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PureMathematicsnØêÆ,2023,13(5),1219-1226
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.135125
Hardy-Littlewood4ŒŽf3Wiener
Amalgam˜m¥k.5
ÇÇǘ˜˜éé駧§šššSSS
∗
§§§MMM777xxx
Ü“‰ŒÆ§êƆÚOÆ§[‹=²
ÂvFϵ2023c411F¶¹^Fϵ2023c512F¶uÙFϵ2023c519F
Á‡
©A^Wieneramalgam˜mi\'X§y²Hardy-Littlewood4ŒŽfM
W(FL
p
,L
q
) k.5ÚWieneramalgam˜m¥f(1,1) 5"
'…c
Hardy-Littlewo od4ŒŽf§Lebesgue˜m§WienerAmalgam˜m
TheBoundednessoftheHardy-Littlewood
MaximalOperatorsinWienerAmalgam
Spaces
YulianWu,XiaochunSun
∗
,GaotingXu
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Apr.11
th
,2023;accepted:May12
th
,2023;published:May19
th
,2023
Abstract
UsingtheembeddingrelationshipofWieneramalgamspaces,weprovedthatthe
∗ÏÕŠö"
©ÙÚ^:ǘ駚S§M7x.Hardy-Littlewood4ŒŽf3WienerAmalgam˜m¥k.5[J].nØêÆ,
2023,13(5):1219-1226.DOI:10.12677/pm.2023.135125
ǘé
Hardy-Littlewo odmaximaloperatorM isboundedfromW(FL
p
,L
q
)toW(FL
p
,L
q
).
Meanwhile,weobtainedthattheHardy-LittlewoodmaximaloperatorMisweak(1,1)
inWieneramalgamspaces.
Keywords
Hardy-Littlewo odMaximalOperator,LebesgueSpaces,WienerAmalgamSpaces
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
Hardy-Littlewood 4ŒŽf 3ØÓ¼ê˜m¥k.5k¯õïÄ. Stein 3©z[1]¥äN
‰ÑHardy-Littlewood4ŒŽf3Lebesgue˜m¥´r(p,p).Úf(1,1).Žf.Kinnunen3
©z[2]¥y²Hardy-Littlewood 4ŒŽf3Sobolev ˜mW
1,p
(R
n
) þk.5, ¿Ñ:
O(J.Diening3©z[3]¥y²Hardy-Littlewood4ŒŽf3C•êLebesgue˜mL
p(·)
(R
n
)
¥k.5. Bennett,DeVoreÚSharpley3©z[4]y²BMO ˜m¥Hardy-Littlewood 4ŒŽ
fk.5.
Wiener 3©z[5]¥Äg½Â¼ê˜ mW(L
1
,L
2
) ÚW(L
2
,L
1
) ;3©z[6][7]¥q½Â
¼ê˜mW(L
1
,L
∞
)ÚW(L
∞
,L
1
),Ù¥amalgam ˜mW(L
p
,L
q
)IO‰ê½Â•
kfk
W(L
p
,L
q
)
=
X
n∈N

