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PureMathematicsnØêÆ,2021,11(5),790-801
PublishedOnlineMay2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.115093
n‘HausdorffŽf\Øª
±±±ùùùDDD
þ°ŒÆêÆX,þ°
ÂvFϵ2021c410F¶¹^Fϵ2021c511F¶uÙFϵ2021c518F
Á‡
©Ì‡ ¼˜n‘Hausdorff Žf3\V V‚¼ê˜m¥k.5'u¼ê¿‡^
‡.·‚Óžn‘ÝHausdorffŽf\Øª"
'…c
Žf§ÝHausdorff Žf§¼ê
TheWeightedInequalitiesofn-DimensionalHausdorff
Operators
HongxiuZhou
DepartmentofMathematics,ShanghaiUniversity,Shanghai
Received:Apr.10
th
,2021;accepted:May11
th
,2021;published:May18
th
,2021
Abstract
Inthispaper,weobtainsomenecessaryandsufficient conditionsfortheboundedness
ofthen-dimensionalHausdorffoperatorsonthetwo-weightedLebesguespaces.The
correspondingresultsfortheadjointofn-dimensionalHausdorffoperatorsarealso
obtained.
©ÙÚ^:±ùD.n‘Hausdorff Žf\Øª[J].nØêÆ,2021,11(5):790-801.
DOI:10.12677/pm.2021.115093
±ùD
Keywords
HausdorffOperator,AdjointHausdorffOperator,Weight
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
Hausdorff Žḟ´dHausdorff 3©z[1] ¥• )ûêÂñ5¯KÚ\,§†²;©Û
k—ƒéX,XFourier?ê¦Ú[2]!ê‘?êHausdorff¦Ú[3] .§3E©Û[4]!NÚ
©Û[5–7]¥k2•A^ÚïÄ.
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ϕ
,Ù½ÂXe:
h
ϕ
(f)(x) =
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∞
0
ϕ(
x
t
)
t
f(t)dt,
Ù¥f•ÐŒ±b½áuSchwartz ¼ê˜m.XJÜ·ϕ,Hausdorff ŽfŒ±C•Nõ²;
Žf.eϕ(t) =
χ
(1,∞)
(t)
t
,Kh
ϕ
=•²;˜‘HardyŽf[8]:
Hf(x) =
1
x
Z
x
0
f(t)dt;
eϕ(t) = χ
(0,1)
(t),h
ϕ
=•˜‘HardyŽfÝŽf:
H
∗
f(x) =
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∞
x
f(t)
t
dt,
ùpx>0.
Š•˜‘/ªí2,Andersen3©z[9] ¥½ÂR
n
þn‘HausdorffŽf
H
Φ
(f)(x) =
Z
R
n
Φ(x/|y|)
|y|
n
f(y)dy,x∈R
n
.
aqu˜‘œ/,XJΦ(y) =
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{|y|>1}
(y)
|y|
n
,@oH
Φ
¬C•n‘HardyŽf[10]:
Hf(x) =
1
|x|
n
Z
|y|<|x|
f(y)dy;
DOI:10.12677/pm.2021.115093791nØêÆ
±ùD
XJΦ(y) = χ
{|y|≤1}
(y),KH
Φ
¬C•n‘HardyŽfÝŽf:
H
∗
f(x) =
Z
|y|≥|x|
f(y)
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n
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n
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|x|

f(y)dy.
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L
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9ÙÝŽflL
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Φ
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∗
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n
þü‡¼ê.éun‘HausdorffŽf,·‚H
Φ
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p
(v)
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q
(u)k.5'u¼êuÚv¿‡^‡.Ù(JXe:
½n1.1.1 <p≤q<∞,f∈L
p
(v),Φ ∈L
∞
ÚΦ,f≥0.e•3~êC
1
>0 ¦
C
1
:= sup
t>0
Z
R
n
u(x)Φ

