Publicacions
Ma emá iques,
Vol
35
(1991),
141-153
.
WEIGHTED
INEQUALITIES
THROUGH
FACTORIZATION
EUGENIO
HERNÁNDEZ
1
.
In oduc ion
and
esul s
In
[4]
P
.
Jones
sol ed
he
ques ion
posed
by
B
.
Muckenhoup
in
[7]
con-
ce ning
he
ac o iza ion o
A
p
weigh s
.
We
ecall
ha
a
non-nega i e
mea-
su able unc ion
w
on
R"
is
in
he
class
A
p
,
1
<
p
<
oo
i
and
only
i
he
Ha dy-Li lewood
maximal
ope a o
is
bounded
on
LP(R',w)
.
In
wha
ol-
lows
LP(X,
w)
deno es
he
class
o
all
measu able
unc ions
de ined
on
X
o
which
Il w
l/P
II
Lo(X)
<
oo,
whe e
X
is
a
measu e
space
and
w
is
a
non-nega i e
measu able
unc ion
on
X
.
I
has
ecen ly
been
p o ed
ha
he
ac o iza ion
o
A
p
weigh s
is
a
pa icula
case
o a
gene al
ac o iza ion
heo em
conce ning
posi i e
sublinea
ope a o s
.
The
case in
which
he
ope a o
is
bounded om
LP(X,
)
o
LP(Y,
u),1
<
p
<
oo, o
u
and
non-nega i e
measu able
unc ions
on
X
and
Y
espec i ely,
is
ea ed
in
[8]
.
The
case
in
which
he
ope a o
is
bounded
om
LP(X,
)
o
L9(X,
u),
1
<p
<
q
<
oo
is
ea ed
in
[3]
.
Ou
i s
esul
is
a
ac o iza ion
heo em
o
weigh s
u
and
associa ed
o
ope a o s
bounded
om
LP(X,
)
o
L9
(Y, u),
whe e
X
and
Y
a e wo,
possibly
di e en ,
measu e
spaces,
and p and q
a e
any
index
be ween
1
and
oo
.
Le
X
and
Y
be
wo
measu e
spaces
and
le
M(X
),
M(Y)
be
he
class
o
measu able
unc ions
de ined
in
X
and
Y
espec i ely
.
An
ope a o
T
de ined
on
a
subse
o
M(X)
wi h
alues
in
M(Y)
is
called
sublinea
i
I
T
(
+
g)
I
<
IT( )I
+
IT(g)I
and
is
called
posi i e
i I 1
<
g
~-->
IT( )I
<
T(g),
o
all
,
g
E
M(X)
which
belong
o
he
domain
o
T
.
Theo em
1
(Fac o iza ion)
.
Le
T
and
T'
be
wo
posi i e
sublinea op-
e a o s de ined
on
subse s
o
M(X)
and
M(Y)
espec i ely
.
Le
E
M(X)
and
u
E
M(Y)
be
non-nega i e
unc ions
and
1
<
p,
q
<
oo
.
Suppose
ha
T
is
bounded
om
LP(X,
)
o
L9(Y,
u)
wi h
no m
IITII
and
T'
is
bounded
om
L9'(Y,u-9'/q)
o
LP'(X, -p'/p)
wi h
no m
IIT'II
.
Then
he e
exis
non-nega i e
unc ions
uo
E
M(X),
o
E
M(Y),
ul
E
M(Y)
and
i
E
M(X)
such
ha
=
uO
p/P,VI,
u
=

O
q/q'ul,
IIuo IIILI(X)
<
1,
IIVOUIIIL~(Y)
<
1,
T(u
o
)
_<
IITIl o
and
T'(u
l )
<
2p/p'IIT'Il l
.
This
heo em
can
be
applied
o a
la ge class
o
ope a o s
o
ob ain he
ac-
o iza ion o
he
associa ed
weigh s
.
The
eade
can
ind se e al
examples
in
[8]
and
[3]
.
14
2

