Publicacions
Ma emá iques,
Vol
37
(1993),
177-208
.
A
bs ac
VECTOR-VALUED
INEQUALITIES
WITH
WEIGHTS
LUZ
M
.
FERNÁNDEZ-CABRERA
AND
JOSÉ
L
.
TORREA(
*
)
This
pape
deals
wi h
he
ollowing
p oblem
:
Le
T
be
a
gi en ope a o
.
Find
condi ions
on
(x)
( esp
.
u(x))
such
ha
IT (x)IPu(x)dx
<
C
J
I (x)IP (x)dx
is
sa is ied
o
some
u(x)
( esp
.
(x))
.
Using
ec o - alued
inequali ies
he
p oblem
is
sol ed
o
:
Ca -
leson's
maximal
ope a o
o
Fou ie
pa ial
sums,
Li lewood-
Paley
squa e
unc ions,
Hilbe
ans o m
o
unc ions
alued
in
U
.M
.D
.
Banach
spaces
and
ope a o s
in
he uppe -hal
plane
.
In oduc ion
This
pape
deals
wi h
he
ollowing
P oblem
A
.
Le
T
be a
gi en
ope a o
.
Find
condi ions
on
(x)
( esp
.
u(x))
such
ha
(0
.1)

¡T
(x)
(Pu(x)dx
<
C
l
(x)
¡P (x)dx
is
sa is ied
o
some
u(x)
( esp
.
(x))
.
This
p oblem
was
s udied
o
di e en
ope a o s
in
[C,
J],
[G,
G], [H,
M,
S]
.
The
ope a o s
ea ed
we e
singula
in eg als,
ac ional
in eg als,
Ha dy-Li lewood
maximal
ope a o
and
ac ional
maximal
ope a o
.
In
all
he
cases
he
me hod
used
was
cons uc i e
.
(*)Pa ially
syppo ed
by
DGICYT
.
PB90-187
.
178

L
.
M
.
FERNÁNDEZ-CABRERA,
J
.
L
.
TORREA
On
he
o he
hand
i
was
hnown
ha
a
"good"
weigh ed
no m
in-
equali y
o
ype
(0
.1),
o
an
ope a o
T,
gi es
a
ec o - alued
inequali y
o ype
(0
.2)
II(E
IT jlP)1/PIIS
<CII(E
I jI
P
)
1
lPlls
.
.7

7
Fo
ins an e,
he ec o
alued
inequali ies
o
he
Ha dy-Li lewood
maximal
ope a o ,
M,
we e
ob ained
in [F,
S]
om
he
es ima e
I
M
.
I
Pu
<
C
I
I
PMu,

1
<p<
oo
.
In
1981,
José
Luis
Rubio
de
F ancia
showed
ha
weigh ed
no m
in-
equali ies
and
ec o
alued
inequali ies
we e
equi alen
in
some
sense,
see
[R
de
F,
1]
.
Using
ha equi alen e,
he
de eloped
a
non
cons uc i e
me hod
in
o de
o
sol e
he
P oblem
A
o
come
ope a o s
.
The aim
o
his
pape
is
o
show
ha
a
sligh
gene aliza ion
o
he
me hod
o
Rubio
de
F ancia
allows
us
o
sol e
he
P oblem
A
o
a huge
amily
o
ope a o s
.
The
gene aliza ion
is
wo
old
:
Fi s
we
shall
conside ec o - alued
e sions
o
inequali y
(0
.1)
and
we
shall
p o e
he
ela ion
wi h
he
co esponding
inequali y
(0
.2),
see
Theo em
(1
.1)
.
We
use
his
ec o - alued
e sion
in
o de
o
:
(a)
sol e
pa ially
P oblem
A
o
Ca leson's
maximal
ope a o
o
Fou ie
pa ial
sums,
see
Theo em
(2
.9)
.
(b)
ind
condi ions
on
(x)
( esp
.
u(x))
such
ha
R
II
H
(x)
II
Éu(x)dx
<
C
R
.II
(x)
II
F (x)dx
is
sa is ied
o
some
u(x)
( esp
.
(x)),
whe e
H
is
he
Hilbe
ans o m
and
E
is
a
U
.M
.D
.
Banach
space,
see
Theo em
(2
.24)
.
In
he
case ha
E
is
a
U
.M
.D
.
Banach
la ice
we
sol e
he
same
p oblem
o
he
ope a o
M&)

s
u
p
-QI
Q
I
.
(y)
I
dy,
x
E
R''
whe e
1
.1
is
he
absolu e
alue
in
E
and
he
sup emum
is
aken
in
he
la ice
o de , see
Theo em
(2
.26)
(c)
sol e
P oblem
A
o
Li lewood-Paley
squa e
unc ions,
see
Theo-
em
(2
.11)
.
VECTOR-VALUED
INEQUALITIES
WITH
WEIGHTS

