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PureMathematicsnØêÆ,2022,12(9),1441-1456
PublishedOnlineSeptember2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.129157
©êg4ŒŽf9Ù†f3Lie+Š^e
2ÂOrlicz-Morrey˜mþO
§§§ÍÍÍ111ŸŸŸ
∗
§§§oooÈÈÈrrr
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2022c87F¶¹^Fϵ2022c96F¶uÙFϵ2022c913F
Á‡
©Äk‰Ñ©o+GŠ^e2ÂOrlicz-Morrey˜mM
Φ,ϕ
(G)½Â§Ùg|^H¨older
Øª±9¼ê ©)•{§©êg4ŒŽfM
α
3d˜mþk .5O§•y²©
êg4ŒŽfM
α
†BMO ¼ê)¤†fM
b,α
lM
Φ,ϕ
(G) M
Ψ,η
(G) þk.5"
'…c
©êg4ŒŽf§†f§2ÂOrlicz-Morrey˜m§BMO(G) ˜m§Lie+
FractionalMaximalOperatorand
ItsCommutatoronGeneralized
Orlicz-MorreySpacesover
LieGroups
LiRui,GuanghuiLu
∗
,XuemeiLi
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Aug.7
th
,2022;accepted:Sep.6
th
,2022;published:Sep.13
th
,2022
∗ÏÕŠö"
©ÙÚ^:Â,Í1Ÿ,oÈr.©êg4ŒŽf9Ù†f3Lie+Š^e2ÂOrlicz-Morrey˜mþ
O[J].nØêÆ,2022,12(9):1441-1456.DOI:10.12677/pm.2022.129157
Â
Abstract
Thisarticle firstgivesthedefinitionofgeneralizedOrlicz-MorreyM
Φ,ϕ
(G) on stratified
LiegroupG;secondprovesthatthefractionalmaximaloperatorM
α
isboundedfrom
spacesM
Φ,ϕ
(G)intospacesM
Ψ,η
(G)bymeansofH¨olderinequalityandthemethod
offunctiondecomposition.Furthermore,theboundednessofthecommutatorM
b,α
generatedbyb∈BMO(G)fromspacesM
Φ,ϕ
(G)intospacesM
Ψ,η
(G)isalsoobtained.
Keywords
FractionalMaximalOperator,Commutator,GeneralizedOrlicz-MorreySpace,Space
BMO(G),LieGroup
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.Úó9̇(J
¯¤±•,Orlicz˜m½ÂŒ±ïá3[1,2]²;Lebesgue ˜m*Ѓþ,§3NÚ©ÛÚ
‡©•§¥åX'…Š^(ë„[3–7]). gd,Nõ©ÙÑ8¥?ØOrlicz ˜m5Ÿ±93ù
˜mþÈ©Žfk .5. ~X, 32012 c, KawasumiÚNakai 3©z[8] ¥Orlicz˜m
þòÈŽfFá“C†˜5Ÿ,•õ'uOrlicz˜mïÄŒ„©z[9–12].
;X, 32004c, NakaiÄg¼Orlicz-Morrey ˜m½Â(„©z[13]).d, Orlicz-
Morrey˜mþÈ©Žfk.5É2•'5. ~X,3©z[14] ¥ŠöïÄOrlicz4ŒŽf,
Orlicz ©êg4ŒŽfÚ©êgÈ©Žf3Morrey ˜mÚOrlicz-Morrey ˜mþk.5. •C,
Yamaguchi ÚNakai 3©z[15] ¥Orlicz-Morrey ˜mþ†f[b,T] Ú[b,I
ρ
] ;5¿‡
^‡.3©z[16]¥,Sawano<O3Orlicz-Morrey˜mþ2©êÈ©ŽfÚ2©ê4
ŒŽfk.5. d, Hasanov 3©z[17] ¥ÄgJÑ2ÂOrlicz-Morrey ˜m½Â, ¿
ù˜mþΦ Œ#Ng‚5ÛÉŽfk.5. 'uÈ©Ž f32ÂOrlicz-Morrey ˜mþ
ˆ«5ŸŒë„©z([18–24]).
Éþã(JÚ©z[25] éu, 3©¥, Šö̇•Ä©ê4 ŒŽf9Ù†f3?¿©
o+þ2ÂOrlicz-Morrey˜mþk.5.
DOI:10.12677/pm.2022.1291571442nØêÆ
Â
·‚Äk£˜'u©Lie+ÐÚïÄ(ë\©z[17,26–28]).bG´†Gƒ'Lie
+,Ù¥G´k•‘,˜•ê•"©Lie“ê,Kù‡•êN´lGGÛ‡©ÓN.
Ïd, x= (x
ij
) ∈R
N
, 1 ≤i≤K
j
, 1 ≤j≤m, N=
P
j=1
K
j
, é?¿g∈G, kg=exp(
P
x
ij
X
ij
).