x
t

q/p

Z
|y|<t
Φ

x
|y|

v(y)
−p
0
/p
|y|
−np
0
dy

q/pp
0
×

Z
R
n
Φ

x
|y|

v(y)
−p
0
/p
|y|
−np
0
dy

q/p
02
dx<∞(1.1)
¤á,Keª¤á

Z
R
n
|H
Φ
(f)(x)|
q
u(x)dx

1/q
≤C
2

Z
R
n
f(y)
p
v(y)dy

1/p
,(1.2)
Ù¥C
2
†C
1
,p,qk'.
eé?¿¢êr>0,•3†xÃ'~êC
3
>0,¦eª¤á
Φ

x
r

Z
|y|<r
v(y)
−p
0
/p
|y|
−np
0
dy≤C
3
Z
|y|<r
Φ

x
|y|

v(y)
−p
0
/p
|y|
−np
0
dy,(1.3)
K
Z
R
n
u(x)Φ

x
r

q/p

Z
|y|<r
Φ
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x
|y|

|y|
−np
0
v(y)
−p
0
/p
dy

q/p
0
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4
´(1.2)¤á7‡^‡,Ù¥C
4
†C
2
,C
3
,p,qk'.
DOI:10.12677/pm.2021.115093792nØêÆ
±ùD
½n1.2.1 <p≤q<∞,f∈L
p
(v),Φ ∈L
∞
ÚΦ,f≥0.e•3~êC
5
>0 ¦
C
5
:= sup
t>0
Z
R
n
u(x)Φ

x
t

q/p

Z
|y|>t
Φ

x
|y|

v(y)
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0
/p
|y|
−np
0
dy

q/pp
0
×

Z
R
n
Φ

x
|y|

v(y)
−p
0
/p
|y|
−np
0
dy

q/p
02
dx<∞(1.4)
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
Z
R
n
|H
Φ
(f)(x)|
q
u(x)dx

1/q
≤C
6

Z
R
n
f(y)
p
v(y)dy

1/p
,(1.5)
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6
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5
,p,qk'.
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7
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Φ

x
r

Z
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v(y)
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0
/p
|y|
−np
0
dy≤C
7
Z
|y|>r
Φ

x
|y|

v(y)
−p
0
/p
|y|
−np
0
dy,(1.6)
K
Z
R
n
u(x)Φ

x
r

q/p

Z
|y|>r
Φ
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x
|y|

|y|
−np
0
v(y)
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0
/p
dy

q/p
0
dx≤C
8
´(1.5)¤á7‡^‡,Ù¥C
8
†C
6
,C
7
,p,qk'.
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∗
Φ
,·‚ÓH
∗
Φ
lL
p
(v) L
q
(u)k.5'u¼ê
uÚv¿‡^‡.Ù(JXe:
½n1.3.1 <p≤q<∞,f∈L
p
(v),f≥0.bΦ ´šK»•¼ê,…Φ ∈L
∞
.e•3~ê
C
9
>0 ¦
C
9
:= sup
t>0
Z
R
n
u(x)|x|
−nq
Φ

t
|x|

q/p

Z
|y|<t
Φ

y
|x|

v(y)
−p
0
/p
dy

q/pp
0
×

Z
R
n
Φ

y
|x|

v(y)
−p
0
/p
dy

q/p
02
dx<∞(1.7)
¤á,Keª¤á

Z
R
n
|H
∗
Φ
(f)(x)|
q
u(x)dx

1/q
≤C
10

Z
R
n
f(y)
p
v(y)dy

1/p
,(1.8)
Ù¥C
10
†C
9
,p,qk'.
eé?¿¢êr>0,•3†xÃ'~êC
11
>0,¦eª¤á
Φ

r
|x|

Z
|y|<r
v(y)
−p
0
/p
dy≤C
11
Z
|y|<r
Φ

y
|x|

v(y)
−p
0
/p
dy.(1.9)
DOI:10.12677/pm.2021.115093793nØêÆ
±ùD
K
Z
R
n
u(x)|x|
−nq
Φ

r
|x|

q/p

Z
|y|<r
Φ

y
|x|

v(y)
−p
0
/p
dy

q/p
0
dx≤C
12
´(1.8)¤á7‡^‡,Ù¥C
12
†C
10
,C
11
,p,qk'.
½n1.4.1 <p≤q<∞,f∈L
p
(v),f≥0.bΦ ´šK»•¼ê,…Φ ∈L
∞
.e•3~ê
C
13
>0 ¦
C
13
:= sup
t>0
Z
R
n
u(x)|x|
−nq
Φ

t
|x|

q/p

Z
|y|>t
Φ

y
|x|

v(y)
−p
0
/p
dy

q/pp
0
×

Z
R
n
Φ

y
|x|

v(y)
−p
0
/p
dy

q/p
02
dx<∞(1.10)
¤á,Keª¤á

Z
R
n
|H
∗
Φ
(f)(x)|
q
u(x)dx

1/q
≤C
14
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Z
R
n
f(y)
p
v(y)dy