E
.
HERNÁNDEZ
Fo
in eg al
ope a o s
wi h
non-nega i e
ke nel,
he
ac o iza ion
heo em
has
a
con e se
o
some
pa icula cases o
p and q
.
Le
k(x, y)
be
a
measu able
non-
nega i e
unc ion
on
X
x
Y
.
Le
us
deno e
by
K
and
K*
he ans o ma ions
:
(K )(y)
=
k(x,y) (x)dx,

(K*g)(x)
=
k(x,y)g(y)dy,
X

Y
he
domain
o
K
being
he
se
o
all
unc ions
E
M(X)
such
ha he
i s
in eg al
exis s
and
is
ini e o
almos
all
y,
and
he
domain
o
K*
being
anal-
ogously
de ined
.
Theo em
2
.
Leí
1
<
q
<p
<
oo
and
E
M(X
),
u
EM(Y)
be
non-nega i e
.
A
necessa y
and
su
cien
condi ion o
K
ío be
bounded
om
LP(X,
)
o
Lq(Y,
u)
is
hai
Me e
exis
non-nega i e
unc ions
u
o
E
M(X),
o
E
M(Y),
u
l
E
M(Y),
i
EM(X)
and
ini e
cons an s
Co,
Cl such
ha
Iluo 1IILl(X)
<
1,
=
u0PIP' l,
u
=
o
g1q'
u,,
K(u
0
)
<
Co o
and
K*(ul)
<
C
l
l
.
Mo eo e
IIKII
<
Colq'Cilq
.
The
case
p
=
q
is
simple
:
Theo em
3
.
Leí
E
M(X)
and
uEM(Y)
be
non-nega i e
.
A
necessa y
and
su
cien
condi ion
o
K
lo
be
bounded
om
LP(X,
)
o
LP(Y,
u)
is
ha
he e
exisi
non-nega i e
unc ions
u
o
E
M(X),
o
E
M(Y),
u
1
E
M(Y),
i
E
M(X)
and
ini e
cons an s Co,
C
l
such
ha
=
uo
PIP'
l,
u
=
o
PIP'ul
K(u0)
<
Co o
and
K*(u
l )
<
Cl l
.
Mo eo e
IIKII
<
Co
lIP'C1IP,
The
case
-
u
-
1
o
heo ems
2
and
3
is
p o ed
in
[1]
.
Ou
p oo
o
hese
heo ems
is
an
adap a ion
o he
p oo
o he
co esponding
esul s
in
[1]
.
In
he
case
p
<
q
he
condi ions
o
heo em
2 a e
no
su icien
o
he
boundedness
o
K
om
LP(X,
) o
Lq(X,
u)
e en
in
he
case
-
u
-
1
(see
[1])
.
Obse e
ha
in
heo em
2
we
only need
he
condi ion
Iluo 1IIL'(X)
<_ 1
while
he
"symme ic"
condi ion
IIUI 0IIL1(Y)
<
1
is
no
needed
.
Nei he
o
hese
is
needed
in
heo em
3
.
Fo
some
applica ions
i is
be e
o
eplace
he
su icien
condi ion
o
heo em
2
by
he
ollowing
one,
whose
s a emen
is
a
gene aliza ion
o
he
su icien
condi ion
o
heo em
3
:
Theo em
4
.
Le¡
1
<
q
<
p
<
oo
and
E
M(X
),
uE
M(Y)
be
non-nega i e
.
Suppose
ha
he e
exis
non-nega i e
mesu able
unc ions
u
o
,
o
,
u,,
i
such
ha
=
uo
PIP
'
l
q
'
IP'
u
=
u, o
-
PIq
'
K(uo) o
1
E
L''(u)
(wi h
L'(u)
no m
equal
o
C
o
)
and
K*(ul) 1
1
E
L''( -P'IP)
(wi h
L''( -P'IP)
no m
equal
o
C
l
),
whe e
-
9
-
.

