Publicacions
Ma emá iques,
Vol
35
(1991,
169-186
.
WEIGHTED
NORM
INEQUALITIES
FOR
GENERAL
MAXIMAL
OPERATORS
C
.
PÉREZ
1
.
In oduc ion
In
[13]
Muckenhoup
p o ed
he
undamen al
esul
cha ac e izing
all
he
weigh s
o
which
he
Ha dy-Li lewood
maximal
ope a o
is
bounded
;
he
su -
p isingly
simple
necessa y
and
su icien
condi ion
is
he
so called
A
P
-condi ion
(see
below)
.
A
di e en
app oach
o
his
cha ac e iza ion
was
ound
by
Jaw-
e h
(c
.
[9])
.
An
ad an age
wi h
his
app oach
is
ha
i
gene alizes o
mo e
gene al
si ua ions
;
o
ins an e,
o
Ha dy-Li lewood
ype
maximal
ope a o s,
ob ained by
eplacing
he
cubes
by
any
collec ion
o
se s
in
R"°,
and
o
spaces
o
homogeneous
ype
.
Fo
a
gene al
in oduc ion,
and
his o ical
commen s
we
e e
o
[7]
.
The
main
pu pose
o
his
pape
is
o
use
some
o
he
esul s
and
echniques
in
[9]
o
u he
in es iga e
weigh ed
no m
inequali ies
o
Ha dy-Li lewood
ype
maximal
ope a o s
.
We
s a
by
in oducing
some
no a ion
.
By
a
basis
!3
in
Rn
we
mean
a
collec ion
o
open
se s
in
R"
.
We
say
ha
w
is
a
weigh
associa ed
o he
basis
13 i
w
is
a
non-nega i e
measu able
unc ion
in
Rn
such
ha
w(B)
=
B
w(y)
dy
<
oo
o
each
B
in
13
.
M 3,w
is
he
co esponding
maximal
ope a o
de ined
by
M ,w (x)
=
s
u
á
w(B)
Áj (y)jw(y)dy
i
x
E
UBE 3
and
MB

,
(x)
=
0
o he wise
.
I
w-
1,
we
jus
w i e
MB
(x)
.
We
say
ha
he weigh
w
belongs
o he
class
A,
,j3,
1
<
p
<
oo,
i
he e
is
a
cons an
c
such
ha
IBI
~B
w(y)
dy)

~BI
JB
w(J)
1-P'
dy)
P-1
<_
c
o
all
B
E
3
.
p'
will
always
deno e
he
dual
o p,
ha
is
17
0

C
.
PÉREZ
In
he
limi
case p
=
1
we
ha e
ha
w
belongs
o he
class
A1,13
i
o
all
B
E
13
;
his
is
equi alen
o
saying
almos
e e ywhe e
x
E
Rn
.
Fo he
o he
limi
case,
p
=
oo,
we
se
I
ollows
om
hese
de ini ions
and
H&lde 's
inequali y ha
i 1<p<q<00
.
In sec ion
(5)
we
shall
use
also
he
ollowing
no a ion
.
The
weigh
w
belongs
o
he
class
A
p u
(dp),
1
<
p
<
oo,
i
hese
is
a
cons an
c
such
ha
and
only
i
11
1B
w(y)
dy)
ess
.SUPB(w
-1
)
<
e
Maw(x)
<
cw(x)
A,,~,a
=
Up>jA,,i3
.
A
p,
a
C
A9,13
(p(B)
1B
w(y)
dp(y)
/
(p(B)
1B
w(y)
'-p'
dp(y»p-'
<
c
o
all
B
EB
.
We
also
deno e
by
M
B
,d,
he
maximal
ope a o
de ined
by
MB,wdp (x)
=
s
up(wp)(B)