Gþàg‰ê¼ê|·|ÏL|g|=(
P
|x
ij
|
2·m!/j
) ÚQ=
P
j=1
jk
j
5½Â, ¡ŠGàg‘. Gþ
*Üξ
r
(r>0)Œ½Â•:
ξ
r
(g) = exp(
X
r
j
x
ij
X
ij
), g= exp(
X
x
ij
X
ij
).
Ù¥d(ξ
r
x) = r
Q
dx.
2022c,©z[25]¥Äg?Ø©ê4ŒŽf9Ù†f3?¿©Lie+þOrlicz-Morrey
˜mþ5Ÿ.Óž,•)éõ'u©Lie+þ2ÂOrlicz-Morrey˜mþƒ'5ŸïÄ.
Gþàg‰ê´lG[0,∞) ½Â3G\{0}þC
∞
ëY¼ê, •3x→ρ(x),¦÷v
e^‡:
(1)ρ(x
−1
) = ρ(x),
(2)é?¿r>0,kρ(ξ
r
x) = rρ(x),
(3)é?¿x,y∈G,kρ(xy) ≤c
0
(ρ(x)+ρ(y)).
Š âþã‰ê, ·‚ÏLB(x,r)={y∈G:ρ(y
−1
x)<r}½Â±x•¥%, r•Œ»¥, ^
B(x,r)
c
= G\B(x,r) L«B(x,r)Ö8, Ù¥c= c(G),
|B(x,r)|= cr
Q
,x∈G,r>0.
©êg4ŒŽfk.5nØ3C•I¼ê˜mþ®²?12•ïÄ,Œë„©z([29–32]).
©êg4ŒŽfŒ(„[25])½Â•:
M
α
f(x) =sup
x>0,r>0
|B(x,r)|
−1+
α
Q
Z
B(x,r)
|f(y)|dy,(1.1)
Ù¥|B(x,r)|L«B(x,r)þHaar ÿÝ.
e¡·‚£Á˜e'uYoung ¼ê˜PÒ.
½Â1.1[25]XJ¼êΦ:[0,∞)→[0,∞)´à,ëY,=lim
r→+0
Φ(r)=Φ(0)=0…
lim
r→∞
Φ(r) = ∞,@o¡Φ•Young ¼ê.
dYoung¼êà5ÚΦ(0) = 0Œ±wÑ:?¿Young¼êÑ•O¼ê.XJ•3s∈(0,∞)
¦Φ(r) = ∞,kr≥s.
-Γ•¤kYoung ¼êΦ 8Ü,¦
0 <Φ(r) <∞, 0 <r<∞.
XJΦ ∈Γ,KΦ 3[0,∞) ¥z‡4«mþýéëY,¿…l[0,∞) Ùg•V.
DOI:10.12677/pm.2022.1291571443nØêÆ
Â
˜‡Young ¼êΦ XJ÷vΦ(2r)≤cΦ(r) ,r>0.Ù¥C>1, @o¡Φ ÷v4
2
-^‡, PŠ
Φ∈4
2
. XJΦ∈4
2
, @oΦ∈Γ. ˜‡Young ¼êΦ e÷vΦ(2r)≤
1
2c
Φ(cr), r≥0, Ù¥c>1,
K¡Φ÷v∇
2
-^‡,PŠΦ ∈∇
2
.
éu˜‡Young ¼êΦ …0 ≤s≤∞, Φ ∈Γž,¡Φ
−1
•Φ_¼ê.…k
Φ(Φ
−1
(r)) ≤r≤Φ
−1
(Φ(r)),0 ≤r<∞,
Ù¥Φ
−1
Œ½Â•:
Φ
−1
(s) = inf{r≥0 : Φ(r) >s}(inf∅= ∞).
w,k
r≤Φ
−1
(r)
˜
Φ
−1
(r) ≤2r,r≥0,
Ù¥
˜
Φ½Â•:
˜
Φ =
(
sup{rs−Φ(s) : s∈[0,∞)},r∈[0,∞),
∞,r= ∞.
e¡‰Ñ©Lie+þk.²þ˜m(= BMO)½Â:
½Â1.2 [32]BMO(G)˜mŒ½Â•:
BMO(G) = {b∈L
1
loc
: kbk
BMO(G)
<∞}
kbk
BMO(G)
:= sup
B
1
|B|
Z
B
|b(x)−b
B
|dx<∞,(1.2)
Ù¥é¤kB⊂Gþ(., …b
B
•b3Bþ²þŠ.