179
The
second
gene aliza ion
is
o
conside abs ac
measu e
spaces,
in-
s ead
o
Rn
.
Wi h
hese
ideas
we
a e
able
o
sol e
P oblem
A
o
ope a-
o s
which
map
unc ions
in
Rn
x
[0,
oo)
in o
unc ions
in
Rn
x
(0,
oo),
see
Theo ems
(2
.16)
and
(2 .17)
.
The
ec o - alued
inequali ies
ob ained
in
his case,
see
Theo em
(3
.16),
can
be
o
independen
in e es
when
wo k-
ing
wi h
ope a o s
ac ing
on
unc ions
de ined
on
he
uppe
hal
plane
.
These
ope a o s include
as
pa icula
cases
Poisson
in eg als,
balayages,
see
(2
.19),
and
some
well
known
maximal
ope a o s,
see
(2
.23)
.
Ou
me hod
gi es
in
all
he
cases
some
ex a
in o ma ion
abou
he
size
o
he
weigh ha
is
ound
.
I
is
an honou
o
us
o
use
ideas
o
ou
iend
and
ad iso
José
Luis
.
Th oughou
his
pape
we
shall
wo k
on
gene al
measu e
spaces
(Y,
d ),
(X,
dp),
whe e
d
and
dp
a e
posi i e
measu es
.
Gi en a
Banach
space
E,
we
shall
deno e
by
L
E
'
(Y,
d ),
LÉ(Y)
o
LÉ(d )
he
Bochne
space
o
E- alued
s ongly
measu able
unc ions
such
ha
.lY
I1 (x)IIÉd (x)
<
+oo
.
Gi en a
posi i e
measu able
unc ion
w(x),
on
(Y,
d ),
we
shall
deno e
by
LÉ(w(x)d (x)),
o
LÉ(w)
he
space
o
E- alued
s ongly
measu able
unc ions
such
ha
Y
li (x)II
Éw(x)d (x)
is
ini e
.
Gi en a
Banach
space
E,
we
shall
deno e
by PÉ,
o
&(E)
he
Banach
space
00
{{an}
C
E
:

IIan1IÉ
<
+
00}
.
n=1
I
E
is
a
Banach
la ice
we
shall
deno e
by
E(P')
he
la ice
{{xn}
:
sup
Ixnl
E
E}
wi h
he
no m
I I
{xn
}
I I
E(e-)
=
II
SUP
Ixn
I
II
E-
n
The
o ganiza ion
o
he
pape
is
as
ollows
:
in
Sec ion
1
we
s a e
and
p o e
he
abs ac
esul s
ha
gene alize
he
p e ious
wo k
o
Rubio
de
Rancia,
in
Sec ion
2
we
gi e
(wi hou
p oo ) he
applica ions ha
sol e
P oblem
A
o
se e al
ope a o s
and
Sec ion
3
is
de o ed
o
he
p oo s
.
180

L
.
M
.
FERNÁNDEZ-CABRERA,
,I
.
L
.
TORREA
1
.
Abs ac
esul s
We
begin
his
sec ion
wi h
a
lemma,
ha
i is
known
o
he
scala
case,
see
[GC,
R
de
F,
VI
.4
.21
.
The
p oo
in
he
ec o - alued
case
is
essencially
he
same,
bu
we
include
i
he e
o
he
sake
o
comple eness
.
(1
.0)
Lemma
.
Le
(Y,
d )
be
a
measu e
space,
le
F
and
G
be
Banach
spaces
.
Assume
ha
0
<
s
<
p
<
oo
and
T
is
a
sublinea
ope a o
which
sa is ies
II(E
IIT jllF)
1'p
iiLs(Y,d )
<
G(
E
II j1iG)1'P,
j

j
Then,
i
o,

he e
exis s
a
nonnega i e
unc ion
w(x)
wi h
II
W
II
Lo-
1
(Y,d )
<
1
and
such
ha
Y
IIT j(x)IIF
,
w(x)d (x)
<
GII j1iG
The
p oo
o
his
lemma
is
based
in
he
ollowing
esul ,
see
[GC,
R
de
F]
.
Mini-max
Theo em
.
Le
A,
B
be
con ex
se s
in
some
ec o spaces
and
assume
ha
B
is
compac
o
a
ce ain
opology
.
Le
0
be
a
unc ion,
0
:
A
x
B
-->
R
U
{+oo},
which
is
conca e
on
A
and
conex
and
lowe
semicon inuous
on
B
.
Then
min
sup
~b
(a,
b)
=
sup
min
O(a,
b)
.
bEB
aEA

aEA
bEB
P oo
o he
lemma
(1
.0)
:
Le
A
and
B
gi en
by
A={E
IIT jIIF
:
jEG,~II jIIG<-1}
B=
{b
E
L°(Y)
:
b(x)
>
0,
IIbil,
<
1}
and
we
de ine
on
A
x
B
he unc ion
0
as
O(a,
b)
=