B
J (y)j
w(y)dp(y)
i
x
E
UBEa
and
Ma

,d
p
(x)
=
0
o he wise,
wi h
(wdF,)(B)
=
B
w(y)
dp(y)
.
One
o he
main
esul s
in
[9] is
he
ollowing
Theo em
.
Theo em
1
.1
(Jawe h)
.
Le¡
1
<p
<
oo
.
Suppose
ha
Ci is
a
basis
and
ha
w
is
a
weigh ,
and
se
a
=w
l-
p'
.
Then
Ma
:
LP(w)
->
LP(w)
Ma
:
LP
,
(o)
-~
LP'(u)
wEA
p
a
M8
,
,,,
:
LP'(w)
-~
LP'(io)
Ma,
o
:
LP(o)
--~
LP(o)
.
Theo eln
1
.1
includes
Muckenhoup 's
esul ,
men ioned
abo e,
ha
o
a
ixed
1
<p<o0
M2
:
LP(dp)
-> Lp'(dp)
WEIGHTED
NORM
INEQUALITIES

17
1
i
and
only
i
dlc
=
w(y)dy,
wi h
w
E
A
P
2
.
He e
Q
is
he basis o
all
open
cubes
in
R"
.
A
key
ac
conce ning
Theo em
1
.1 is
ha
he
p oo
comple ely
a oids
he
(di icul )
"Re e se
H6lde
inequali y
."
Acknowledgemen s
.
The
con en
o
his
pape
is
pa
o
my
Washing on
Uni e si y
Ph
.
D
.
Thesis
.
I
would
like
o
exp ess
my
deepes
g a i ude
o
my
eache
Bjó n
Jawe h
o his
guidance
and
all
his
eaching
.
I
also
would
like
o
hank
R
.
Howa d
and
A
.
R
.
Schep
om
he
Uni e si y
o
Sou h
Ca olina,
o
se e al
con e sa ions
conce ning
hei
wo k
in
[8]
.
The
e e ee
has
made
se e al use ul
obse a ions
o
which
I
am
g a e ul
.
Finally,
i
is
a
g ea
honou
o
me
o
dedica e
his
wo k
o
he
memo y
o
José Luis
Rubio
de
F ancia
.
He
in oduced
me
o
he
ield
o
Ha monic
Analysis,
and,
la e ,
always
kindly
suppo ed
and
encou aged
me
.
2
.
One-weigh
heo y
I is
a
undamen al
ac
ha
M2,
.
is
bounded
in
LP(w)
o
each
1
<
p
<
oo,
i
he weigh
w
is
doubling
(c
.
[7]
p
.144
)
.
In
pa icula
MQ,
.
is
bounded
i
w
is
a
A,,,2
weigh
.
R
.
Fe e man
in
[3]
and
B
.
Jawe h
and
A
.
To chinsky
in [11]
(also
c
.
[7]
p
.463)
p o ed
ha
he
weigh ed
s ong
maximal
ope a o
M-
R,,,,
ha
is
he
weigh ed
maximal
ope a o
associa ed
wi h
he basis
13
=
R
o
all
ec angles
in R'°
wi h
sides
pa allel
o
he coo dina e
axes,
is
also
bounded
in
LP(w)
whene e
he weigh
w
belongs
o
he
class
A,,
.-
.
The
p oo
o
his
esul
is
based
on
a
geome ic
co e ing
lemma
which
goes
back
o he
wo k
o
Có doba
(c
.
[1])
.
In
his
sec ion
N e
show
ha
hese
esul s
a e
pa icula
phenomena
o
a
gene al
ac
.
Theo em
2
.1
.
Le
Ci
be
a
basis
.
The
ollowing
s a emenis
a e
equi alen
.
i)
Fo
each
1
<
p<
oo,
a ad
whene e
w
E
A
P
,5
(1)

Mi3
:
LP(w)
--,
LP(w)
;
ü) o
each
1
<
p
<
oc,
and
whene e
w
E
A
.,13
(2)

Mj3,
.
:
L
P
(w)
-+
LP(w)
.
P oo
:
Assume
ha
he
basis
8
sa is ies
ii)
.
Fix
1
<
p <
oo,
and
le
w
E
A,
j3
.
By
deno ing
o
=w
1-
P',
we
ha e
ha
o
E
A
P
,
L3
C
A,,
.j3,
and
hus
M8,