‰½˜‡¼êb∈BMO(G), †fM
α,b
Ú[b,M
α
]Œ©O½Â•:
M
α,b
(f)(x) = sup
B
|B|
−1+
α
Q
Z
B
|b(x)−b(y)||f(y)|dy,(1.3)
[b,M
α
](f)(x) = sup
B
|B|
−1+
α
Q
Z
B
(b(x)−b(y))|f(y)|dy.(1.4)
NõÆöïÄM
b,α
Ú[b,M
α
] NA5,=[b,M
α
] ŒdM
b,α
››, ë•©z[33–36], ©•
I‡é†fM
b,α
?15ŸO.
e¡‰ÑLie+Š^eOrlicz˜m½Â.
½Â1.3 [25]éu˜‡Young ¼êΦ, Orlicz˜mŒ½Â•:
L
Φ
(G) =

f∈L
1
loc
(G) :
Z
G
Φ

|f(x)|
λ

dx<∞forsomeλ>0

,
DOI:10.12677/pm.2022.1291571444nØêÆ
Â
Ù¥L
Φ
loc
(G)´é?¿¥B⊂GÑkfχ
B
∈L
Φ
(G)¼êf8Ü.
L
Φ
(G)´˜‡Banach˜m,Ù˜m‰ê•:
kfk
L
Φ
(G)
= inf

λ>0 :
Z
G
Φ

|f(x)|
λ

dx≤1

.(1.5)
ŠâOrlicz˜m½Â,·‚k
Z
Ω
Φ(
|f(x)|
kfk
L
Φ
(Ω)
)dx≤1,(1.6)
Ù¥kfk
L
Φ
(Ω)
= kfχ
Ω
k
L
Φ
±9kfk
WL
Φ
(Ω)
= kfχ
Ω
k
WL
Φ
.
½Â1.4[37]ϕ(x,r) ´G×(0,∞) þ˜‡Œÿ¼ê, ¿…Φ ´?¿Young ¼ê, K
2ÂOrlicz-Morry˜mM
Φ,ϕ
(G)Œ½Â•:
M
Φ,ϕ
(G) = {f∈L
Φ
loc
(G) : kfk
M
Φ,ϕ
(G)
<∞},
Ù¥
kfk
M
Φ,ϕ
(G)
=sup
x∈G,t>0
ϕ(x,t)
−1
Φ
−1
(|B(x,t)|
−1
)kfk
L
Φ
(B(x,t))
.(1.7)
Óž,f2ÂOrlicz-Morry˜mWM
Φ,ϕ
(G)Œ½Â•:
kfk
WM
Φ,ϕ
(G)
=sup
x∈G,t>0
ϕ(x,t)
−1
Φ
−1
(|B(x,t)|
−1
)kfk
WL
Φ
(B(x,t))
<∞.(1.8)
5P1.5(1)XJΦ(r) = r
p
,1 ≤p<∞,@o2ÂOrlicz-Morry ˜mM
Φ,ϕ
(G)du2Â
Morry˜mM
p,ϕ
(G).
(2)XJϕ(r)=Φ
−1
(r
−Q
),@o2ÂOrlicz-Morry ˜mM
Φ,ϕ
(G)duOrlicz˜mL
Φ
(G).
©̇½nLãXe
½n1.6[Spanne-.(J]Φ,Ψ•Young ¼ê,0 <α<Q.…ϕ
1
,ϕ
2
∈Γ
Φ
.
(1)
r
α
Φ
−1
(r
−Q
) ≤CΨ
−1
(r
−Q
),Φ ∈∇
2
.(1.9)
@o^‡
sup
r<t<∞
ϕ
1
(t)
Ψ
−1
(t
−Q
)
Φ
−1
(t
−Q
)
≤Cϕ
2
(r),(1.10)
´M
α
lM
Φ,ϕ
1
(G)M
Ψ,ϕ
2
(G)þk.¿©^‡,Ù¥r>0,…C´†rÃ'~ê.
(2)ϕ
1
´˜‡A??4~¼ê,…Φ ∈∇
2
,K^‡
ϕ
1
(r)r
α
≤Cϕ
2
(r),
´M
α
lM
Φ,ϕ
1
(G)M
Ψ,ϕ
2
(G)k.7‡^‡,Ù¥r>0,…C´†rÃ'~ê.
DOI:10.12677/pm.2022.1291571445nØêÆ
Â
(3)^‡
r
α
Φ
−1
(r
−Q
) ≤Cϕ
−1
(r
−Q
),r>0,
÷veØª:
sup
r<t<∞
ϕ
1
(t)
Ψ
−1
(t
−Q
)
Φ
−1
(t
−Q
)
≤Cϕ
1
(r)r
α
,r>0.
Ù¥Φ ∈∇
2
,ϕ
1
´A??4~¼ê,…C>0´†rÃ'~ê.@o^‡
ϕ
1
(r)r
α
≤Cϕ
2
(r)
´M
α
lM
Φ,ϕ
1
(G)M
Ψ,ϕ
2
(G)k.¿‡^‡.