~I
T
j
(x)
II
Fb(x)"d (x),
B
is
con ex
and
weakly
compac ,
A
is
con ex
;
0
is
con ex
and
lowe
semicon inuous
in
B,
see
VI
.4
.3
in
[GC,
R
de
F],
and
0
is
linea
on
A,
he e o e
by
he
MinimaxTheo em
we
ha e,
min
sup
O(a,
b)
=
sup
min
O(a,
b)
<
bEB
aEA

aEA
bEB
<
sup
II(E
IIT jII
F)lIpllLs(Y,d )
<_
aEA

Cp
.
j
VECTOR-VALUED
INEQUALITIES
WITH
WEIGHTS

181
The e o e, he e
exis
bo
E
B
such
ha ,
o
e e y
aE
A,
we
ha e
he
p oo
o
he
lemma
inishes
by
choosing
w(x)
=
bo(x)-°'
.
(1
.1)
Theo em
.
Le
(Y,
d )
be
a
measu e
space,
F
and
G
be
Ba-
nach
spaces,
and
{Ak}"
be
a
sequence o
disjoin
se s
in
Y
such
ha
Uk
o
Ak
=
Y
.
Assume
ha
0
<
s
<
p
<
oo
and
T
is
a
sublinea
ope a o
which
sa is ies

11T
;(x)IIFbo(x)"d (x)
<
CP,
.7
(1
.2)

II(EIIT
;IIF)
I
IPIIL-(A,,d )
<Gk(EII j1I
G
)IIP,
kE0,1,
.
.
.
.7

7
whe e
o
each
k,
Ck
is
acons an
depending
on
G, F,
p
and
s
.
Then
he e
exis s
a
posi i e
unc ion u(x)
on
Y
such
ha
(1
.3)

(
.lY
IIT (x)IIFU(x)d (x))
1
IP
<
CII JIG
holds,
whe e
C
is
a cons an
depending
on
G,
F,
p
and
s
.
Mo eo e ,
gi en a double sequence
{ak}k
á
such
ha
and
o,
=
(p~~,
u
can
be
ound
such
ha
¡¡u
-1
xAkIIL-
i
(Ak,d )
~
(ak1Ck)P
.
P oo
.
.
Gi en
k,
we
de ine
he
ope a o
Tk
by
Tk
(x)
=
T
(x)
MA,
(x),

xE
A
k
.
Tk
sa is ies
Ak
such
ha
llw
-l
llL , -
1
(Ak,d )
<
1
and
A
IITk (x)IIFwk(x)d (x)
<_
Ckli lic*
k
k=0
II(E
IITk
;
IIFWPII
L-(Ak,d )
<
Ck(E
II
;
IIG)lIP,
7
ak
<
+oo
hen
by
lemma
(1
.0),
he e
exis s
a
nonnega i e
unc ion
wk
de ined
on

182

L
.
M
.
FERNÁNDEZ-CABRERA,
J
.
L
.
TORREA
+oo
We
de ine
u(x)
by
u(x)
=
y~akCk
p
wk
(x)XA
k
(x),

x E
Y
;
hen
0
l
IIT
(x)
II
Fu(x)d (x)
_
akC
k
p
J

IITj(x)
II
Fw(x)d (x)
<_
ak
ll
II
+00

+00
G
=
0

Ak

0
=Cll
. llG,
and
¡¡u-1XAk
IIL
-
1(Ak,d )
=
(ak
1
Ck)plluwkIlL-
-1
(Ak,d )
:5
(ak
1
Ck)
p
.
We
in oduce
i s
some
no a ion
.
o
weigh s
w(x)
in
R
n
.
2
.
Applica ions
Gi en
1
<
p
<
oo
and
0
<
-y
<n
we
shall
de ine
he
ollowing
classes
(2
.1)

Z
P
, .
y
=
{w
:

w(x)(1
+
Ixl)(''
-n
)pdx
<
+oo}
Rn
(2
.2)

D
p
,_
y
=
{w
:
L

w(x)
1-
p
'
(
1
+
IXI)(-y
-n)p'
dx
<
+oo}
Rn
(2
.3)

Dp
,
,
y
=
{w
:
sup
R('Y
-n
)p

W
1-
p
(x)dx
<
+oo},
R>1
l-I<R
when
y=
0,
we
shall
w i e
simply
Z
p
,
D
pand
D*
p
,
Rema k
.
I
is
easy o
check
ha
i
p
<
q
hen
D
p
C
Dp
C
D9,
D
p
¢
Dq
and
DP
~¿
D
q
,
in ac
he
weigh
(x)
=
(1
+
IxIn)-1
be
longs
o
DT,
1
<
<
oo
bu
1
D
q
o
any
q,
1
<
q
<
oo
.
Finally
he weigh
w
p
(x)
=
IXI
n
(p
-1
)
(1
+
.
IxIn)
-
p(1
+
I
log
IXII)2(p-1)
belongs
o
D
p
bu
w
p
¢
Dp
1
i
pl
<
p
.
(2
.5)
(2
.6)
VECTOR-VALUED
INEQUALITIES
WITH
WEIGHTS