,
:
L
P'
(w)
->
LP'(w)
Mj3
o
:
LP(o)
->
LP(u)
.
Applying
now
Theo em
1
.1
we
ge
Ms
:
LP(w)
-i
LP(w)
M 3
:
L
P'
(o)
-
L
P'
(u),
17 2

C
.
PÉREZ
which
in
pa icula
gi es us
i)
.
Assuming
now
i),
we
ix
1
<
p <
oo,
and
we
ake
w
E
A..
.
Suppose
ha
w
E
A
q
,
1
<
q
<
oo
.
The e
a e
wo
cases
.
a)q<p'
;
b) q
>
p'
.
In
he
i s
case
we
ha e
ha
w
E
AP
,,
which
means
ha
w'
-
PE
A
p
.
Hence,
by
hypo hesis,
Me
:
LP1(w)
->
LP'(w)
M13
:
LP(w
1-
P)
-
LP(wl-P)
.
And
applying
again
Theo em
1 .1
we
ob ain
MB,w
:
LP(w)
->
LP(w)
.
In
he
second
case
we
ha e
ha
MB
:
Lq(w)
-+
Lq(w)
M 3
:
Lq
'
(w
l-
q
'
)
-+
Lq'(wl-q')
.
since
w
l-
9
'
E
A
q
,
Now
by
using
Theo em
1
.1
we
ge
Since
we
always ha e
ha
we
can
in e pola e
o ge
M 3,w
:
Lq'(w)
->
Lq'(w)
.
M3,
.
:
L'(w)
->
L'(w),
M 3,,,
:
LP(w)
-+
LP(w),
o
e e y
q'
< p
<
oo,
and
hence
p'
<
q
.
This
concludes he
p oo
.
Example
2
.2
.
Le
MR
be
he
Có dóba-Zygmund
maximal
ope a o
.
The
basis
R
de ining
his
maximal
ope a o
is
o med
by
hese
ec angles
in
R
3
wi h
sides
pa allel
o he
coo dina e
axes
whose
sideleng hs a e o
he
o m
{s,
,
s }
.
I
has
been
shown
by
R
.
Fe e man
(see
o
ins an e
[4] )
ha
MR
is
bounded
in
LP(w),
o
each
1
< p
<
oo,
i
and
only
i
w
E
A
P
,R
.
Hence, by
Theo em
2
.1
MR,
,:
L
P
(w)
-+
LP(w)
.
o
each
1
<
p
<
oo,
whene e
w
E
A
.,R
In
iew
o
all
hese
impo an
examples
wemake
he
ollowing
de ini ion
.
De ini ion
2
.3
.
We
say
ha
he
basis
13
is
a
Muckenhoup
basis
i
o
each
1<p<oo,ande e ywEA,a
M,3
:
LP(w)
->
LP(w)
.
Wi h
his
de ini ion,
Theo em
2
.1
can
be
s a ed
as ollows
Ci
is
a
Muckenhoup
basis
i
and
only
i
WEIGHTED
NORM
INEQUALITIES

17
3
Ms,
w
:
LP(w)
-->
LP(w),
o
each
1
<
p
<
oo,
and
whene e
w
E
A,,,s
Nex
esul
is
an
ex ension
o
Lin's esul
(c
.
[12])
o
he s ong
maximal
ope a o
o
any
maximal
ope a o
whose
basis
is
a
Muckenhoup
basis
.
Co olla y
2
.4
.
Suppose
¡ha¡
8
is
a
Muckenhoup
basis
.
Le
1
<
p
<
oo,
and
suppose
ha
w
E
A,,,s,
hen
he ollowing
Fe e man-S ein
inequali y
holds
(4)

n
Ms (y)
P
w(y)dy
C
c

n
(y)PMs
w(y)dy
.
IR

R
P oo
..
Suppose
ha
w
E
A,,s, o
some
1
<
<
oo
.
The
ollowing
poin -
wise
inequali y
hen
ollows
easily
om
Hólde 's
inequali y
and om
he
A,
.,
s-
condi ion
(5)