½n1.7[Adams-.(J]0<α<Q, Φ•Young¼ê,ϕ∈Γ
Φ
A??4~,η(t)=
ϕ(t)
β
,Φ(t)
β
= Ψ(t)±9Ψ(t) = Φ(t
1/β
),Ù¥β∈(0,1).
(1)Φ ∈∇
2
,Øª
t
α
ϕ(t) ≤Cϕ(t)
β
,t>0,(1.11)
´M
α
lM
Φ,ϕ
(G)M
Ψ,η
(G)k.¿©^‡,Ù¥C´†rÃ'~ê.
(2)Øª
t
α
≤Cϕ(t)
β−1
,t>0,
´M
α
lM
Φ,ϕ
(G)M
Ψ,η
(G)k.7‡^‡,Ù¥C´†rÃ'~ê.
(3)Øª
t
α
≤Cϕ(t)
β−1
,t>0,
´M
α
lM
Φ,ϕ
(G)M
Ψ,η
(G)k.¿‡^‡,Ù¥C´†rÃ'~ê,…Φ ∈∇
2
.
5P1.8Šâ½Â1.6±9Ún3.1, ·‚•α= 0ž, M´lM
Φ,ϕ
(G)M
Ψ,η
(G)k..
½n1.9[Spanne-.(J]0 <α<Q, b∈BMO(G)…Φ,Ψ •Young ¼ê.
(1)Φ ∈∇
2
,ϕ(t),Ù¥ϕ
1
∈Γ
Φ
9ϕ
2
∈Γ
Ψ
.÷ve^‡:
r
α
Φ
−1
(r
−Q
)+sup
r<t<∞
(1+
t
r
)Φ
−1
(t
−Q
)t
α
≤CΨ
−1
(r
−Q
)
KØª:
sup
r<t<∞
ϕ
1
(t)(1+
t
r
)
Φ
−1
(t
−Q
)
Ψ
−1
(t
−Q
)
≤Cϕ
2
(r)
´M
b,α
(G)lM
Φ,ϕ
1
(G)M
Ψ,ϕ
2
(G)þk.¿©^‡.
(2)ϕ
1
A??4~,…Ψ ∈4
2
,KØª:
ϕ
1
(t)t
α
≤Cϕ
2
(t)
´M
b,α
(G)lM
Φ,ϕ
1
(G)M
Ψ,ϕ
2
(G)þk.7‡^‡.
DOI:10.12677/pm.2022.1291571446nØêÆ
Â
(3)ϕ
1
A??4~, Φ∈4
2
∩5
2
±9Ψ∈4
2
, ^‡(1.9)´M
b,α
(G) lM
Φ,ϕ
1
(G) 
M
Ψ,ϕ
2
(G)þk.¿‡^‡.
þã(JŒ±æ^†©z[4]¥Ún6.1aqy²•{5y²,•{'å„,d?Ø2Kã.
½n1.10[Adams-.(J]0 <α<Q, b∈BMO(G)…Φ,Ψ•Young ¼ê.
(1)Φ ∈4
2
∩5
2
,Ψ ∈4
2
,Ù¥ϕ
1
∈Γ
Φ
…ϕ
2
∈Γ
Ψ
.^‡
r
α
Φ
−1
(r
−Q
) ≤CΨ
−1
(r
−Q
)
´M
b,α
(G)lM
Φ,ϕ
1
(G)M
Ψ,ϕ
2
(G)þk.¿©^‡.
(2)ϕ
1
A??4~,…Ψ ∈4
2
,K^‡
ϕ
1
(t)t
α
≤Cϕ
2
(t)
´M
b,α
(G)lM
Φ,ϕ
1
(G)M
Ψ,ϕ
2
(G)þk.7‡^‡.
(3)ϕ
1
A??4~,Φ ∈4
2
∩5
2
…Ψ ∈4
2
,^‡(1.9)´M
b,α
(G)lM
Φ,ϕ
1
(G)
M
Ψ,ϕ
2
(G)þk.¿‡^‡.
©¥,CL«†Ì‡ëêÃ'~ê,ÙŠ3ØÓ/•ŒUئƒÓ.
2.ý•£
3!¥,•y²Ì‡½n,·‚Äk£˜Ún.
Ún2.1 [22]Φ•˜‡Young ¼ê.
(1)eMl˜mL
Φ
(G)˜mWL
Φ
(G)k.ž,Øª
kMfk
WL
Φ
(G)
≤C
0
kfk
L
Φ
(G)
¤á,Ù¥C
0
´†fÃ'~ê.