1/p
,
Ù¥C
14
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13
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15
>0,¦eª¤á
Φ

r
|x|

Z
|y|>r
v(y)
−p
0
/p
dy≤C
15
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|y|>r
v(y)
−p
0
/p
Φ

y
|x|

dy.
K
Z
R
n
u(x)|x|
−nq
Φ

r
|x|

q/p
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|x|
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0
/p
dy
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q/p
0
dx≤C
16
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16
†C
14
,C
15
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½n1.1y²Äk`²HausdorffŽfH
Φ
½Â´Ün.¯¢þ,|^H¨olderØª
(1/p+1/p
0
= 1)
H
Φ
(f)(x) =
Z
R
n
Φ(x/|y|)
|y|
n
f(y)dy
=
Z
R
n
f(y)v(y)
1/p
Φ(x/|y|)
1/p
Φ(x/|y|)
1/p
0
v(y)
−1/p
|y|
−n
dy
≤

Z
R
n
f(y)
p
v(y)Φ(x/|y|)dy

1/p

Z
R
n
Φ(x/|y|)v(y)
−p
0
/p
|y|
−np
0
dy

1/p
0
.
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p
(v) ÚΦ∈L
∞
,¤±mý1˜‡È©´k•.qdb^‡(1.1),Œ•mý1
‡È©'ux´A??k•.
DOI:10.12677/pm.2021.115093794nØêÆ
±ùD
e¡y²¿©5.|^4‹IC†
Z
R
n
|H
Φ
(f)(x)|
q
u(x)dx=
Z
R
n

Z
S
n−1
Z
∞
0
Φ

x
t

f(ty
0
)
1
t
dtdσ(y
0
)

q
u(x)dx.(2.1)
d(2.1),|^H¨older Øª(1/p+1/p
0
= 1)
Z
R
n
u(x)

Z
S
n−1
Z
∞
0
Φ

x
t

f(ty
0
)
1
t
dtdσ(y
0
)

q
dx
≤
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R
n
u(x)

Z
S
n−1
Z
∞
0
Φ

x
t
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0
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1
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0
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S
n−1
Z
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t
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0
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0
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0
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1
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t
0
Z
S
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x
s
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0
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0
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s
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0
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0
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0
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w,
h
1
(x,t) =

Z
|y|<t
Φ

x
|y|

v(y)
−p
0
/p
|y|
−np
0
dy
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1/pp
0
.
-z= h
1
(x,t)
pp
0
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S
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t
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0
)
−p
0
/p
t
−np
0
+n−1
h
1
(x,t)
−p
0
dtdσ(y
0
)
=
Z
h
1
(x,∞)
pp
0
h
1
(x,0)
pp
0
z
−1/p
dz
= p
0

Z
∞
0
Z
S
n−1
Φ

x
t

v(ty
0
)
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0
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t
−np
0
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0
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0
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p
0q/p
0
Z
R
n
u(x)

Z
∞
0
Z
S
n−1
Φ

x
t

f(ty
0
)
p
v(ty
0
)t
n−1
h
1
(x,t)
p
h
1
(x,∞)
p
2
/p
0
dσ(y
0
)dt

q/p
dx
≤p
0q/p
0

Z
∞
0
Z
S
n−1
f(ty
0
)
p
v(ty
0
)t
n−1

Z
R
n
u(x)Φ

x
t

q/p
×h
1
(x,t)
q
h
1
(x,∞)
pq/p
0
dx

p/q
dσ(y
0
)dt

q/p
.
db^‡(1.1),Œ±•

Z
R
n
|H
Φ
(f)(x)|
q
u(x)dx

1/q
≤C
1/q
1
p
01/p
0

Z
R
n
f(y)
p
v(y)dy

1/p
.
DOI:10.12677/pm.2021.115093795nØêÆ
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ey7‡5.b(1.2)¤á,Ké?¿¢êr>0,eãØª


Z
R
n
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




Z
|z|<r
Φ

x
|z|

|z|
n
f(z)dz




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[1]Hausdorff,F.(1921)SummationsmethodenundMomentfolgenI.MathematischeZeitschrift,
9,74-109.https://doi.org/10.1007/BF01378337
[2]Hardy, G.H. (1925)Notes on SomePointsin the Integral Calculus, LX. An Inequality between
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