MB(XE)(x)
<_
c(Ms,w(XE)(X))l1
.
Since
l3
is
a
Muckenhoup
basis,
Theo em
2
.1
yields
( 6
)

Ms,w
:
LP(w)
-->
LP(w),
and
oge he
wi h
(5)
we
easily
conclude
ha o
each
measu able
E
and
each
0
<
A
<
oo
he
ollowing
inequali y
holds
:
w
(Ms(XE)
>
A)
<
c(A)w(E)
.
Since
also
Ms
:
LP(R")
->
LP(R'),
we
a e
now
in
a
posi ion
whe e we
can
p oceed
as in
he
p oo
o
Lemma
7
.1
in
[9],
o
conclude
he
p oo
o
he
Co olla y
.
Rema k
2
.5
.
We
poin
ou
ha
o (4)
o hold,
we
do
no
need
o
assume
ha
w
E
A,,
.,s
;
(7) o
some
0
<
A
<
1
would be
su icien
.
Fo he
sake
o
comple ness
we
obse e ha
Muckenhoup
bases
sa is y
Jones'
ac o iza ion
heo em
.
We
jus
s a e
he
esul
and
e e
he eade
o
[9],
Co olla y
6
.1,
o he
p oo
unde
a
weake
condi ion
on
he
basis
.
P oposi ion
2 .6
.
Suppose
ha
Ci is
a
Muckenhoup
basis,
and
le
1
<
p
<
oo
.
Suppose
ha
w
E
A
P ,s
.
Then
he e
a e
weigh
.s
w1,w2
E
Al,s
such
ha
w=
WJw2-P
.

3
.
Vec o - alued
inequali ies
I
is
well
known
by
now
ha
he e
is
an
in ima e
connec ion
be ween
ec o
alued
inequali ies
and
weig hed
no m
inequali ies
(c
.
[7]
Chap e
5)
.
In
his
sec ion
we
shall
use he
esul s
om
he
p e ious
one
o
ob ain
an
ex ension
o
he
classical
Fe e man-S ein
ec o - alued
inequali y
(c
.
[2])
lIq

lIq
IMQ ilq~

<
CP,q
P

~~
I i
q
q-O

4O

[,P
whe e
1
<
p
<
oo
and
1
<
q
<_
oo,
o
any maximal
ope a o
ML3
whose
basis
is
a
Muckenhoup
basis
(c
.
he
de ini ion
abo e)
.
We
would
like
o
poin
ou
ha
(8)
has
played
a
undamen al
ole
in
he analysis
made
in he ecen
wo ks
[5]
and
[10]
.
We
shall
assume
h oughou
he
sec ion
ha
X3 is
a
Muckenhoup
basis
.
Fo
a
ix
1
<
p <
oc,
and
mo i a ed
by
he
me hod
in oduced
by
J
.
L
.
Rubio
de
F ancia
(c
.
sec ion
5 .5
in
[7]
and
sec ion 6 in
[9] ),
we
deno e
by R13
he
ope a o
Rp
:
LP(R')
->
LP(R")
°°

,
.
-~
1
:
MB
(2K)'
,
whe e
MI
=
Id
and
M,'
3
is
he
i h
i e a e
o
he
ope a o
M,3
.
K
is
he
no m
o
M13,
as
an
ope a o
on
LP(R")
.
Al hough
RL3
is
poin wise
la ge
han
ML3,
i
p ese es
mos
o
i s
p ope ies,
namely
i)
u<RL3u
IIRi3
UJILP(R^)
:~
2IIuJILP(R^)'
Fu he mo e,
RB
has he
p ope y
ha
iii)
Ri3
E
A
l
, 3
o
each
ELP(R")
.
The
las
s a emen
is
no
ue
o
MB
as he
ollowing
a gumen
shows
.
Conside
he
Ha dy-Li lewood
maximal
ope a o
117
=MQ,
and
ake
any
posi i e
unc ion
cp
E
L
l
(R")
.
We
shall
show
ha
D lW
is
no
e en an
A,,,,
weigh
.
Indeed,
suppose
ha
Mcp
E
A,,
o
some
1
<
p
<
oo
.
Then
(Mco)
1-P
'
E
A
p
,
and, by
Muckenhoup 's
heo em,
M (y)"
M~p(y)
1-P'
dy
C
c~