(2)M3L
Φ
(G)þk.ž,…=Φ ∈∇
2
Øª:
kMfk
L
Φ
(G)
≤C
0
kfk
L
Φ
(G)
¤á,Ù¥C
0
´†fÃ'~ê.
Ún2.2 [19]0 <α<Q, Φ,Ψ •Young ¼ê…Φ ∈Γ, Φ∈∇
2
.@o^‡:
r
α
Φ
−1
(r
−Q
) ≤CΨ
−1
(r
−Q
)
´M
α
lL
Φ
(G)L
Ψ
(G)þk.¿‡^‡,Ù¥r>0,C´†rÃ'~ê.
Ún2.3 [19]0 <α<Q, b∈L
1
loc
(G)…Φ ∈Υ,Ψ •Young ¼ê.
DOI:10.12677/pm.2022.1291571447nØêÆ
Â
(1)Φ ∈∇
2
…^‡(3.1) ¤á,@o^‡b∈BMO(G) ´M
b,α
lL
Φ
(G)L
Ψ
(G)þk.
¿©^‡.
(2)Ψ
−1
(r
−Q
).r
α
Φ
−1
(r
−Q
), @o^‡b∈BMO(G)´M
b,α
lL
Φ
(G) L
Ψ
(G) þk.
7‡^‡.
(3)Φ ∈∇
2
±9Ψ
−1
(r
−Q
) ≈r
α
Φ
−1
(r
−Q
),@o^‡b∈BMO(G) ´M
b,α
lL
Φ
(G)
L
Ψ
(G)k.¿‡^‡.
Ún2.4 [29]E⊂G´k•Haar ÿÝ,…Φ •Young¼ê,K:
kχ
E
k
L
Φ
(G)
= kχ
E
k
WL
Φ
(G)
=
1
Φ
−1
(|E|
−1
)
.
Ún2.5 [29]éuYoung ¼êΦ ±9?¿¥B,k:
Z
B
|f(y)|dy≤2|B|Φ
−1
(|B|
−1
)kfk
L
Φ
(B)
.
Ún2.6f,g•EþŒÿ¼ê, …E⊂G•Œÿ8. éuYoung ¼êΦ 9ÙÖ¼ê
˜
Φ,
k:
Z
E
|f(x)g(x)|dx≤2kfk
L
Φ
(E)
kgk
L
˜
Φ
(E)
.
Ún2.70 <α<Q, Φ,Ψ •Young ¼ê.e•3˜‡~êC¦:
r
α
Φ
−1
(r
−Q
) ≤CΨ
−1
(r
−Q
).
Kéu?Ûf∈L
Φ
loc
(G)…B= B(x,r), ·‚k:
kM
α
fk
WL
Ψ
(B) ≤
1
Ψ
−1
(r
−Q
)
sup
t>r
Ψ
−1
(t
−Q
)kfk
L
Φ
(B(x,t))
.
Ún2.8 [4]B
0
= B(x
0
,r
0
),K|B
0
|
α
Q
.M
α
χ
B
0
(x)
,Ù¥?¿x∈B
0
.
3.½n1.6Ú½n1.7y²
du†|^ 2ÂOlicz-Morrey ˜m½Ây²k.5•3˜½(J, ¤±·‚Äk‡
©êg4ŒŽf3Olicz˜mþ˜‡Øª,=k.5y²xù.
Ún3.10<α<Q, …Φ,Ψ •Young ¼ê. Φ
−1
ÚΨ
−1
÷v^‡(1.10),Ké?¿
f∈L
Φ
loc
(G),k:
kM
α
fk
L
Ψ
(B)
≤
1
Ψ
−1
(r
−Q
)
sup
t>r
Ψ
−1
(t
−Q
)kfk
L
Φ
(B(x,t))
.
y²Ún3.1éu?¿f∈L
Φ
loc
(G), f= f
1
+f
2
, Ù¥f
1
=fχ
2k
0
B
, f
2
=fχ
(2k
0
B)
c
.@
DOI:10.12677/pm.2022.1291571448nØêÆ
Â
o
M
α
f(z) = M
α
f
1
(z)+M
α
f
2
(z).
ŠâÚn2.3,·‚k
kM
α
f
1
k
L
Ψ
(B)
≤kM
α
f
1
k
L
Ψ
(G)
≤Ckf
1
k
L
Φ
(G)
= Ckfk
L
Φ
(B(x,2k
0
r))
,
Ù¥
kfk
L
Φ
(B(x,2k
0
r))
= kfk
L
Φ
(B(x,2k
0
r))
sup
t>2k
0
r
Ψ
−1
(t
−Q
)Ψ(t
−Q
)
≤
kfk
L
Φ
(B(x,2k
0
r))
Ψ
−1
(r
−Q
)
sup
t>2k
0
r
Ψ
−1
(t
−Q
)
≤
1
Ψ
−1
(r
−Q
)
sup
t>2k
0
r
Ψ
−1
(t
−Q
)kfk
L
Φ
(B(x,t))
.
z•B¥?¿˜:,B(z,t)∩(G\2k
0
B) 6= ∅ž,@ot>r.Óž,XJy∈B(z,t)∩(G\2k
0
B),
·‚ÒŒ±t>ρ(y
−1
z)≥
1
k
0
ρ(x
−1
y)−ρ(x
−1
z)>2r−r=r,Ù¥k
0
´÷vρ(yz)=
k
0
(ρ(y)+ρ(z))~ê.