183
Analogoulsy,
gi en
1
<
p
<
oc, 0
<_
-y
<
nand
a
measu e
p
on
he
uppe
hal
plane
R++
1
,
we
shall
conside
he
ollowing
classes
o
weig hs
w
(x,
)
in
R++
1
,
(2
.4)
Zp,7(dp,)
=
{w
:

R'+1
w(x,
)(1
+
+
I
x1)(
7-n
)Pdp(x,
)
<
+oo}
DP,7(dp)
=
{w
:
R'+1
w(x~
)1-p'(1
+
+
I
xI)(-
,-n)P
dp(x,
)
<
+oo}
D*
, ,
y
(dp)
=
{w
:
sup
R(-1-n)p'

w(x, )
1-
P
dp(x, )
<
+oo},
R>1
(xI+ <R
when
7
=
0
we
shall
w i e
Zp
(dp,),
D
p
(dp)
and
D*
(dp)
.
A
.
Pa ial
sum
ope a o s
.
Conside
a
homogeneous
unc ion
o
deg ee
0 in
Rn,
p(X)
=
q(x'),
which
we
assume
o be
o
class
C(
1
ou side
he
o igin
and
sa is ying
he
cancella ion
p ope y
Fo
each
~
E
Rn,
we
de ine
he
ke nel
k
g
(y)
=
e
2
n
2
C'yQ(y')IyI
-n
,
and
he
co esponding
ope a o
Tg (x)
=p
.
.
kg (x
-
y) (y)dy
=
=
e27 2l
.xp
.
.

q((x
-
y)
,
)IX
-
yI
-n
e
-27,i
~
*
y (y)dy,
Rn
hen,
we
de ine
he
ope a o
and
conside
he
inequali y
SZ(x')d (x')
=
0
.
l-II=l
T*
(x)
=
sup
TI
(x)
I
(2
.7)


IT* (x)IPu(x)dx
<_
C
nI (x)IP (x)dx
.
R

R
We
ha e
he
ollowing
184

L
.
M
.
FERNÁNDEZ-CABRERA,
J
.
L
.
TORREA
(2
.8)
Theo em
.
Le
1
<
p
<
oo
.
(i)
I
uE
Z
p hen (2
.7)
holds o
some
,
such
ha
a E
Z
p
,
o
n<1
.
(ii)
I
E
D
pn
Dp
l
,
o
some
p1
<
p,
hen
(2
.7)
holds
o
some
u,
such
ha
u«
E
D
p
,
o
a
<
1
.
When
n
=
1and
Q(y)
=
-
1
sign
(y)
hen
To
=
H
(Hilbe
ans o m)
and
he
pa ial
sum
ope a o s
o
he
Fou ie
Se ies
can
be
exp essed
in
e ms
o
{T },ER
:
Fo
each
in e al
I
=
[a,
b]
SI (x)
=
(1)e2-i-ld1
=2(Tb (x)
-
Ta (x))
.
I
In
his
case
T*
is
Ca leson's
maximal
ope a o ,
see
[C],
[H],
and
we
ha e
(2
.9)
Theo em
.
Le 1
<p<
oo
.
(i)
uE
Z
p
i
and
only
i
(2
.7)
holds
o
some
,
(ii)
I
E
D
p
n
Dp
l
,
o
some
p1
<
p,
hen (2
.7)
holds
o
some
u
.
Con e sely
i
(2
.7)
holds
o
some
u
hen
E
D
p
.
(iii)
bo h
(i)
and
(ii)
a e
ae
i
we
eplace
T*
by S*
.
B
.
Li lewood-Paley
ope a o s
.
Le
cP
E
S(Rn)
be
such
ha
supp(0)
C
{1
E
Rn
:
2
<_

<_
2},
cp(
)
=
1
in
a
neighbou hood
o
111
=
1,
and
We
de ine
he
ollowing
ope a o
S*
(x)
=
sup
¡Si
(x)
j
<
CT*
(x)
I
c0(2
k
j)
=
1
o
all
l
;

0
.
kEZ
g (x)
=
(E
I'Pk
*
(x)12)1/2,
kEZ
whe e
Wk(x)
=
2
k
n,P(2
k
x),
and we
conside
he
inequali y
(2
.10)

~n
.~
IG
(x)
I
pu(x)dx
<
C
IR
.~
I
(x)
I
p
(x)dx
hen
we
ha e
VECTOR-VALUED
INEQUALITIES
WITH
WEIGHTS