(y)
"
MW(y)dy
.
R^

R^
and
o
each
>
n
_1
.
WEIGIITED
NORM
INEQUALITIES

175
By
aking
=
cp
we
see
by
Lebesgue's
di e en ia ion
heo em,
ha
he
igh
hand
side o
he
inequali y
is
ini e,
while
he
le
hand
side
is
in ini e
l
.
Lemma
3
.1
.
Leí
1
<
p,
<
oo,
and
leí
B
be
a
Muckenhoup
basis
.
Then
o
each
nonnega i e
unc ion
u
E
Us>1L
9
(R
n
)
we
ha e
I

M[3 (y)
Pu(y)dy
~
c

n
(y)
P
Reu(y)dy,
R

R
Rn
MI3 (y)P
(Ri3(u
))
(y)1/'
dy
<
c
%Rn
(y)P
u(J)dy,
P oo
..
The
i s
inequali y
ollows
om
abo e
ema ks
and
(4)
.
Fo he
second
we
obse e
ha
(RL3
U)-11
=
[(R5
u
)
1
/ (P
-1
)J
1-P
EAP, 1, and
his,
in
u n, ollows
om
he
ac
ha
i
w
E
A1,8
hen
w
óEA1, 3,
i
0
_<
6
<
1,
and
om
he
easy
pa o (2
.6)
.
Finally,
(9)
ollows
om
he
de ini ion
o
Muckenhoup
basis
and om
abo e
ema ks
abou
RB
.
Rema k
3
.2
.
We
poin
ou
ha
o
he case
B=
Q
he
ollowing
inequali y
holds
:
(10)

IR-
M (y)P
M(u)(y)1/
<
cJR-
(y)
P
u
y
,
o
1
< p
<
oo
and
>
1
1
.
The
esul
is
alse
i
=
P
1
1
,
(c
.
[14])
.
P_-
We
may
hink
o
R
;
as
being
he
igh
subs i u e
in he
gene al case
.
Once
we
ha e
his
lemma
hen
he
ec o - alued
inequali y
o
M
;
ollows
om
Theo em
5
.2
Chap e
5
in
[7]
.
Theo em
3
.3
.
Leí
1
<
p
<
oo,
and
1
<
q
_<
oo
.
I
B
is
a
Muckenhoup
basis,
hen
1/q

1/q
~~
I
MCi i
I
ql

<
C
P,q

~~
I
i q
q-o

La
i-o
Ly
Rema k
3
.4
.
Al hough
we
ha e men ioned
ha he
heo em
ollows
om
Lemma
3
.1,
i
is
o
be
men ioned
ha
we
jus
need
he
i s
hal
o
i
.
Indeed,
his
and
a
s anda
p ocedu e
would
gi e he case
q
<
p
.
Now,
since
he
heo em
is
ob ious
o
q
=
p,
and
also
o
q
=
oo,
he
case
1
<p
<
q
<
oo
is
ob ained
by
in e pola ion
.
In
he
same
spi i
we
make
he
ollowing
obse a ion
abou
he
class
A
1
,1
;
.
1
Following
he
me hod
in
[16]
i is
possible
o
p o e
ha
Mcp
may
no
be e en
doubling
.
We
a e
indeb ed
o
F
.
So ia
o
showing
his
o
us
.
176

C
.
PÉREZ
Lemma
3
.5
.
Assume
ha
B
is
a
Muckenhoupi
oasis
.
Le¡ 0
<
q
<_
1,
and
le
w
E
UP>1LP(Rn)
.
Then
i
and
only
i
he e
is
a
posi i e
unc ion
gE
Up>1LP(Rn),
such
ha
o
some
la ge
enough
cons an
A
(12)
Since
ob iously
w.IZZ~
w<
w
E
Al,B
0
00
(
Mb9
)9
1/y
E
Ai
P oo
:
By
i e a ion
Mgw
<_
c'w,
o
some
cons an
c
.
Then
pu ing
A
c2
1
/
9
and g
=w
we
ge
<
2w
.
(12)
ollows
.
To
p o e
he
con e se,
le
G
deno e
he
igh
hand
side
o (12)
.
Since
w
',Z~
G,
i is
enough
o deal
wi h
G
.
We
i s
show
ha
G
is
in
UP>1LP(Rn)
.
Indeed,
suppose
ha
g
E
LP(R
n
),
1
<
p
<
oo
.
Then
IpliiD
=
I