Z
n+1
n
|f(t)|
p
dt

q
p
!
1
q
.
AO/,p=qž,W(L
p
,L
p
)=L
p
.ep
1
≥p
2
,q
1
≤q
2
,KW(L
p
1
,L
q
1
)→W(L
p
2
,L
q
2
);e
p
1
≤p
2
,q
1
≤q
2
,KW(FL
p
1
,L
q
1
) →W(FL
p
2
,L
q
2
).
Feichtinger 3©z[8]¥òWiener JÑamalgam˜mW(L
p
,L
q
)í2˜„ÿÀ+Ú˜
„ÛÜ•¼ê˜m(½˜m)¥, ¿¡T˜m•Wieneramalgam ˜m. ?˜Ú, Feichtinger ïÄ
Wieneramalgam˜mi\'X!òȆéó5Ÿ±9EŠ(J.
Heil 3©z[9]¥½Â\Wieneramalgam˜m, ¿y²\Wieneramalgam ˜mƒ
A5Ÿ.
‘XWieneramalgam ˜mïá, ²;Žf3Wieneramalgam ˜mþk.5•kéõ
DOI:10.12677/pm.2023.1351251220nØêÆ
ǘé
ïÄ.Cordero,D
0
Elia ÚTrapasso3©z[10]¥ïÄ'uaτ–‡©ŽfOp
τ
(a) 3˜m
ÚWieneramalgam ˜m¥ëY5. Ÿ²Úÿí3©z[11]¥y²üa È©Žf3·
܉ê˜mþk.5.Cunanan ÚTsutsui 3©z[12]¥|^ªÇ˜—©)ŽfÚ4ŒØª
ïÄWieneramalgam ˜m,Žfk.5, „ÑIOWieneramalgam ˜mÚˆ•É5
Wieneramalgam ˜mƒmi\'X. ©ïÄHardy-Littlewood 4ŒŽf3W(FL
p
,L
q
) ¥
k.5ÚWieneramalgam˜m¥f(1,1)5.
2.ý•£
ÎÒ½Â:éu?¿x∈R
n
, ½Â|x|
2
=x·x, Ù¥x·y´R
n
þIþÈ.C
∞
0
(R
n
) ´R
n
þäk;|8ጇ¼ê˜m. S(R
n
) ´R
n
þ1w„ü¼ê˜m. S
0
(R
n
) ´S(R
n
) ÿÀé
ó˜m,•¡…O©Ù˜m.
ef,g∈L
2
,Kü‡¼êf,gSȽ•
<f,g>=
Z
R
n
f(t)g(t)dt.
§*ÐS
0
×S•Œ±d<·,·>L«.
1 ≤p≤∞, Fourier-Lebesgue˜m½Â•
FL
p
(R
n
) = {f∈S
0
(R
n
) : •3h∈L
p
(R
n
),
ˆ
h= f}.
ef∈L
1
(R
n
),KfFourierC†½Â•
ˆ
f(ξ) = Ff(ξ) =
R
R
n
e
−2πix·ξ
f(x)dx.
²£ŽfÚ^=C†½Â•
T
h
f(t) = f(t−h)ÚM
ω
f(t) = e
2πiωt
f(t).
w,F(T
h
f) = M
−h
ˆ
f, F(M
ω
f) = T
ω
ˆ
f, M
ω
T
h
= e
2πihω
T
h
M
ω
.
ef∈L
p
(R
n
),g∈L
q
(R
n
),1 ≤p,q≤∞, ½Âf,gòÈ•
(f∗g)(x) =
R
R
n
f(x−y)g(y)dy.
ÎÒB
1
→B
2
L«¼ê˜mB
1
ëYi\¼ê˜mB
2
¥.
k‰ÑHardy-Littlewood 4ŒŽfÚWieneramalgam ˜m½Â.
½Â2.1[13]ef∈L
1
loc
(R
n
),f¥%Hardy-Littlewood 4Œ¼êMf½Â•
Mf(x) = sup
r>0
1
|B(x,r)|
Z
B(x,r)
|f(y)|dy,
Ù¥B(x,r) ´±x•¥%, r•Œ»m¥, =B(x,r)={y∈R
n
:|x−y|<r},({PB(0,r) •
B(r)).ŽfM: f7→Mf¡•Hardy-Littlewood 4ŒŽf.
DOI:10.12677/pm.2023.1351251221nØêÆ
ǘé
Hardy-Littlewood 4Œ¼êMf•Œ±ÏLòÈ/ªL«.
½Â2.1[13]Pv
n
•R
n
¥ü ¥B(1) ÿÝ,@o
Mf(x) = sup
r>0
(|f|∗ϕ
r
)(x),
Ù¥ϕ(y) =
1
v
n
χ
B(1)
(y),ϕ
r
(y) =
1
r
n
ϕ(
y
r
).
½Â2.2[7]éu?‰f∈W(L
1
,L
∞
)(R
n
),'uI•¼êg∈W(L
∞
,L
1
)(R
n
)ážFourier
C†(STFT)½Â•
V
g
f(x,ω) :=
Z
R
n
f(t)g(t−x)e
−2πiωt
dt=<f,M
ω
T
x
g>,(x,ω) ∈R
n
.
w,kV
g
f(x,ω) =
\
(f·T
x
¯g)(ω),(x,ω) ∈R
n
.eg∈S…f∈S
0
,KV
g
f3R
2n
þ´˜—ëY.
½Â2.3[8]g∈C
∞
0
(R
n
)…÷vkgk
2
=1,1≤p,q≤∞,½ÂWieneramalgam˜m
W(FL
p
,L
q
)Xe