Then,
K
is
a
bounded
ope a o
om
LP(X, )
o
Lq(Y,U)
wi h
no m
less
han
o
equal
o
Co
I
q
,
C1
,
IP
.
Fo
he cases
q
=
1
o
p
=
oo,
which
a e
no
co e ed
by
he
abo e
heo ems,
we
ha e
he
ollowing
sa is ac o y
esul
:
WEIGHTED
INEQUALITIES
THROUGH
FACTORIZATION

143
Theo em
5
.
(A)
I
1
_<
p
<
oo,
a
necessa y
and
su
cien
condi ion
o
K
o be
bounded
om
LP(X, )
o
L
1
(Y,u)
wi h
no m
IIKII
is
11
JY
k(x,
y)u(y)dylI
LP'(X, -n'ln)
:5
IIKII
(B)
I
1
<_
q
<
oo,
a
necessa y
and
su cien
condi ion
o
K
o be
bounded
om
L'(X,
)
o
L9(Y,
u) wi h
no m
IIKII
is
11
JX
k(x,y)
-1
(y)dylIL9(Y,«)
~
IIKII
In
his
heo em
L'(X, )
=
{ E
M(X)
:
Il ll
.
<
oo}
.
Examples
o
ope a o s
o
which
hese
heo ems
can
be
applied
a e
he
ol-
lowing
:
he
Ha dy
ope a o
and
i s
dual
x
he
ac ional
in eg al
ope a o
T (x)=101
(y)dy

,x>0,
T
*
(x)
=
J
~ (y)dy

,x>0
;
(Ia )(
x
)
=
I
n
(
x
-
y)l
yl
`dy,

xE
Rn,
0
<
a
<
n
R
which
is
sel -adjoin
;
he
Riemann-Liou ille
ope a o
1

j'

(Y)

«dy,
(Ta )(x)
=
I (-)

(x
-
Y)
,-C,
Laplace
ans o m
00
£
(x)
=

e-y (y)dy
0
and
he
mul idimensional
Ha dy
ope a o
x>0,a>0
;
x~
x
n
T
.Ax11
... ,
xn
)

(yl,
.
. .
,
y
n
)dy
n
.
..
dyl
.
The
p oo s
o
heo ems
1
o
5
will
be
gi en
in
sec ion
2
.
Applica ions
will
be
gi en
in
sec ion
3
.
These
a e
conce ned
wi h weigh ed
inequali ies
o
some
o
he
abo e
ope a o s
.
I
would
like
o
hank
B
.
Jawe h
o
calling
my
a en ion
o
[1],
which
u ned
ou
o
be
he
s a ing
poin
o
his
esea ch
.
14 4

E
.
HERNÁNDEZ
2
.
P oo s
o
Theo ems
1
o 5
To
p o e
heo em
1
we
need
he
ollowing
lemma
which
can
be
ound
in
[1]
.
Lemma
2
.1
.
Le
B
be
a
Banach
space
and
P
a con ex cone
in
B
.
By
calling
his
cone
"posi i e",
B
will be
aken
as
an
o de ed
Banach
space
.
Le
us
suppose o
B
and
P
ha
e e y
bounded
inc easing
sequence
in
P
con e ges,
mo e
p ecisely
:
{ n}
C
P,
n+l
-
n
E
P,
Il nll
<
M
<
oo =>
n
-
E
P
Le¡
S
be
a
ans o ma ion
de ined
in
B
such
ha
S(P)
C
P,
S
is
nondec easing
( ha
is,
,
g,
g
-
E
P
=>
Sg
-
S
E
P),
S
is
con inuous
and
11
II
<
1
=>
Ils ll<Co<oo
.
Then
he e
exis s
a
E
P,
a
7É
0,
llall
<
1
such
ha
2C
o
a
-
Sce
E
P
To
p o e
heo em
1
we
ake
B=
LP(X
),
P
such
ha
E
P
<=>
(x)
>
0a
.e
.
and
S

-

T'