185
(2
.11)
Theo em
.
Le
1
<
p
<
oo
(i)
I
u E
Z
p
,
hen
(2 .10)
holds
o
some
,
such
ha
«
EZ
p
o
n<1
.
(ii)
I
E
D
p
,
hen (2
.10)
holds
o
some
u,
such
ha
ua
E
D
p
o
a<1
.
C
.
Ope a o s
on
he
uppe
hal
plane
.
We
shall
conside
he
uppe
hal
plane
R++1
=
Rn
x
[0,
00)
.
Gi en
a
measu e
d
on
R++
1
,
we
shall
say,
as
usual,
ha
d
is
a
Ca leson
measu e
i
he e
exis s
a
cons an
C
such
ha
o
any
cube
Q
in
Rn,
(Q)
<_
CIQ1
whe e
Q
=
{(x,
)
:
x E
Q,
0
<_ <_
side
leng h
o
Q}
and
IQI
s ands
o
he
Lebesgue
measu e
o
Q
.
Gi en
a
measu e
dp
on
R++1,
we
shall
conside
he
ollowing
ope a o s
:
(2
.12)
Gene alized
ac ional in eg als
.
Le
0
<
7
<n
and
K
y
(x,
,
u)
=
Cn
(IXI
-}-
+
u)?'-n,

xE
R
n
,

,
u E
[0,
00)
.
Then we
de ine,
o
compac ly
suppo ed
on
R++
1
Tp,y
(XI
)
=
R'+1
Ky(x
-
y,
,
u)
(y,
u)dp(y,
u),

x
E
R
n
,

y
?
0
.
I
is
a
compac ly suppo ed
unc ion
on
Rn
and
K
.
(x,
)
=
cn(1x1
+
)y -
n,
xERn,
E
[0,
00), we
de ine
T-
y
(x,
)
=
I

K
y
(x
-
y,
)
(y)
dy,

(x,
)
E
Rn+
l
Rn
(2
.13)
Gene alized
Poisson
in eg als
and
Balayages
.
Le
K
be
a
unc ion
K
:
Rn
x
Rn
x
[0,
00)
x
[0,
00)
-->
R+
ha
sa is ies
K
(x,
y,
,
u)
C
(I
x
-
y1
+
+
u)n'

x,
yE
R
n
,
,
u
E
[0,
oc),
we
de ine
he ope a o
T
p
,,o
(x,
)
=
P
.V
.
IR-+
1
K(x,
y,
,
u)
(y,
u)dp(y,
u),

(x,
)
E
R+
+1
.
192

L
.
M
.
FERNÁNDEZ-CABRERA,
J
.
L
.
TORREA
whe e
in
he
las
inequali y
we
ha e used
ha
E
D
P
,
in
pa icula
we
ha e
1
/P

1/P
sup
EIT í'(x)IP

<C
~Il j~~LP
XESk

j

j

£1 (
)
(

)
1
hence
ge
II

1
: IT
.
IP

IILs(S
i,)
<C2kn/s

~II iIILPl( )
(
j

j
On
he
o he
hand,
as
we
said
be o e,
T
maps
L'P(Pl)
in o
weak-Ll
he e o e
we
use
Co la 's
inequali y
(see
[GC,
R
de
F,
V
.
2
.8])
and
we
1/P

1/P
II

E
IT
j
IP

IILs(S,)
<
ISkll
/S-1
11

E
IT
j IP

Leak-L
l
(SO
(
j

(
j
<
CpISki
1/S-1
111
E
Il ,llél

IILl
j
)
1/P

1/P
<
CpISkl
l/S-1

n
(
j

~
Il
. j(x)~~11

(x)áx

xI<2k+1
(x)1-p,dx
R
I
)
1/P
<
Cp2kn/s

II
J
II
Le
l
(
)
j
whe e
in
he
las
inequali y
we
ha e used
ha
E
D
P
.
This
inishes
he
p oo
o
(i)
.
In
o de
o
p o e
(ii)
we
decompose
again
each
unc ion
as
be o e
= + l
.
Since
~ j
K(x,
y)
II
Q-
<_
CI
X
I
-
n
and
E
D
p
,
we
ha e
analogously
as o
T, ha
1
/P

1/P
II

IIT
j
"IIe-

IIL'(Sk)
<_
C2knls

~I ALP( )
j

j

On
he
o he
hand
T
maps
LQP(R
n
)
in o
Lép(e-)(Rn),
1
<
,
p
<
oo,
he e o e
by
Hólde 's
inequali y
we
ha e
1/p

1/p
11

EIIT jllIILs(Sk)

ISkls

¡¡

E
IIT jIIllL (Sk)
(
j

(
j
<CISkIl
ll
I
I
:I
j lp

JIL-
.7
<
CIS
k
I1/S

I j
l
p

(x)d

(x)-(P)(-,)~
~
~
11
.
j

IXI<2k+1
Now
we
choose
such
ha
2
=
p1 and by
using
he
hypo hesis
E
Dp
l
,
we
ha e
VECTOR-VALUED
INEQUALITIES
WITH
WEIGHTS