f∈L
p
loc
(R
n
) : kfk
W(FL
p
,L
q
)
= kkfT
x
gk
FL
p
k
L
q
x
<∞

,
T½Â¿Ø•6ugÀJ.,,eB,Cþ•Banach˜m,KWieneramalgam˜mW(B,C)Œ
aq½Â.
e¡0©Ì‡^Ún.
Ún2.1[9]B
i
,C
i
(i= 1,2,3)´Banach˜m,W(B
i
,C
i
)•ƒAuB
i
,C
i
Wiener amalgam
˜m.
(1)éu?¿f
1
∈B
1
,g
1
∈B
2
,•3~êc
1
>0, ¦
kf
1
∗g
1
k
B
3
≤c
1
kf
1
k
B
1
kg
1
k
B
2
,
…éu?¿f
2
∈C
1
,g
2
∈C
2
,•3~êc
2
>0, ¦
kf
2
∗g
2
k
C
3
≤c
2
kf
2
k
C
1
kg
2
k
C
2
,
K•3~êc>0 ,¦éu?¿f∈W(B
1
,C
1
),g∈W(B
2
,C
2
),k
kf∗gk
W(B
3
,C
3
)
≤ckfk
W(B
1
,C
1
)
kgk
W(B
2
,C
2
)
.
(2)eB
1
→B
2
…C
1
→C
2
,KkW(B
1
,C
1
) →W(B
2
,C
2
).
AO/,éu1 ≤p
i
,q
i
≤∞,i= 1,2,…p
1
≥p
2
,q
1
≤q
2
,Kk
W(L
p
1
,L
q
1
) →W(L
p
2
,L
q
2
).
(3)éu?¿u∈W(B
1
,C
1
)∩W(B
2
,C
2
)…θ∈[0,1], Ku∈W(B
3
,C
3
)…
DOI:10.12677/pm.2023.1351251222nØêÆ
ǘé
kuk
W(B
3
,C
3
)
≤kuk
θ
W(B
1
,C
1
)
kuk
1−θ
W(B
2
,C
2
)
(4)eB
0
,C
0
©O´Banach˜mB,CéóÿÀ˜m,…C
∞
0
3BÚC¥þÈ—,KW(B,C)
éó˜mW(B,C)
0
= W(B
0
,C
0
).
Ún2.2[13]Hardy-Littlewood4ŒŽfM´f(1,1) .Žf.=•3~êC= C
n
,¦é?
¿λ>0 9f∈L
1
(R
n
)k
|{x∈R
n
: Mf(x) >λ}|≤
C
λ
kfk
L
1
.
Ún2.3e1 ≤p,q≤∞,K
W(FL
p
,L
q
)∗W(FL
∞
,L
1
) →W(FL
p
,L
q
),
=•3˜‡~êC,¦éu?¿f∈W(FL
p
,L
q
),g∈W(FL
∞
,L
1
),k
kf∗gk
W(FL
p
,L
q
)
≤Ckfk
W(FL
p
,L
q
)
kgk
W(FL
∞
,L
1
)
.
ydYoungØªkL
q
∗L
1
→L
q
.
ŠâÚn2.1(1)Œ•,·‚•Iy:FL
p
∗FL
∞
→FL
p
.
FL
p
∗FL
∞
= F(L
p
·L
∞
) →FL
p
,=FL
p
∗FL
∞
→FL
p
.(Øy.
3.½n9y²
Äk‰ÑHardy-Littlewood 4ŒŽf3Wieneramalgam ˜mW(FL
p
,L
q
)¥k.5.
½n3.1éu1<p,q<∞,ef∈W(FL
p
,L
q
),KHardy-Littlewood4ŒŽfM´
W(FL
p
,L
q
) →W(FL
p
,L
q
)k..=
kMfk
W(FL
p
,L
q
)
≤Ckfk
W(FL
p
,L
q
)
.
yϕMf(x) = sup
r>0
(|f|∗ϕ
r
)(x),dFourierC†Ä5ŸÚCauchy-Schwartz Øª
kϕ
r
(x)k
W(FL
∞
,L
1
)
= kkϕ
r
(x)T
y
g(x)k
FL
∞
x
k
L
1
y
≤kkϕ
r
(x)T
y
g(x)k
L
1
x
k
L
1
y
≤