T( -1/P)

qlq'
u

]
P'/p
-P1/P2
( )
[
(C
IITII

)
and
apply
lemma
2
.1
.
Obse e
ha he
boundedness
o
T
and
T' implies
IIS lILD(X)
<
IIT'IIP'/PIL IILP(X)
so
ha Il 1I
<
1
=>
IIS 1I
<
JIT'l1P'/P
.
Hence
he e
exis s
a
qÉ 0,
a
>_
0,
a
E
LP(X)
wi h
no m
less
han
o
equal
o 1
and
S(a)
<
2l1T'1IP
/P
a
.
The
p oo
o
heo em
1 is
inished
by
aking
and
l
=
uó
/P'
.
uo
=
a
-1
/P,
o
=
T(uo)/IITII,
ul
=
(T(uo)/IITII)1/1'u
Theo ems
2
and
3
will
be
es ablished
once
we
p o e
he
su iciency
o
he
condi ions, since
he
necessi y ollows
immedia ely
om
heo em
1
.
To
p o e
heo em
2
ake
E
LP(X,
),
g
E
Lq'(Y,
u-
q'/q)
and
use
Holde 's inequali y
wi h
,
p
and
q',
whe e
=
1
-
o
ob ain
(2
.2)
WEIGHTED
INEQUALITIES
THROUGH
FACTORIZATION

145
1/T
~X
Y
k(x,y) (x)9(y)dxdyl
C
(L
1Y
k(x,y)u1(y)uo(x)dxdyl
1/p
.
(
/
k(x,y)u1(y)uo(x)-pl"I (x)Ipdxdy/
X
Y
1/q'
.
~
/
k(x,y)uo(x)u1(y)-q'lglg(y)Iq~dxdy)
X
Y
Using
K*(u
1 )
_<
C
1
1
and
Iluo 1IIL1(X)
<
1
he
i s
ac o
en
he
igh
hand
side o
he
abo e
inequali y
is
bounded
by
Cl
/
.
Using
K*
op/p'

(u
1
)
<
C
1
1
and
=
u

1
we
deduce
ha
he
second
ac o
is
bounded
by Cl
/p
ll JI
L (X, )-
Finally,
using
K(u0)
_<
Co o
and u
=
o
'/"u,
he
hi d ac o
can
be
majo-
a ed
by
Cól
q
II9I6'(Y,
.-199)
.
Pu ing
hese
es ima es
oge he
we
ob ain
JX Yk(-,y) (x)g(y)dxdy
<Có/q'Cl/gli lILP(X
; )IIgIIL
F om
he e
he
desi ed
esul
ollows
.
The
same
a gumen
applies
o
p
=
q,
ha
is
o
heo em
3,
excep
ha
in
his
case
1
--
1
-
ñ
=
0,
and
hence
he
i s
ac o
on
he
igh
hand
side
o (2
.2)
does
no
appea
.
Thus
heo em
3
does
no
equi e
he
condi ion
IIuo 1IIL1(X)
<
1
.
To
p e e
heo em
4
le
ELP(X,
) and g E
L9
(Y,
u -
q'1q)
.
F om
=g-
we
deduce
,
-{-
ñ
=
,
so
ha
we
can
apply
Holde 's
inequali y
wi h
indices
p/ '
and
q'/ '
o
ob ain
I
k(x,y) (x)9(y)dxdy
X Y T
/p
<
CJxJYk(x,y)u1(y)uo(x)-plq'I (x)Ip/ 'dxdy)
l
'/q'
.
CJx
/Y
k(x,
y)uo(x)u1(y)-q'lpl9(y)I
q'l
dxdy'

=
(I)
.(II)
.
Using
Holde 's
inequali y
wi h
index
,
oge he
wi h
=
uoplp
'
q
'
lp'
and
K*(u
1
) 1
1
E
L (
-
p
'
1
p)
we
ob ain
'/ p
(I)
<
C x_

[K*u1)(x)] 1
(x) -p'1p(x)dx)

146

E
.
HERNÁNDEZ
1/p
'