(Mgg)9I

=
~
0

Ai

~
Lp/4
=
E

n(
mÁi(Y)
)
u(y)dy,
R
o
some u
E
L(Pl9)'(Rn)
wi h
uni
no m
.
The e o e,
by
he
abo e
ema ks
and
by
i e a ing
(4),
he
las
exp ession
is
domina ed
by
00
C
M
~
(y)
)
Reu(y)dy<
g(y)9
RBu(y)dy
.
R
.n
i
i=0

i=0
He e
F
deno es
he
smalles
cons an
o
which
(4)
holds
.
Finally,
by
aking
A
=
2
1
/
9
F,
using
Hdlde 's
inequali y,
and
ii)
abo e
ob ain
ha
JIGIlL,

<
2
1
/9
11911LI
,
We
shall
p o e
now
ha
G
E
A
1
,13
.
Fo
each
B
E
Ci
we
shall
see
ha
1

G
(y)

<AGx
IBI
e

(y)
y
_

( ),
WEIGIITED
NORM
INEQUALITIES

177
a
.
e
.
x
E
B
.
Indeed,
by
Minkowski's
in eg al
inequali y
wi h
=
9
>
1
_

l
1/
=
[
1

¡

-

Msg(y)
q
(
1

B
G(y)
dyl

¡B¡
JB

1

A

)~
1

7
Még(y)

`
1/

oo
(
M~
+1

x
l
9
=
<

B

As

dy

<
A9

As
g()
A G(x
)9
.
$=o

;=o
4
.
An
al e na i e
o mula ion
o
Muckenhoup 's
heo em
In
his
sec ion
we
gi e
a
di e en
c i e ion
o
decide
whe he
he
ope a o
M
13
is
bounded
on
LP(w),
assuming
ha he basis
is
a
Muckenhoup
basis
.
In
pa icula
his
esul
applies o
he
case
X3
=
Q,
p o iding
a
di e en
cha ac-
e iza ion o
Muckenhoup 's
heo em
.
This
app oach
is
inspi ed
by
he
esul s
in
[8]
.
Theo em
4
.1
.
Le¡
1
<
p
<
oo,
and
suppose
ha
13 is
a
Muckenhoup
basis
.
Le
w
be
a
weigh
o
13
.

Then
(13)

Mí3
:
LP(w)
-->
LP(w)
i
and
only
i
(14)
Me
(w(M6go)
P-1
)
<
c
w
go
1,-1,
o
some
nonnega i e
measu able
unc ion
g
o
,
wi h
B
g
o
(y)
dy
<
co
o
each
B
in
13,
and
o
some
posi i e
cons an
c
.
P oo
..
Assuming
(13)
i is
s anda d
o
see
ha
w
E
A
P
L3,
and
hence
o
=
W1-P'
E
A
P
,
, 3
.
Since
Li
is
a
Muckenhoup
basis
Mg
:
LP(w)
->
LP(w),
and
M,3
:
LP
,
(o)
--->
LP'(o)
.
Hence
and
which
implies
Si
:
LP(Rn)
->
LP(Rn)
-->
w1/P
M13
(W-11P
)
S
2
:
LP'(Rn)
,
LP'(Rn)
~,
w-1/P
A4
-
L
;
(w
1
IP
),
T
:
LP'(R')
->
LP'(Rn)
184