Z
R
n
|ϕ
r
(x)|
2
dx

1
2

Z
R
n
|g(x−y)|
2
dx

1
2





L
1
y
≤C
1






Z
R
n
|g(x−y)|
2
dx

1
2





L
1
y
,
Ù¥C
1
=

R
R
n
|ϕ
r
(x)|
2
dx

1
2
=
π
1
2
v
n
r
n
.
DOI:10.12677/pm.2023.1351251223nØêÆ
ǘé
ey




R
R
n
|g(x−y)|
2
dx

1
2



L
1
y
ŒÈ.
dCauchy-SchwartzØªk






Z
R
n
|g(x−y)|
2
dx

1
2





L
1
y
=
Z
R
n

Z
R
n
(|g(x−y)|χ
B(x,α)
(y))
2
dx

1
2
χ
B(α)
(y)dy
≤


Z
R
n






Z
R
n
|g(x−y)χ
B(x,α)
(y)|
2
dx

1
2





2
dy


1
2

Z
B(α)


χ
B(α)
(y)


2
dy

1
2
≤C
2
kgk
W(L
2
,L
2
)
,
Ù¥χ
B(x,α)
(y)•I•¼ê,α>0,B(x,α) ´±x•¥%, α•Œ»¥.
kgk
W(L
2
,L
2
)
= kgk
L
2
= 1,¤±




R
R
n
|g(x−y)|
2
dx

1
2



L
1
y
≤C
2
.
¤±kϕ
r
(x)k
W(FL
∞
,L
1
)
≤C(C= C
1
C
2
),=ϕ
r
(x) ∈W(FL
∞
,L
1
).
Ïd,ŠâÚn2.3Œ•
kMfk
W(FL
p
,L
q
)
≤kϕ
r
k
W(FL
∞
,L
1
)
kfk
W(FL
p
,L
q
)
≤Ckfk
W(FL
p
,L
q
)
.
½ny.
íØ3.2éu1 <q≤∞, ef∈W(L
1
,L
q
),¯g∈W(L
∞
,L
1
), …Hardy-Littlewood4ŒŽf
M´L
p
→L
p
k.,KŽfM•´W(L
1
,L
q
) →W(FL
∞
,L
q
)k..
y-Q= [0,1)
n
L«ü •N,K
kMfk
W(FL
∞
,L
q
)
= kkMf(x)T
y
g(x)k
FL
∞
x
k
L
q
y
≤kkMf(x)T
y
g(x)k
L
1
x
k
L
1
y
=

Z
R
n

Z
R
n
|Mf(x)T
y
g(x)|dx

q
dy

1
q
=
Z
R
n
X
k∈Z
n
Z
Q+k
|Mf(x)T
y
g(x)|dx
!
q
dy
!
1
q
≤
Z
R
n
X
k∈Z
n
sup
Q+k
T
y
g(x)
Z
Q+k
|Mf(x)|dx
!
q
dy
!
1
q
DOI:10.12677/pm.2023.1351251224nØêÆ
ǘé
=
Z
R
n
X
k∈Z
n
sup
Q+k
T
y
g(x)
!
q

Z
Q+k
|Mf(x)|dx

q
dy
!
1
q
≤
Z
R
n
X
k∈Z
n
sup
Q+k
T
y
g(x)
!
q

Z
Q+k
|f(x)|dx

q
dy
!
1
q
≤





X
k∈Z
n
sup
Q+k
T
y
g(x)





L
∞




Z
Q+k
|f(x)|dx




L
q
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ë•©z
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