,˜•¡,XJy∈B(z,t)∩(G\2k
0
B),·‚kρ(x
−1
y) ≤ρ(y
−1
z)+ρ(x
−1
z) <t+r<2t≤2k
0
t.
¤±,
M
α
f
2
(z) = sup
t>0
1
|B(z,t)|
1−
α
Q
Z
B(z,t)∩(2k
0
B)
c
|f(y)|dy
= sup
t>0
1
|B(z,t)|
1−
α
Q
|B(x,2k
0
t)|
1−
α
Q
|B(x,2k
0
t)|
1−
α
Q
Z
B(z,t)∩(2k
0
B)
c
|f(y)|dy
≤Csup
t>r
1
|B(x,2k
0
t)|
1−
α
Q
Z
B(x,2k
0
t)
|f(y)|dy
≤C sup
t>2k
0
r
1
|B(x,t)|
1−
α
Q
Z
B(x,t)
|f(y)|dy.
|^Ún2.5,·‚k
kM
α
f
2
k
L
Ψ
(B)
≤C
1
Ψ
−1
(r
−Q
)
sup
t>r
t
α−Q
|B(x,t)|Φ
−1
(|B(x,t)|
−1
)kfk
L
Φ
(B(x,t))
≤C
1
Ψ
−1
(r
−Q
)
sup
t>r
Ψ
−1
(t
−Q
)kfk
L
Φ
(B(x,t))
.
(ܱþO,Œ±:
kM
α
fk
L
Ψ
(B)
≤C
1
Ψ
−1
(r
−Q
)
sup
t>r
Ψ
−1
(t
−Q
)kfk
L
Φ
(B(x,t))
.
DOI:10.12677/pm.2022.1291571449nØêÆ
Â
y²½n1.6(1)ÏL|^Ún3.1 ±9(1.10), ·‚k
kM
α
fk
M
Ψ,ϕ
2
(G)
=sup
X∈G,r>0
ϕ
−1
2
(x,r)Ψ
−1
(r
−Q
)kM
α
fk
L
Ψ
(B)
≤Csup
X∈G,r>0
ϕ
−1
2
(x,r)sup
t>r
Ψ
−1
(t
−Q
)kfk
L
Φ
(B(x,t))
≤Csup
X∈G,r>0
ϕ
−1
2
(x,r)sup
t>r
Ψ
−1
(t
−Q
)Φ(t
−Q
)ϕ
1
(x,t)kfk
M
Φ,ϕ
1
(G)
≤kfk
M
Φ,ψ
1
(G)
,
(2)-B
0
=B(x
0
,t
0
) …x∈B
0
, K(ÜÚn2.8 ·‚kr
α
0
≤M
α
χ
B
0
(x)
. 2orlicz‰ê, …
M
α
lM
Φ,ϕ
1
(G)M
Ψ,ϕ
2
(G)´k.,Œ±:
r
α
0
.Ψ
−1
(|B
0
|
−1
)kM
α
χ
B
0
k
L
ψ
(B
0
)
.ϕ
2
(x
0
,t
0
)kM
α
χ
B
0
k
M
Ψ,ϕ
2
(G)
.ϕ
2
(x
0
,t
0
)kχ
B
0
k
M
Φ,ϕ
1
(G)
.
ϕ
2
(x
0
,t
0
)
ϕ
(
x
0
,t
0
)
.
(3)½n1nÜ©y²Œ±Šâ½n1˜Ü©Ú1Ü©.
y²½n1.7(1)-B:= B(x,r) ´˜‡±x•¥%, ±r•Œ»¥.e¡·‚òf©)•:
f:= f
1
+f
2
= fχ
B
+fχ
G\2B
.
ÏLŽfM
α
g‚5,k
M
α
f(x) ≤M
α
f
1
(x)+M
α
f
2
(x) = D
1
+D
2
.