C
.
PÉREZ
whe e
and
We
iew
he
sum
Ek
j
Pk
jgk
,
j
,
as
an
in eg al
on
a
measu e
space
(X,
)
buil
o e
he
se
X
=
{k,
j},
assigning
o
each
(k,
j)
he
measu e
[¿k,j
.
Fo
A
>
0,
se
Then
We
can
es ima e
p(F(, ))
as ollows
9
Mk,j
- (Ek,j)
(IBkj
~
~
gk,i
~(y)dy)
l
1
'


(Y)

d

p
9k,j
=
(a(Bk
j)

B
k
,
i
-(Y)

(y) yl
P(A)
=
{(k,
.7)
:
gk,j
>
A},
G(, )
=
U(k,j)E (a)Bk,j-
gk
/.i
F k j
=

Aglp

dA
k,
j
Pk,j
(k,j)EF(a)

MI
;
(o-XB,k,i )(J)
9
(J)dy
<
_
(k,j)E (A)
Ekj
<
cu(G(A))q/p
<
c ({J
E
R'
:
DIB, ( la)(J)
p
>
, })9/p
He e
we
ha e used
he
hypo hesis
on
AIB
Q
in
he
hi d
inequali y
.
Finally
by
making
a
change
o a iables
we
ob ain
J

AIB (J)
g
(J)dy
<
<
e

Ao
,
({J
E
R
n
:
11IB
o( w)(J)
>
A})1/pl9

_
=
lo (
CIIMB,o( lU)II
L(p,9)'
concluding
he
p oo
.
As a
consequence
o
his
heo em
we
can deduce
he
ollowing
cha ac e iza-
ion
.

(24)

Mí3
:
LP(u)
->
Lq( )
i
and
only
i
(25)
WEIGHTED
NORM
INEQUALITIES

185
Co olla y
6
.2
.
Le 1
<
p
<
q
<
oo,
and
le
( ,
u)
be
a
couple
o
weigh s
.
Suppose
ha
MB
o
:
LP(o)
->
LP(o),
whe e
o
=
u
1-
'
.
Then
1/q
U
MH
(oXG(a))(y)q
(y)dy)

<
cu(G(ñ)1/P
o
e e y
se
G
which
is
a
union
o
se s
in
8
.
P oo
..
By
se ing
=
UXG(a)
in
(24)
we
eadily ge (25)
.
To
p o e
he
con e se
we
use
Theo em
6
.1,
ha
p
<
q,
and
ou
hypo hesis
on
Mí
3
,
,
IIMS
IIL
, (
,,
)
:5
V
II
MI3,o
( l~)II
L ,a(
o
)
<
<
CIIMI3,o( IQ)IILP(o)
<_
CII lUIILp(o)
-CII lILP(u)
As
a
consequence
o
his
esul
we
can
ob ain
Sawye 's
cha ac e iza ion
o
hose
couple
o
weigh s
( ,
u) o
which
he
Ha dy-Li lewood
is
bounded
om
LP(u)
o
Lq( )
.
We
jus s a e
he
esul since
he
p oo
is
like
he
gi en
o
he
case
p
=
q
in
[7]
p
.432,
wi h
some
ob ious
modi ica ions
.
Co olla y
6
.3
.

La
1
<
p
<
q
<
oo,
and
le ( ,
u) be
a
couple o weigh s,
ando,=
u'
-
P'
.
Then
(26)

M
:
LP(u)
->
Lq( )
i
and
only
i
1/q
(27)

(IQ
M(-XQ)(y)
q
(y)dy)

<
co(Q)1/P
o
e e y
cabe
Q
.
Re e ences
1
.

A
.
CÓRDOBA,
On
he
Vi a
co e ing
p ope ies
o a
di e en ia ion
oasis,
S udia
Ma h
.
57
(1976),
91-95
.
2
.

C
.
FEFFERMANAND
E
.
M
.
STEIN,
Some
maximal
inequali ies,
Ame
.
J
.
Ma h
.
93
(1971),
107-115
.
3
.

R
.
FEFFERMAN,
S ong
di e en ia ion
wi h
espec
o
measu es,
Ame
.
T
.
Ma h
.
103
(1981),
33-40
.
186

C
.
PÉREZ
4
.