193
II
IIT j1Ie-

p
)
IILs(Sk)
<_
C2kn/s

ll
JllLp( )

.
j

j
his
inishes
he
p oo
o
(ii)
.
P oo o
Theo em
(2
.8)
:
S
k
Mo eo e u
can
be
ound
such ha
Aswe
said
abo e
in
o de
o
p o e
(i) i is
enough
o
p o e
(3
.2)
.
We
obse e
ha
by
he
las
P oposi ion
he
ope a o
T
sa is ies
(1
.2)
wi h
F=
R,Ak
=
Sk,
Ck
=
C2
k,
/'
and
G
=
LP
l
( ),
hen
by
Theo em
(1
.1)
he e
exis s
a
weigh
u
sa is ying
(3
.2)
.
<
(ak12kn/s)p
L
(1
+
lxI)np'
dx
<
E
2-Knp
~K
U(X)q-
1
dx
<
L
1
00

Q(

1
7
<
572
-Knp'
(
a
K12kn/s)p(q-1)
(2kn)
9-1
K=o
wi h
Q
=
(P)'
and
ak
such ha
E
aP
<
oo
.
The e o e
i
we
ake
q,
1
<
q
<
oo,
such
ha
q
-
1
<
p'
-
1,
we
ha e
194

L
.
M
.
FERNÁNDEZ-CABRERA,
J
.
L
.
TORREA
bu
-p'+
P (q
-
1)
+

_a
1
1

,

p')
+
(q
-
1)
<
0,
hen
i
we
chose
e
-1
aK
=
2K,E,
wi h
e
small
enough,
we
ge
ha
u«
E
D
P
wi h
a
=
P
.
Analogously
in
o de
o
p o e
(ii)
we
obse e ha
by
using
P oposi ion
(3
.3)
(ii)
T
sa is ies
(1
.2)
wi h
F=
Q°°,
Ak
=
Sk,
C
k
= C2
kn
/
s
and
G
=
LP( ),
hen
Theo em
(1
.1)
can
be applied
as
be o e
.
P oo
o
Theo em
(2
.9)
:
The
su icien
condi ions
on
(i)
and
(ii)
ha e
been
p o ed
in
Theo em
(2
.8)
.
In
o de
o
ob ain he necessa y
condi ions
we
obse e
ha
T
*
(x)
?
IH (x)I,
whe e
H
is
he
Hilbe
ans o m,
hen
he
condi ions
in
o de o
ha e
(2
.7)
a e
also
necessa y
in
o de
o
ha e
(3
.4)

~R
IH (x)IPu(x)dx
<
CJR
~ (x)IP (x)dx,
bu
i
is
well
known
ha
in
o de
o
ha e
(3
.4)
o
some
u
( esp
.
some
)
i is
necessa y
ha
E
D
P
( esp
.
u E
Z
P
),
see
[GC,
R
de
F]
.
Finally
o
p o e
(iii)
we
obse e
ha
as
S*
(x)
<CT*
(x)
we
ha e
ha
he
su icien
condi ions
o
weig hs
in
o de
o
ha e
(2
.7)
a e
also
su icien
o
(3
.5)

~S*
(x)
¡Pu(x)dx
<
C
l
I
(x)
IP (x)dx
.
IR

R
On
he o he
hand
obse e
ha
an
inequali y
o
he
ype
(3
.5)
implies
ha
o
any
in e al
I
C
R,
he
inequali y
l
R
ISI
(x)
I
Pu(x)dx
<C
R
I
(x)
I
P
(x)dx
holds
wi h
C
independen
o
I
and
hence
(3
.4)
holds
and
he
necessa y
condi ions
again
a e
he
some
ha
hey
a e
o
he
Hilbe
ans o m
.
B
.
Li lewood-Paley
ope a o s
.
Ou
idea
is
o
p o e
Theo em
(2
.11)
ollowing
he
lines
o
he
p oo
o
Theo em
(2
.8)
.
We
conside
he
PZ- alued
ope a o
T (x)
=
{Wk
*
(x)}kEZ
VECTOR-VALUED
INEQUALITIES
WITH
WEIGHTS

195
whe e
Wk
a e
he
unc ions
de ined
in
sec ion
2
pa
B
.
I is
clea
ha
G
(x)
=
II
T
(x)
IIQ2
and
hen
we
a e
going
o
deal
wi h
he
ollowing
inequali y
(3
.6)