I (x)Ip 1(x) '*)p,
/p
uo
(
x
)-p
lq'dx)
x
Using
again
Holde 's
inequali y
wi h
he
same
index,
oge he
wi h
u
u
l
o
p/
q'
and
K(uo) o
1
E
L'(u)
we
ob ain
(B)
The
p oo
is
analogous
.
1/p
=
C1'/p
(L
I
(x)Ip (x)dx
)
'
/ g
,
(II)
<(
[K(uo)(y)]
o(y)`u(y)dy)
1/q
'
'
(l
YIg(y)I9
u1(Y)
-q, 'lp
o(y)
u(y)-
/ dy)
1
/q~
=
Co'/q,
(/
I
g(y)I
q'
u(y)
-q'lg
dy)
Y
The
desi ed
esul
ollows
by
pu ing
hese
wo
es ima es
oge he
.
This
inishes
he
p oo
o
heo em
4
.
We
now
p o e
heo em
5
.
(A)
Su ciency
.
Fo
E
LP(X,
)
and
g
E
L'(Y,
u
-1
)
we
ha e
k(x,
y)
(x)g(y)dxdy
_<
IIglIL-(Y,u-1)
ix

Y
/
k(x,
y)I
(x)I
u(y)dxdy
~
x
Y
IIgIIL-(Y,u-1)
{
u
k(x,y)u(y)dy)
pl
(x)
-p'lp
dx
X
Y
Necessi y
.
The
boundedness
o
K
implies
o
all
E
LP(X,
),
g
E
L'(Y,
u
-1
)
.
Wi h
g
-
u
we
ob ain
(x)
(/
k(x,y)u(y)dy)
dx

<_
IIKIIII lILD(X, )
~
X

Y
o
all
E
LP(X,
)
.
Thus
he
esul
ollows
.
1/p'
1/p
I (x)Ip (x)dx}

_<
IIsIIL~(Y,u-1)IIh~~II IILy(X, )
.
X
L
lyk(x,y) (x)g(y)dydx1
`IIhIIII ilLD(X, )II9IIL-(Y,u-1)
3
.
Applica ions
Conside
he
in eg al
ans o ma ions
(3
.2)
B
l
=
(3
.3)

B
z
=
WEIGHTED
INEQUALITIES
THROUGH
FACTORIZATION

147
x
(K )(x)
=

k(x,y) (y)dy,
(h
*
)(x)
=
J

k(y,x) (y)dy
0o

x
de ined
on
he
eal
line,
whe e
k(x,
y)
is
a
nonnega i e
measu able
unc ion
de ined
on
A =
{(x, y)
E
R
2
:
y
<
.
x}
.
Gi en
wo
non-nega i e
measu able
unc ions
u and
de ined
on
he
eal
line,
we
w i e
(u,
) E
W
l
(K,
p,
q),
1
<
q<p<ooi
{1-00
I~
/
OOk(x,y)u(x)dx)1/q
00 y
-00
and
(u,
)EW
2
(K,p,q),1<q<p<ooi
y

1/q

1/
,
k(y,
z) (z)
-
P
1Pdz
)

]
(y)-PIPdy
l
00
y
00

~
k(x,
y)u(x)dx

1/P

y
J

CJ-

k(y,
z) (z)-P'1Pdz

1/P

~

u(y)dy
1/
j
J
<
+oo
whe e
=
1
-
P
.
Obse e
ha
in
he
limi ing case
p
=
q,
(3
.2)
and
(3
.3)
become
~/P
y

i/P
,
(3
.4)

B
=
sup
o
~~

k(x,
y)u(x)dx
p

(

k(y,
z) (z)
-
P
'1
Pdz~

<
+oo
y
Y>

00
P oposi ion
3
.1
.
Le
K
be
he
in eg al
ans o ma ion
de ined
by (3
.1),
whe e
k(x, y)
>_
0
is
nondec easing
in
y
and
noninc easing
in
x
.
I
(u,
)
E
W
;
(Is,
p,
q),
1
<
q
<_
p
<
oo,
j
=
1, 2,
hen
(
~~
I(h )(x)Iqu(x)dxlllq
_<
C

I (x)IPV(x)dx~1/P
<
+oo
148

E
.
HERNÁNDEZ
whe e
C
<_
(gB1)TI
/P(pT
2
)
/ ',
P oo
.
Since
>
y
implies
k(x,
)
>
k(x, y)
he
no m
o
he
unc ion
00

