Publicacions
Ma emá iques,
Vol
37
(1993),
429-441
.
Abs ac
NORM
INEQUALITIES
FOR
OFF-CENTERED
MAXIMAL
OPERATORS
RiCHARD
L
.
WHEEDEN
I
Su icien
condi ions
a e
de i ed
in
o de
ha
he e
exis
s ong-
ype
weigh ed
no m
inequali ies
o
some
o -cen e ed
maximal
unc ions
.
The
maximal
unc ions
a e
o
Ha dy-Li lewood
and
ac ional
ypes
aken
o e
s a like se s
in
Rn
.
The
su icien
con-
di ions
a e
close o
necessa y
and
ex end
some
p e iously
known
weak- ype
esul s
.
1
.
In oduc ion
In
[CWW],
weigh ed
no m
inequali ies
a e
de i ed
o
some
in eg al
and
maximal
ope a o s
associa ed
wi h
s a like
se s in
Euclidean
space
1[8''x
.
Ou
aim
now
is
o
ex end
he
esul s
which
deal
wi h
analogues
o
he
Ha dy-Li lewood
and
ac ional
maximal
unc ions
.
The
si ua ion
we
will
conside
is
closely
adap ed
o
bo h
he
geome y
o
he
se s
used
in
he
de ini ions
o
hese
unc ions
and
o
he
ela ionship
be ween
hese
se s
and
he
poin
a
which
he
maximal
unc ions a e
o med
.
The
s udy
o
a e ages
o
unc ions
o e
se s
o he
han
balls
o
cubes
has
a
long
his o y,
and
o
some
o he
weigh ed
esul s
we
e e
o
[J]
and
[P3],
and
he
e e en es ci ed
he e
.
To
be
mo e
p ecise,
gi en
0
<
p,
<
n
and
wo
(possibly
un ela ed)
se s
S
and
E
in
Rn,
we
conside
he
maximal
unc ion
MS,E,w (x)
=

sup

,
`
-n

l
.
(y)
¡
dy,
>0,
ZER
n
 _
,
+ S
xEz+ E
whe e
z+ E
deno es
he
se
{z+ ~
:
~E
E},
and
simila ly
o
z+ S
.
We
a e
mos
in e es ed
in
he
case
when
S
is
s a like
abou
he
o igin
and
E
is
bounded
.
I
M
=
0, (1
.1)
is
a special
case
o
an
ope a o
conside ed
1
Suppo ed
in
pa
by
NSF
g an
DMS91-04195
.
43
0

R
.
L
.
WHEEDEN
in
[Co ],
al hough
he
no maliza ions
a e
di e en ,
and
o
0
<
,
<n
i
was
s udied
in
[CWW]
.
A
simple
example
occu s
by
w i ing
x
=
(x',
xn)
wi h
x'
=
(x1,
. . . ,
x
n
_1)
and
choosing
E
o
be
he
uni
cube
Q1
cen e ed
a 0
and
S
o be
he
unbounded
se
(1
.2)
S,={x
:Ixnl<min{1,
Ix
l7
}},
-y>0
.
I
we
pick
co so
ha
Q1
C
S
.
y
,
hen
he
equi emen
in
(1
.1)
ha
x E
z+ E
simply
amoun s
o
equi ing
ha
x
belong
o
he
cube-like
cen al
po ion
o
z+ S
.
y
as
opposed
o
lying
a he
ou
in
he
"ski "
o
z+ S,
y
.
The
weak- ype
beha io
o
(1
.1)
was
desc ibed
in
[CWW],
and
we
now
wan
o
s udy
i s
s ong- ype
beha io
.
Fo
pu poses
o
compa ison,
we
de ine
he
cen e ed
maximal
unc ion
(1
.3)

