
−1
)|η
−1
ξγ
−1
|
α−Q
−2G(ξ)|η
−1
ξ|
α−Q
)dη|
≤Ckgk
q
0
−1
L
∞
(Ω)
|
Z
Ω
(G(ξγ)|ηγ|
α−Q
+G(ξγ
−1
)|ηγ
−1
|
α−Q
−2G(ξ)|η|
α−Q
)dη|
≤Ckgk
q
0
−1
L
∞
(Ω)
Z
Ω
|G(ξγ)|ηγ|
α−Q
+G(ξγ
−1
)|ηγ
−1
|
α−Q
−2G(ξ)|η|
α−Q
|dη
≤Ckgk
q
0
−1
L
∞
(Ω)
(
Z
Ω
G(ξγ)|ηγ|
α−Q
dη+
Z
Ω
G(ξγ
−1
)|ηγ
−1
|
α−Q
dη+2
Z
Ω
G(ξ)|η|
α−Q
dη)
≤Ckgk
q
0
−1
L
∞
(Ω)
Z
Ω
(|ηγ|
α−Q
+|ηγ
−1
|
α−Q
+|η|
α−Q
)dη
= Ckgk
q
0
−1
L
∞
(Ω)
[
Z
Ω
(|ηγ|
α−Q
+|ηγ
−1
|
α−Q
−2|η|
α−Q
)dη+3
Z
Ω
|η|
α−Q
dη]
≤Ckgk
q
0
−1
L
∞
(Ω)
(
Z
Ω
||ηγ|
α−Q
+|ηγ
−1
|
α−Q
−2|η|
α−Q
|dη+3
Z
Ω
|η|
α−Q
dη).
(3.17)
©)
Z
Ω
||ηγ|
α−Q
+|ηγ
−1
|
α−Q
−2|η|
α−Q
|dη=
Z
Ω∩{|η|≥2|γ|}
||ηγ|
α−Q
+|ηγ
−1
|
α−Q
−2|η|
α−Q
|dη
+
Z
Ω∩{|η|≤2|γ|}
||ηγ|
α−Q
+|ηγ
−1
|
α−Q
−2|η|
α−Q
|dη.
DOI:10.12677/aam.2022.1141931777A^êÆ?Ð
•ZV
y3,·‚ò駂?1ŘO.˜•¡,Šâ·K3¥ª(2.4),·‚UO
Z
Ω∩{|η|≥2|γ|}
||ηγ|
α−Q
+|ηγ
−1
|
α−Q
−2|η|
α−Q
|dη
≤C
Z
Ω∩{|η|≥2|γ|}
|γ|
2
|η|
α−Q−2
dη
= C|γ|
2
Z
Ω∩{|η|≥2|γ|}
|η|
α−Q−2
dη
≤C|γ|
α
.
(3.18)
,˜•¡,Ï•|η|≤2|γ|,ù¿›X|η|≤2|γ
−1
|,¤±d ·K2¥ª(2.2),ŒíäÑ:•3˜‡~
êC≥1,¦
|ηγ|≤C(|η|+|γ|) ≤C(2|γ|+|γ|) = 3C|γ|
±9
|ηγ
−1
|≤C(|η|+|γ
−1
|) ≤C(2|γ
−1
|+|γ
−1
|) = 3C|γ
−1
|.
Ï,·‚Œ±O
Z
Ω∩{|η|≤2|γ|}
||ηγ|
α−Q
+|ηγ
−1
|
α−Q
−2|η|
α−Q
|dη
≤
Z
Ω∩{|η|≤2|γ|}
|ηγ|
α−Q
dη+
Z
Ω∩{|η|≤2|γ|}
|ηγ
−1
|
α−Q
dη+2
Z
Ω∩{|η|≤2|γ|}
|η|
α−Q
dη
≤
Z
Ω∩{|ηγ|≤3C|γ|}
|ηγ|
α−Q
d(ηγ)+
Z
Ω∩{|ηγ
−1
|≤3C|γ
−1
|}
|ηγ
−1
|
α−Q
d(ηγ
−1
)+2
Z
Ω∩{|η|≤2|γ|}
|η|
α−Q
dη
≤C|γ|
α
.
(3.19)
Ïd,éÜ(3.18)Ú(3.19),·‚U
Z
Ω
||ηγ|
α−Q
+|ηγ
−1
|
α−Q
−2|η|
α−Q
|dη≤C|γ|
α
.(3.20)
‘,ò(3.16)Ú(3.20)“\(3.17)¥,Œ
|g(ξγ)+g(ξγ
−1
)−2g(ξ)|≤Ckgk
q
0
−1
L
∞
(Ω)
|γ|
α
,
•Ò´`,α= 1ž,sup
ξ,γ
|g(ξγ)+g(ξγ
−1
)−2g(ξ)|
|γ|
α
<∞.ddŒíÑg∈Γ
1
(Ω).
Ïd,(Üœ/iÚœ/ii,·‚Œ±g∈Γ
α
(Ω),Ù¥0<α≤1.,,·‚d·K5Œ,
0 <α≤1ž,kg∈Γ
α
(
¯
Ω) ⊂C
α
2
(
¯
Ω).Úny. 2
½n1y²ŠâÚn2ÚÚn3,·‚Œ±Ñ:é?¿‰½λ∈R,•§(1.1)Ñk˜‡
)f∈L
∞
(
¯
Ω)∪C(
¯
Ω).AO/,0 <α≤1ž,)f∈Γ
α
(
¯
Ω) ⊂C
α
2
(
¯
Ω).½ny.2
DOI:10.12677/aam.2022.1141931778A^êÆ?Ð
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4.o(
3©¥,·‚®²3g.q
α
<q<2œ/e,é?¿‰½λ∈R,•§(1.1)Ñk
˜‡)f∈L
∞
(
¯
Ω)∪C(
¯
Ω).AO/,0<α≤1ž,)f∈Γ
α
(
¯
Ω)⊂C
α
2
(
¯
Ω).Ødƒ,„k
˜'uTš‚5È©•§¯Kk–)û,'X:H
n
k.«•þ†°(‡•HLSØªk
'‘K˜š‚5È©•§(1.1))•35,ùp0<q<1,α>Q,0<β<α−Q,
Q= 2n+2´H
n
àg‘ê,λ∈R,Ω ⊂H
n
´˜‡1wk.•…G(ξ)´
¯
Ω¥šKëY¼ê.
ë•©z
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