R
.
FEFFERMAN,
"Mul ipa ame e
Fou ie
Analysis,"
Annals
o
Ma h
.
S u-
dies
112,
Beijing
Lec u es
in
Ha monic
Analysis,
P ince on
Uni e si y
P ess,
P ince on,
1986,
pp
.
47-130
.
5
.

M
.
FRAZIER
AND
B
.
JAWERTH,
A
disc e e
ans o m
and
decomposi ions
o
dis ibu ion
spaces, o
appea
in
J
.
Func
.
Anal
.,
also
in
MSRI
epo s
00321-89
and
00421-89
(1988)
.
6
.

E
.
GAGLIARDO,
On
in eg al
an o ma ions
wi h
posi i e
ke nels,
P oc
.
Ame
.
Ma h
.
Soc
.
1
6
(1965),
429-434
.
7
.
J
.
GARCIA-CUERVA
AND
J
.
L
.
RUBIO
DE
FRANCIA,
"Weigh ed
no m
inequali ies
and
ela ed opics,"
No h
Holland
Ma h
.
S udies
116,
No h
Holland,
Ams e dam, 1985
.
8
.

R
.
HOWARD
AND
A
.
R
.
SCHEP,
No ms
o posi i e
ope a o s
on
LP-spaces,
P oc
.
Ame
.
Ma h
.
Soc
.
10
9
(1990),
135-146
.
9
.

B
.
JAWERTH,
Weigh ed
inequali ies
o
maximal
ope a o s
:
linea iza ion,
localiza ion,
and
ac o iza ion,
Ame
.
J
.
Ma h
.
108
(1986),
361-414
.
10
.

B
.
JAWERTH,
C
.
PÉREZ
AND
G
.
WELLAND,
"The
posi i e
cope
in
T iebel-
Lizo kin
spaces
and
he
ela ion
among
po en ial
and
maximal
ope a o s,"
Con empo a y
Ma hema ics,
Ha monic
Analysis
and
Pa ial
Di e en ial
Equa ions
(M
.
Milman
and
T
.
Schonbek
edi o s)
107,
Ame
.
Ma h
.
Soc
.,
P o idence,
1990
.
11
.
B
.
JAWERTH
AND
A
.
TORCHINSKY,
The
s ong
maximal
unc ion
wi h
espec
o
measu es,
S udia
Ma h
.
80
(1984),
261-285
.
12
.
K
.
C
.
LIN,
Ha monic
Analysis
on
he
bidisc,
Thesis,
U
.
C
.
L
.
A
.
(1984)
.
13
.
B
.
MUCKENHOUPT,
Weigh ed
no m
inequali ies
o
he
Ha dy-Li lewood
maximal
unc ion,
T ans
.
Ame
.
Ma h
.
Soc
.
16
5
(1972),
207-226
.
14
.
C
.
PÉREZ,
Weigh ed
no m
inequali ies
o
po en ial
and
maximal
ope a-
o s,
Thesis,
Washing on
Uni e si y
(1989)
.
15
.

E
.
T
.
SAWYER,
A
cha ac e iza ion
o a
wo
weigh
no m
weigh
inequali y
o
maximal
ope a o s,
S udia
Ma h
.
75
(1982),
1-11
.
16
.

F
.
SORIA,
A
ema k
on
A
l
-weigh s
heo y
o
he
s ong
maximal
unc ion,
P oc
.
Ame
.
Ma h
.
Soc
.
10
0
(1987),
46-48
.
17
.
J.-O
.
STRUBERG
AND
R
.L
.
WIIEEDEN,
F ac ional
in eg als
on weigh ed
HP
and
LP
spaces,
T ans
.
Ame
.
Ma h
.
Soc
.
287
(1983),
293-321
.
Depa men
o
Ma hema ics
Campus
Box
1146
Washing on
Uni e si y
Sain
Louis,
Missou i
63130
U
.S
.A
.
cu en
add ess
:
Depa men
o
Ma hema ical
Sciences
New
Mexico
S a e
Uni e si y
Las
C uces,
NEW
MEXICO
88003