Šâ©z[38]HedbergE|,éN´:
D
1
= M
α
f
1
(x) ≤Cr
α
Mf(x).
y3·‚5OD
2
,(ܪ(1.1)ÚÚn2.5,·‚k:
M
α
f
2
(x) = sup
t>0
1
|B(x,t)|
1−
α
Q
Z
B(x,t)
|f
2
(y)|dy
≤Csup
t>0
t
α
|B(z,t)|
Z
B(x,t)
T
{G\2B(z,r)}
|f(y)|dy
≤Csup
r<t<∞
t
α
|B(x,t)|
Z
B(x,t)
|f(y)|dy
≤Csup
r<t<∞
t
α
Φ
−1
(|B(x,t)|
−1
)kfk
L
Φ
(B(x,t))
.
DOI:10.12677/pm.2022.1291571450nØêÆ
Â
d,(ܽÂ1.4,ª(1.11)±9éD
1
ÚD
2
O,·‚Œ±íäÑ:
M
α
f(x) ≤Cr
α
Mf(x)+Csup
r<t<∞
t
α
Φ
−1
(|B(z,t)|
−1
)kfk
L
Φ
(B(x,t))
≤Cr
α
Mf(x)+Ckfk
M
Φ,ϕ
(G)
sup
r<t<∞
t
α
ϕ(t)
≤C[ϕ(r)]
β
[ϕ(r)]
−1
Mf(x)+C[ϕ(r)]
β
kfk
M
Φ,ϕ
(G)
≤C[ϕ(r)]
β−1
Mf(x)+C[ϕ(r)]
β
kfk
M
Φ,ϕ
(G)
≤min

Cϕ(r)
β−1
Mf(x),Cϕ(r)
β
kfk
M
Φ,ϕ
(G)

≤sup
s>0
min

Cs
β−1
Mf(x),Cs
β
kfk
M
Φ,ϕ
(G)

≤C(Mf(x))
β
kfk
1−β
M
Φ,ϕ
(G)
,
(ܽÂ1.3,·‚k:
Z
B
Ψ
(Mf(z))
β
kMfk
β
L
Φ
(B)
!
dz=
Z
B
Φ
Mf(z)
kMfk
L
Φ
(B)
!
dz≤1,
'uþãØªþ(.,éN´:
k(Mf)
β
k
L
Ψ
(B)
≤kMfk
β
L
Φ
(B)
.(3.1)
ÏL½Â1.4,Ún2,1±9ª(3.1),·‚k:
kM
α
fk
M
Ψ,η
(G)
=sup
x∈G,t>0
η(x,t)
−1
Ψ
−1
(|B(x,t)|
−1
)kM
α
(f)k
L
Ψ
(B(x,t))
≤Csup
x∈G,t>0
η(x,t)
−1
Ψ
−1
(|B(x,t)|
−1
)k(Mf(x))
β
k
L
Ψ
(B(x,t))
kfk
1−β
M
Φ,ϕ
(G)
≤C

sup
x∈G,t>0
ϕ(x,t)
−1
Φ
−1
(|B(x,t)|
−1
)kMfk
L
Φ
(B(x,t))

β
kfk
1−β
M
Φ,ϕ
(G)
≤Ckfk
M
Φ,ϕ
(G)
.
(2)-B
0
= B(x
0
,t
0
)…x∈B
0
.ŠâÚn2.8,·‚k:
r
α
0
≤CM
α
χ
B
0
(x).
Ïd,éþãØªOrlicz‰ê,·‚k:
r
α
0
≤CΨ
−1
(t
−Q
0
)kM
α
χ
B
0
k
L
Ψ
(B
0
)
≤Cη(t
0
)kM
α
χ
B
0
k
M
Ψ,η
(G)
≤Cη(t
0
)kχ
B
0
k
M
Φ,ϕ
(G)
≤Cϕ(t
0
)
β−1
.
DOI:10.12677/pm.2022.1291571451nØêÆ
Â
(3)½n1nÜ©y²Œ±Šâ½n1˜Ü©Ú1Ü©.
4.½n1.10y²
•y²½n1.10,·‚I‡±eÚn:
Ún4.1 [25]f∈L
1
loc
(G),Ke^‡d:
(1)b∈BMO(G) …f
−
∈L
∞
(G).
(2)•3t∈[1,∞),kØª:
sup
B
k(f(·)−M
B
(f)(·))χ
B
k
L
t
(G)
kχ
B
k
L
t
(G)
≤C.
(3)éu¤kt∈[1,∞),(2) ¥Øª¤á.
Ún4.2 [25]b∈L
1
loc
(G)…B
0
:= B(x
0
,r
0
),KeØª¤á:
r
α
0
|b(x)−b
B
0
|≤CM
b,α
χ
B
0
(x),∀x∈B
0
.
Óž,·‚I‡éBMO(G)˜m?1£ã,Œ„ë•©z[39].