Ms,/,
(x)
=
sup
"
-n
1
(y)
¡
dy,

0
<
p
<
n,
>o
.+ s
which
co esponds
o choosing
E
o
be
he
emp y
se
in (1
.1)
.
Bo hweak
and
s ong- ype
esul s
o
(1
.3)
a e
de i ed
in
[CWW],
and
o
pu
ou
esul s
in
pe spec i e,
we
ecall
hese
o
he
model
case
S
=
S
.
y
gi en
in
(1
.2)
.
Fo
a
>_ 1
and
ixed
-y
>
0,
we
associa e
he
linea
ope a o s
S
Q
x
=
(ax1,
.
.
.
.
axn
_1,
a-7xn)
and
he
ec angles
R
a
=
Sa
Ql
.
These
ec angles
a e
na u ally
adap ed
o
S
.
y
since
UR,,,CS,C
UKR
2
;
a>1
o
some
geome ic
cons an
.
Gi en
a
ec angle
R, we
deno e
by
B(R)
he
collec ion
o
all
ansla es
and
dila es o
R,
Le
.,
B(R)
=
{z
+
R
:
z
E
R
n
,
>
0}
.
Le
1
<p <
q
<
co, p'
=
p/(p
-
1),
and
w(x),
(x)
be
nonnega i e
locally
in eg able
weigh
unc ions
.
Deno e
o,
=
-1
y(P
-1
)
.
I
is
p o ed
in
[CWW]
ha
i
he
cen e ed
maximal
ope a o
MS,,,
sa is ies
he
weak- ype
es ima e
w{x
:
MS
'
,
m
(x)
>
A}
<-
(
C
A
II
IIP,
)
q
wi h
c
independen
o
and
A,
A
>
0,
hen
IRIñ-1w(R)'u(R)PL
<
CIRa1ñ_1
INEQUALITIES
FOR
OFF-CENTERED
OPERATORS

431
o
all
R
E
B(R
a
)
and
all
a
>
1
.
He e
we
ha e used
he
s anda d
no a ions
w(A)
=
A
w
dx,
II llp,
=
(

I (x)Ip (x)dx
l
p
,
and
c o a
cons an
which
may
be
di e en
a
di e en
occu ences
.
Con-
e sely,
he
weak- ype
es ima e
holds
i
he e
exis s
a
mono one
unc ion
C(a),
a
>
1,
such ha
(1
.4)

IRA
ñ
-l
w(R)
4
o,(R)
-'
<
C(a)¡Ra
l
Ty
-1
,

R
E
,13(R
a
),
and
(1
.5)

C(a)
aa
<
oo
.
1
Mo eo e ,
we
ha e
he
s ong- ype
es ima e
II
MS,,jil
q,w
:5
CII
p, ,
1
<
p
<
q
<
oo,
i
he e
exis s
>
1 so
ha
(1
.6)

IR¡
1

p
w
(
R
)
1
(

<'
IR¡
IR
o
all
R
E
B(R
a
),
a
_>
1,
and
C(a)
is
a
mono one
unc ion
which
sa is ies
(1
.5)
.
O
cou se,
(1
.6)
is
s onge
han
(1
.4)
due
o
H51de 's
inequali y
.
Fo
he
uncen e ed
ope a o
(1
.1),
only
a
weak- ype
es ima e
is
p o ed
in
[CWW]
.
To
desc ibe
i ,
le
6*
be
de ined
by
S*(x)
_
(ax1,
. .
.
,
axn_1, xn)
o
a
_> 1,
and
le
R*
=
6*Q1
.
No e
o
u u e
e -
e ence ha
R*
is
he
smalles
ec angle
which
con ains
bo h
R
a
and
Q1
.
To
each
R
E
B(R
a
),
associa e
a
ec angle
R*
as
ollows
:
i
R=
z
+
Ra,
hen
R*
=
z
+
R*
.

Thus
he
pai
(R,
R*)
is
a
join
ansla ion
and
dila ion
o
(R
a
,
Ra*)
by
he
same
z,
.