IR
.
IIT (x)IIé2u(x)dx
<
CL
.n
I (x)Ip (x)dx,
in
ins ead
o
(2
.10)
.
I
is
known
ha
T
is
gi en
by
he
2
2
- alued
ke nel
K(x,
y)
=
{W
k
(x
-
y)}kEZ
ha
sa is ies
IIK(x,y)11£2
<CIx-
y
l
-
n
.
Mo eo e
he
ope a o
de ined
by (
j
)
j
->
(T i
)
j
is
bounded
om
L
.1P(R
n
)
in o
weak
-L,lP(e2)(Rn
)
,
1
<
p
<
oo,
see
[R
de
F,
R,
T]
.
We
can
conside
also
he
adjoin
ope a o
T,
ac ing
on
2
2
- alued
unc-
ions,
(X)
=
( k(x))k,
and
de ined
by
T
(x)
=T
(( k)k)(x)

Wk
*
(x),
k
T
is
de ined
by
he
0=
G(P
2
,
C)- alued
ke nel
K(x,
y)
=
{'Pk(y
-
x)Ík
and
again
he
ope a o
T((
j
)
j
)
is
bounded
om
LQP(e2)
in o
weak
-
LlP,
1<p<oo
.
The e o e
(i)
and
(ii)
o
Theo em
(2
.11)
a e
equi alen
s a emen s
and
i is
enough
o
p o e
(ii)
.
In
o de
o
p o e
(ii)
we
need
he ollowing
(3
.7)
P oposi ion
.
Le
E
D
p
,
1
<
p
<
oo
and
le
s
<
1
<
p
.
Then
we
ha e
11(1
:
IIT j1Ie2WpIILs(Sx)
:5
Cs,p
2xn/
s(E
II
II
L
P( ) )l
ip
7
K
=
0,
1,
2,
. .
.,
whe e
SK
a e
h se s
de ined
in
(3
.3)
.
The
p oo
o
his
P oposi ion
ollows
he
pa e n
o
he
p oo
o
P opo-
si ion
(3
.3)
.
Once
we
know
P oposi ion
(3
.7)
he
p oo
o
Theo em
(2
.11)
can be
buil
as
he
one
o
Theo em
(2
.8)
.
196

L
.
M
.
FERNÁNDEZ-CABRERA,
J
.
L
.
TORREA
C
.
Ope a o s
on
he
uppe
hal
plane
.
Ou
goal
is
o
es ablish
inequali y
(1
.2)
in his
con ex
.
We
shall
deno e
by
F(x)
he
cone
o
ape u e
one
whose
e ex
is x,
xE
Rn,
Le
.
F(x)
=
{(Y,
)
E
R++i
:

IX
_
y
I
<
}
.
(3
.8)
De ini ion
.
Gi en
a
posi i e
measu e
dp
on
R++
1
,
we
de ine
Al,
(x)
=

(y,
)
dp(n,
)

xE
R
n
.
(x)
(3
.9)
P oposi ion
.
Le
1
<
p
<_
oc
and
dp
be
a
measu e
on
R++1
.
Then
A~
is
a
bounded
linea
ope a o
om
L',(R++1,
dp)
in o
Lép
(Rn,
dx)
.
P oo
.
A
P
,
is
a
posi i e
linea
ope a o ,
hen
i is
enough
o
p o e
ha
A
wmaps
L'
(R+'-'-',
dp)
o o
L
l
(Rn,
dx),
bu
JI
A
W
11L1(d5)
=

I

(y,
)
dp(n,
)
jdx
R
(-)
Rn+1
C R~
X
,
(
.)
(y,
)
¡
(y,
)
I
dx)
d~ y'
)
-

n
1
<
R++1
( n
g(y, )
dx)
l
(y,
)id (y,
)
<
Cnil ilL'(R++1,d )
(3
.10)
Rema k
.
Gi en
a
posi i e
measu e
dp
on
R+
+1
,
we
can
de ine
he
ope a o
Al (x)
=

l (yl )l
d~ n~
)
,
x
E
Rn
.
The
ope a o
Al
is
ela ed
wi h
" en
spaces",
see [C,
M,
S]
and
[R,
T2]
.
I
can
be
showed
ha
A1
maps
Ls(R++1,
dp)
in o
LS(Rn,
dx),1
<
s
<
oo,
i
only
i
dp
is
a
Ca leson
measu e,
see
[R,
T2]
.
The e o e,
since
M
j
1)
(x)
=
Al
( )
(x),
we
ha e
ha
A
w
maps
L', (R++1,
dp)
in o
LQP
(R
n
,
dx),1
<
s
<
oo,1
<
p
<
oo
i
and
only
i
dp
is
a
Ca leson
measu e
.
VECTOR-VALUED
INEQUALITIES
WITH
WEIGHTS