00

1/9
'
1
y

,
1/9'
k
( ,
y)
~~

k(x,
)u(x)dx~

u( )d '
~~

k(y,
z) (z)-P
1Pdz)
y


j
0o
in
LT(
-
P'1P)
is
bounded
by
he
no m
o
he
unc ion
~o
(Y)
=
00

00

1
/9
'

1

y

,
k( ,
y)

k(x,
y)u(x)dx~

u( )d
l
~~

k(y,
z) (z)-P
1Pdzl
y


/

l00
in
he
same
space
.
In eg a ing
by
pa s
we
ob ain
Taking
he
abo e
inequali y
can
be
w i en
as
K*(u
l
)
1
1
E
L (
-
P'/P)
wi h
no m
no
exceeding
qB
l
.
Since
<
y
implies
k( , z)
>
k(y,
z),
he
no m
o he
unc ion
/
w
k(x,
y)u(x)dx
l
1/P
(I-Y
.
k(y,
) ( )-P'1P
~,1
y
in
L''(u)
is
bounded by
he
no m
o
he
unc ion
WP11U( - 9P)
<
qBl
.
u
l
( )
=

k(x, )u(x)dx~
-1/
u( ),
y
1
(y)
_
~~

k(y,
z) (z)_P'lndzl
_
1
/P
k( ,
z) (z)
-
P
'1
Pdzl

d
00

1
/P y


1/P
k(x,
y)u(x)dx
l

k(y,
) ( )
-
P
'1
P
k(y,
z) (z)
-
P'
1Pdz'

d
y

00
V00
in he
same
space
.
In eg a ing
by
pa s
we
ob ain
11011U(u)
<
p'B2
.
Taking
he
abo e
inequali y
can
be
w i en
as
K(u
o
)
.
-1
E
L'(u)
wi h
no m
no
ex-
ceeding
p'B
2
.
The
p oo
o
p oposi ion
3
.1
is
inished
by applying
heo em
4
i
q
<
p
and
heo em
3
i
q
=
p
.
Rema ks
.

1
.
Fo he
case
q
=
1,
(u,
)
E
W,
(K,
p,1)
is
an
equi alen
condi ion
o
he
boundedness
o
he
ope a o
K
de ined
by
(3
.1)
om
LP(X,
)
o
L
1
(Y,u)
.
This
ollows
om
pa
(A)
o
heo em
5
.
2
.

Fo
he
case
p
=
oo, (u,
)
E
W
2
(K,
oo,
q)
is
an
equi alen condi ion
o
he
boundedness
o
K
om
L'(X,
)
o
L9(Y,
u)
.
This
ollows
om
pa
(B)
o
heo em
5
.
3
.
A
esul
simila o
p oposi ion
3
.1
.
can
be ob ained
o
K*
.
De ails
a e
le
o
he
in e es ed
eade
.
4
.
The
ope a o s
de ined
by
(3
.1)
ha e
been
s udied
in
[2]
;
he
condi ions
imposed
on
he
weigh s
u
and
o
ob ain
weigh ed
inequali ies
o
K
a e
di e en
om
hose
used
he e
.
Fo he
pa icula
case o he
Ha dy
ope a o
T
(x)
=
o
(y)dy,
N
T
i
(T,
p, q)
becomes
(3
.5)
Bi

00
~

y0

00
0
U)
1/9
(,y
-P,/P

1/9'
j

1/
<
+
00
=

,
~

)

(y)-P./Pdy
l
0
W2
(T,
p, q)
becomes
WEIGHTED
INEQUALITIES
THROUGH
FACTORIZATION

149
(3
.6)

B2
=
{
00

1
/P

y
-P,
/P
)
1/P'
j
u(y)dy

1/
[(

(1

}

<+
oo,
0
y00

00
and
when
p
=
qwe
ha e
( )
=
(1-

k(y,
z) (z)
-P'1P
dz)

( )
-
P
,
/P
o( )
=
(1
00
k
(x,
)u(x)dx)
ilP
Y>
.
(u)

y
U
In
his
case
we
shall
show
ha
he
condi ion
W
2
(T,
p, q)
is
implied
by
W
1
(T,
p, q)
and
ha
his
condi ion
is
also
necessa y
o
he
boundedness
o
T
.