I
is
p o ed
in
[CWW]
ha
i
1<p<q<ooand
(1
.7)

w{x
:
MS
7
,Q,m (x)
>
A}
<_
C
Cli
. llp,

q
hen
IR1
ñ
-1
w(R*)9Q(R)P'
<
CIRaiñ-1
43
2

R
.
L
.
WHEEDEN
o
all
pai s (R,
R*),
R
E
13(R),
and
all
a
>
1
.
Con e sely,
suppose
1
<
p
<-
q
<
oo
and
he e
is
a
mono one
unc ion
C(a)
such
ha
{RI ñ
-1
w(R*)9Q(R)P'
<
C(a)IRalñ-1
o
all
R
E
B(R
a
)
and
all
a
>_ 1
.
I
C(a)
also
sa is ies
(1
.5)
hen
he
weak- ype
es ima e
(1
.7)
holds
.
E en
in
he
unweigh ed
case
w
=
=
1, i
ollows
ha
he
esul s
o
he
cen e ed
and
uncen e ed
maximal
ope a o s associa ed
wi h
S
. y
a e
di e en
.
In
ac ,
i is
easy
o
check
ha
he
condi ions
hen
equi e
1/q
=
1/p
-
p,/n,
ha
he
cen e ed
maximal
unc ion
is
s ong- ype
o
y
>n-
1 i 1
<
p
<
n/p
, ,
bu
ha
e en weak- ype
o
he
uncen e ed
maximal
unc ion
equi es
p
>

y

(>
y
-
(n
-
1)
1
1
-
ñ)
a
posi i e
esul
being
gua an eed
when
s ic
inequali y
holds
.
Fo he
model
case
S
=
S
y
,
we
will
p o e
he
ollowing
s ong- ype
esul
o
(1
.1)
.
Theo em
1
.
Le
y
>
0 and
S
y
be
de ined
by
(1
.2)
.
Le
0
<
<
n,
1
<
p
<_
q
<
oo
and assume
he e
exis s
>
1
so
ha
(
.8)

IR¡!-lw(R*)1

1

u'dx
p,
<-C(a)IR
.I!-1
1

p

9

IR¡
IR
o
all
R
E
13(R
a
)
and
all
a
>_
1,
whe e
C(a)
is
a
mono one
which
sa is ies
(1
.5)
.
Then
~IMS,,Q1,w liq,w
<
CII IIp,
.
unc ion
A
esul
o
gene al
s a like
S
is
gi en
in
Sec ion
3
.
Condi ion
(1
.8) is
analogous
o
(1
.6)
o
he
cen e ed
maximal
unc ion
.
To
p o e
Theo em
1,
we
use
a
co e ing
echnique
gi en
by
C
.
P
.
Calde ón
in
[Ca]
oge he
wi h
a esul
we
now
desc ibe
.
Le
13
be
he
amily o
all
ansla es
and
dila es o
a
ixed
ec angle
Rz3
(Le
.,
13
=
13(R,3)
in
ou
p e ious
no a ion)
.
O
cou se R13
is
no
uniquely
de e mined
by
13
bu
i s
eccen ici ies
( a ios
o
edgeleng hs)
a e,
and
we
may
assume
wi hou
loss
o
gene ali y
ha
i s
i s
edgeleng h
is 1
.
Thus,
o
example,
we
may
iew
he
basic
ec angle
in
13(R
a
)
as
ha ing
edgeleng hs
1,
. . . ,
1,
a
--
í
-1
a he
han
a,
.
. . ,
a,
a
--
í
.

To
each
R
E
C3,
associa e
a se
(no
necessa ily
a
ec angle)
R*
so
ha
he
ollowing
holds
:
(1
.9)

I
R
l
,
R
Z
E
13
and
R
1
C
R
Z
hen
Ri
C
R2
.
Fo
example,
he
pai s (R,
R*)
o join
ansla es
and
dila es
o
(R
o
,
R*)
de ined
ea lie
ha e
his
p ope y
.
Mo e
gene ally,
i
R*
is
de ined
o
be
any
ec angle
con aining
R
13
,
and
gi en
R
E
13,
R=
z
+
R,3,
we
de ine
R*
=
z
+
R*
hen
he
pai s (R,
R*)
sa is y
(1
.9)
.
Fo
such
a
collec ion
o
pai s
and
0
<a<
1,
de ine
(1
.10)