Ún4.3f∈BMO(G)±9Φ ∈4
2
•Young ¼ê,K
kfk
BMO(G)
≈sup
B
Φ
−1
(|B|
−1
)kf(·)−f
B
k
L
Φ
(B)
,
Ù¥þ(.L«¤k¥B⊂G.
e¡é©êg4ŒŽf†fOŒ„©z[40],=©êg4ŒŽf››'X.
Ún4.40<α<Q±9b∈BMO(G).K•3~êC>0¦é?¿x∈GÚ
f∈L
1
loc
(G),k:
M
b,α
f(x) ≤Ckbk
BMO(G)
[M(M
α
f)(x)+M
α
(Mf)(x)].
y²½n1.10(1)ÏL(ܽn1.7(1)ÚÚn4.4, ·‚Œ±
kM
b,α
fk
M
Ψ,ϕ
2
(G)
≤Ckbk
BMO
kM(M
α
f)+M
α
(Mf)k
M
Ψ,ϕ
2
(G)
≤Ckbk
BMO
(kM
α
fk
M
Ψ,ϕ
2
(G)
+kMfk
M
Ψ,ϕ
2
(G)
)
≤Ckbk
BMO
kfk
M
Φ,ϕ
1
(G)
.
DOI:10.12677/pm.2022.1291571452nØêÆ
Â
(2)-B
0
= B(x
0
,t
0
).Šâ½Â1.4,½n2.4,4.29½n4.3,·‚k:
t
α
0
≤C
kM
b,α
χ
B
0
k
L
Ψ
(B
0
)
kb(·)−b
B
0
k
L
Ψ
(B
0
)
≤C
1
kbk
BMO
kM
b,α
χ
B
0
k
L
Ψ
(B
0
)
Ψ
−1
(|B
0
|
−1
)
≤C
1
kbk
BMO
ϕ
2
(t
0
)kM
b,α
χ
B
0
k
M
Ψ,ϕ
2
(G)
≤Cϕ
2
(t
0
)kχ
B
0
k
M
Φ,ϕ
1
(G)
≤C
ϕ
2
(t
0
)
ϕ
1
(t
0
)
.
(3)½n1nÜ©y²Œ±Šâ½n1˜Ü©Ú1Ü©.
Ä7‘8
[‹ŽpÆ‰ï‘8(2020A-010);Ü“‰ŒÆ“c“‰ïUåJ,‘8(NWNU-
LKQN2020-07).
ë•©z
[1]Birnbaum,Z.andOrlicz,W.(1931)
¨
Uberdieverallgemeinerungdesbegriffesderzueinander
konjugiertenpotenzen.StudiaMathematica,3,1-67.https://doi.org/10.4064/sm-3-1-1-67
[2]Orlicz,W.(1932)
¨
UbereinegewisseKlassevonR¨aumenvomTypusB.Bull.Int.Acad.Pol.
Ser.A,8,207-220.(Reprintedin:CollectedPapers,PWN,Warszawa,1988,217-230)
[3]Aberqi,A.,Bennouna,J.andElmassoudi,M.(2021)NonlinearEllipticEquationswithMea-
sureDatainOrliczSpaces.Ukrains’kyiMatematychnyiZhurnal,73,1587-1611.
https://doi.org/10.37863/umzh.v73i12.1290
[4]Bourahma, M.,Benkirane,A.andBennouna, J.(2021)ExistenceofRenormalizedSolutionsfor
a Class of NonlinearParabolic Equationswith Generalized Growth in Orlicz Spaces. Khayyam
JournalofMathematics,7,140-164.
[5]Mohamed,M.M.A.(2021)NonlinearQuadraticVolterra-UrysohnFunctional-IntegralEqua-
tionsinOrliczSpaces.Filomat,35,2963-2972.https://doi.org/10.2298/FIL2109963M
[6]Bourahma,M.,Benkirane,A.andBennouna,J.(2020)ExistenceofRenormalizedSolutions
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[7]Zhang, C.,Li,C.andMeng,F.(2022)GlobalAttractorsin OrliczSpacesforReaction-Diffusion
Equations.AppliedMathematicsLetters,123,ArticleID:107294.
https://doi.org/10.1016/j.aml.2021.107294
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Â
[8]Kawasumi,R.andNakai,E.(2020)PointwiseMultipliersonWeakOrliczSpaces.Hiroshima
MathematicalJournal,50,169-184.https://doi.org/10.32917/hmj/1595901625
[9]Asadzadeh,J.A. andJabrailova,A.N.(2021) OnStability ofBases FromPerturbedExponen-
tialSystemsinOrliczSpaces.AzerbaijanJournalofMathematics,11,196-213.
[10]Chamorro,D.(2022)MixedSobolev-LikeInequalitiesinLebesgueSpacesofVariableExpo-
nentsandinOrliczSpaces.Positivity,26,5-21.https://doi.org/10.1007/s11117-022-00882-5
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