M
a
(x)
=
sup
IR¡'
-
'

I
(y)¡
dy
.
REC3
IR
R
*Bx
O
cou se
his
depends on
13
and on
he
choice
o
he
se s
R*,
al hough
o
simplici y
ou
no a ion
does
no
e lec
his
dependence
.
We
will
need
he
ollowing
esul
.
Theo em
2
.
Le
1
<
p
<
q
<
co and
0
<_
a
<
1,
and
le
M,
be
de ined
by
(1
.10),
assuming
ha
(1
.9)
holds
.
I
he e
exis s
>
1
such
ha
(1
.11)

IRI
a

P
w
(
R*
)°
C
IR¡

o,'
dx/_

p

<
C
o
all
R
E
, 3,
hen
INEQUALITIES
FOR
OFF-CENTERED
OPERATORS

433
wi h
a
cons an
C
which
is
a
mul iple
depending
on
a, n, p,
and
no
on
13
o
,
o
he
cons an
in (1 .11)
.
We
no e
ha
he
condi ion
IIMa 1I9,w
<
CII IIp,
I
RI
a-l
w(R*)
9o,(R)
p~
<
c,

R
E
I3,
q,
bu
is
necessa y
o
(1 .12)
(e en
o
he
co esponding
weak- ype
esul ),
as
can
be
seen
by
choosing
=
XRU
in (1
.12)
and
using
a
s anda d
a gumen
.
In
case
R*
=
R
and
R
is
a
cube,
Theo em
2
is
due
o
C
.
Pé ez
[P1],
[P2]
.
Ou
p oo
will
be
modeled
on
ideas
in
[SW]
and
is
gi en
in
Sec ion
2
.

43
4

R
.
L
.
WHEEDEN
The
p oo
o
Theo em
2
uses
some
ideas
om
[SW]
.
The
de ails
which
a e
ei he
he
same
o nea ly
he
same
as
ones
he e
will
be
omi ed
.
Le
X3
=
13(R
L3
)
be
a
amily
a
ec angles
R
as in
he
in oduc ion,
wi h
associa ed
se s
R*
which
sa is y
(1
.9)
.

Le
e
l
,
.
.
.
,
en
,
(el
=
1,
say)
be
he
edgeleng hs
o R13,
and
le
B
dy
be
he
co esponding
g id o
dyadic
ec angles
o
he
o m

1
and
o z
ER',
de ine
2
.
P oo
o
Theo em
2
[
m,el
(ml
+
1)el

[m
¿
e
7¿
(m
.,+
1)e
n
,
l
2i
'

2i

X
.
. .
X

2i
'

2i
o
j,
ml,
. .
.
,
m,,
=
0,
±1,
±2,
...

.

Each
ec angle
in
B
d
y
is
also in
1i
.
De ine
M
.
d
y
(x)
=

sup

IR1

1

I
(y)¡
dy,
REI3dy
IR
R'gx
Máy,z
(x)
=

Sup

I
R+
zI
a
-1

I
(Y)¡ d
.
RE
13dy

R+z
O
cou se,
IR
+
zi
=
IR¡
.
(R+z)`Dx
Lemma
(2
.1)
.
I
1
<
q
<
oo
and
w
is
a
weigh ,
hen
JIM« lIq,
.
<
C
Sup
IIMa
y
'
z
IIq,w
ZERn
wi h
c
depending
on
a
and
n
bu
no
on
B
o
.
P oo
..
We
a gue
as in
[W] and [SW], and
ea lie
[FS]
.
The
impo an
pa
o
he
a gumen
is
as
ollows
.
Fix
R
E
. 3
and
conside
he
collec-
ion
o
hose
Rl E
.13
dy
whose
edgeleng hs
a e
abou
wice hose
o R,
espec i ely,
and
hink
o
Rn
as
pa ioned
in o
he
union
o such
Rl
.
O
cou se,
IR,¡
Pz
:~
IR¡ o
each
Rl
wi h
cons an s
o
equi alen e
depending
only
on n
.
A
simple
geome ic
a gumen
using
ansla ions
shows
ha
o
each
Rl,
I{zER
n
:RCR1+z}I>cIR11
wi h
c
>
0
depending
only
on
n
.
Also,
wi h
R
s ill
ixed,
he
se s
{z
E
Rn
:
R
C
Rl
+
z}
a e essen ially disjoin
o
di e en
(nono e lapping)
Rl
.
Le
E(R
1 )
=
{z
E
Rn
:
R
C
R
l
+
z,
R*
C
(R
l
+
z)*},
INEQUALITIES
FOR
OFF-CENTERED
OPERATORS

435
and
no e
by
(1
.9)
ha
E(R1)
is
he
same
as
he
se
{z
E
R'
:
R
C
R,
+z}
abo e
.
Also
i
x
E
R*
and
z
E
E(R1),
hen
(2
.2)

IRIa
-1
R
I
(y)I
dy
<_
cI
R1
+
zia
-1
Rl+z
I
(y)
I
dy
<cMá
y,
'
(x)
since
IR,
+zi
=
IR,¡
z-
IRI,
R
CR1+z
and x
E
(R1+z)*
.
The
cons an
c
depends
only
on
n,
a
.
The
key
poin s
o
obse e
a e
ha
i
we
deno e
SZ
=
UE(R
1
),
hen
R
1
he
inequali y
be ween
he
i s
and
hi d
e ms
in (2
.2)
holds
i
x
E
R*
and
zE
9,
ha
IE(R1)I
>_
cIR1I
o
each
R1,
and
ha
he
E(R1)
a e
essen ially
disjoin
o
di e en
Rl
.
The
es
o
he
p oo
hen
p oceeds
as in
[SW]
o
[W],
and
is
omi ed
.
To
p o e
Theo em
2,
i
is
enough
by
Lemma
(2
.1)
o
p o e
he
ana-
logue
o
(1 .12)
o
each
Máy,z
,
wi h
a
cons an
independen
o z
.
I
we
eplace
by
o
, ,
his
amoun s
o
showing
ha
(2
.3)

IIM«
y'z
( U)Ilq,w
<ClI IIp,a
wi h
c
equal
o
a
mul iple
depending
only
on
a,
n,
p
and
q
o
he
cons an
in
(1
.11)
.
To
p o e
(2
.3),
ix
z
and
>
0,
and
o
k
=
0,
l, 2,
. . . .
le
S2
k
=
{x
E
R'
:
M
d
,,
y,z
( o )(x)
>
2k,}
.
Then
x
E
SZ
k
i
and
only
i
heie
exis s
R
E
B
d
y
such
ha
x
E
(R
+
z)*
and
(2
.4)

IRI"

dy
>
2kn
.
R+z
In
pa icula ,
i
R
E
13d"
and
(2
.4)
holds
hen
(R+z)*
C
1?
k
.
Le
{R
jk }j
be
he
maximal
(wi h
espec
o
inclusion)
ec angles
in 13
d
y
which
sa is y
(2
.4)
;
hei
exis ence
is
assu ed
i
has
compac
suppo ,
which
we
may
assume
o
be
he
case
wi hou
loss
o
gene ali y
.
By
maximali y,
he
{R
j
k
+
z}j
a e
nono e lapping
o
each
k
.
Mo eo e ,
i
R~
is
he
nex
la ges
dyadic
ec angle
con aining
R
.~,
hen
IR~Ia-1

dy
<
2kn
R~
-I-z
by
maximali y,
so
ha
since
I
R~
I
=
2n
I
R~
I,
we
ha e
(2
.5)

2kn
<
IR
;
la-1

dy
<
2n(1-n)2kn
.
Rk+z
436

R
.
L
.
WHEEDEN
We
claim
ha
(2
.6)
SZk
=
U(R
jk
+
z)*
.
j
We
ha e
al eady
obse ed
ha each
(R
i
+
z)*
mus
lie
in
Qk
.
On
he
o he hand,
i
x E
Qk
he e
is
a
dyadic
R
wi h
x
E
(R
+
z)*
such
ha
(2
.4)
holds
.
Thus
R
C
Rh
o
o
some
jo (since
R
is
maximal
o no ),
and
consequen ly
(R
+
z)*
C
(R~
+
z)*
by
(1
.9)
.
Hence,
xE
(R
3
~
o
+
z)*
and
he
claim
ollows
.
By
(2
.6),
whe e
E
j'
=
(R
j
+
z)* S2k
+
1
.
Thus
II
M
a
d
`
( o
,
)Ilq,w
whe e
o
a
ec angle
R,
II
May,z( o,)Ilq,w
<
2'l
Qk Qk+l
=
U[(Rj
+
z)
*
9
k+l1
=
U
E
j
[M
.,,,( a)(x)14w(x)
dx
E
9
k
k Qk+1
E2
(k+1)ngw(Ek)
kj
q
<
2nq
1
:
(IR
j
k
j`
l
 Qdy~
w(E
j
)
k j

R~
+z
=2nq1
:w(Ejk)[IRkI«-lA(Rjk
+z)]q
.
k,j
1

q
dy
l
,
A(R~
+z)
.

R
3
+z
u
A(R)
=
IR¡
T
(
Q
dy)
T
.
R
We
es ima e
he
las
sum
by
using
hypo hesis
(1
.11)
o
he
ec angles
R~k
+
z
and
he
ac
ha
E
j
k
C
(R
jk
+
z)*,
ob aining
ha
(2
.7)
q
A(R~
k
+
z)
P

k

I

u
dy
l

,
A(R
j
+
z)

R~+z
INEQUALITIES
FOR
OFF-CENTERED
OPERATORS

437
whe e
c is
he
cons an
in (1
.11)
.
The
emainde
o
he
p oo
is
based
on
using
he
nex
lemma
o
es i-
ma e
he
sum
in
(2
.7)
.
Lemma
2
.8
.
Le
{Ri}jEI
be
a
collec ion
o
ec angles
om
a
ixed
dyadic
g id
(e .g
.,
om
B
dy
+
z
o
ixed
z),
le
~3
_>
1,
and
le
{ai}iEI
be
posi i e
numbe s
which
sa is y
(i)
a(Ri)
<_
coa¡
(ii)

E

¿
<
coaQ
j
:RjCR
;
o each
i,
wi h
co
independen
o
i
.
Then
i
1
<p
<
oc
and
q
=
pp,
II

1
II

aá
(-
I lo,dy)q
LiEl al Ri
9
<_
ell llp,a,
wi h c
depending
on
co,
p
and
q,
bu
no
on
o
he
pa icula
g id
.
The
p oo
is
i ually
he
same
as
ha
o
Lemma
(2
.10)
o
[SW],
which
deals
wi h
he
case
o dyadic
cubes,
and
is
he e o e
omi ed
.
I
we
apply
Lemma
(2
.8)
o
he
sum
in
(2
.7)
and
no e ha
o,(R)
<
A(R)
by
lldlde 's
inequali y,
we
immedia ely
ob ain
(2
.3)
om
(2
.7)
i
we
e i y
(2
.9)

A(R
y
~
+
z
)g1p
<
cA(R-
+
z)q/p
k,j
:R~
CR-1
o
each
R'
and
0
<p<
q
<
oc,
wi h
c
independen
o
l,
m
and
z
.
We
a gue
as in
he
p oo
o
(2
.11)
in
[SW]
.
Using
he
simple
inequali y
a
¡
<
(~
ai)qlp,
q
>
p, al
>
0,
we
may
p o e
jus
he
case
q
=
p
.
I
Rk
is
a
p ope
subse
o
Rm
hen
whe e
he second
inequali y
ollows
om
he
maximali y
o
R~
.
The e-
o e,
we
mus
ha e
k
>_
m
in (2
.9),
and we
may
ew i e
he
le
side
o
(2
.9)
(wi h
q/p
=
1)
as
(2 .10)
2
mn
<
IR¿
l
a-1

Q
dy
<
2kn
Rm+z
A(R
7
~
+
z)
.
=m
j
:R
;
CR-