197
(3.11)
P oposi ion
.
Le
du
be
a
measu e
on
R++
1
.
The
ollowing
inequali ies
hold
:
Whe e
M
y
,
T,
y
and
P
a e
he
ope a o s
de ined
on
(2
.14),
(2 .12)
and
(2 .18)
.
By
C
n
we
deno e
a
cons an
no
necessa ily
he
same
a
each
ocu en e
.
P oo
.
Le
B
=
B(xo,
),
xo
E
R
n,
>
0,
be
a ball in
Rn
.
I
(x, )
E
B
hen
I
x
-
xo
I
+
<
,
in
pa icula
we
ha e
IBI¡
n-1JB
I (y,u)Idw(y,u)=enIBI
n-1
B
I (y,u)I
(~
J//
B(y,u)dz)dp(y,u)
=
en
I
BI
n
-1

RTy'
1
1
R
XB
(y,
u)XB(Y,u)
(z)
l
(y,
u)
l
dzd'(y,
u)
<
CniBIn-1
BAl,(I 1)(z)dz
<
cnM,(Al, )(x, )
.
In
o de
o
p o e
(3
.13),
we
obse e
ha
Tw,7
(x,
)
I
=
¡en

(y,
u)

1

dz

dM(y,
u)
I
R7+,
(Ix-yI
+ +u)n-7
(un
B(Y,U)

)
<
Cn

I
(y,
u)
I

n-7
XB(y,u)
(z)
dzda(y,
u)
Rn
+1
IRA
(IX
-
yI
+
+
u)

,un
bu
i
z
E
B(y,
u),
hen
I
x
-
zI
+
<
I
x
-
yI
+u+
and
we
ha e
Ti
.,7
(x,
)
I
:5
:
cn
R
:

R-
(Ix
I
zI
y
+
)n-~
X
B(y,u)
(z)
dzdU
n
u)
+
-c
nR~
(IxA'zll_

+) ))
-~dz=enTy(A~I 1)(x, )
.
The
p oo
o
(3
.14)
es
analogous
.
The
ollowing
Theo em
can be
ound
in
[R,
T1]
.
(3
.12)
M,,,-,
(x, ) <_
C
.M-
y
(AI,
I
I)
(x,
),
(x,
)
E
R+
+1
0<y<n
(3
.13)
IT
.,
.y (x, )I
<-C
.T7(A,
.I I)(x, ),
(x, )ER++1
0<y<n
(3
.14)
I
T,,o
(x,«
I
G
CnP(A
W
I
I)
(x,
),
(x,
)
E
R++
1
.

198

L
.
M
.
FERNÁNDEZ-CABRERA,
J
.
L
.
TORREA
(3 .15)
Theo em
.
Le
1
<
p
<
oo,
and
d
be
a
Ca leson
measu e on
R+
+1
,
hen
he
ope a o s
,M
.,,
wi h
0
<
"
y
<
n,
T
y ,
wi h
0
<
-y
<
n and
P
a e
bounded
om
LPp
(R
n
,
dx)
in o
weak-Lé,7(R++
1
,
d )
and
om
LQ
P
(Rn,
dx)
in o
Lép
(R
+
+
i,
d ),
n
n~
,
<
S
<
00, q
=-
l
+
T
.
Taking
in o
accoun
his
heo em
and
P oposi ion
(3
.11)
we
shall
be
able
o
p o e
he
ollowing
(3
.16)
Theo em
.
Le
1
<
p
<
oo, 0
<
,y
<n
and
d
be a
Ca leson
measu e
in
R++
1
.
Gi en
a
measu e dp
in
R++
1
hen
he
ope a o s
Mand
T,,,,-,
a e
bounded
om
L'
P
(R++i,
d[ )
in o
weak
L,p
"
(R++
1
,
d )
.
Mo eo e
i
dp
is
a
Ca leson
measu e,
hen
he
ope a o s
M,,,7
and
TI
.,
.,
a e
bounded
om
LQp
(R++1,
d~)
in o
L'
P
(R++1,
d
)
,
q
=
ñ+
.
P oo
..
The
i s
pa
o
he
heo em
is
a
di ec
consequence
o
(3
.9),
(3
.11)
and
(3
.15)
.
The
second
pa
is
a
consequence
o (3
.10),
(3
.11)
and
(3
.15)
.
(3
.17)
Rema k
.
The
p oo
o
Theo m
(3 .15)
is
based
in
he
heo y
o
ec o - alued
Calde ón-Zygmund
Ke nels
.
Tha
p oo
is
no
a alaible
o
he
case
o
Theo em
(3
.16)
.
Now
we
can
p o e
inequali y
(1
.2)
o
hese ope a o s
.
(3
.18)
P oposi ion
.
Le
0
<
s
<
1
<
p
<
oo, 0
<
,y
<n
and
le
d
a
Ca leson
measu e
in
R++
1
.
Le
SK,
K
=
0, 1, 2,
. .
.,
be he se s in
R++
1
de ined
by
So
={(x, )ER+
1,

Ixl+ <1}
SK
=
{(x,
)
E
R++1,

2
K-1
<
I
XI
+
<
2
K
},
K
=
1,
2,
...
(i)
I
E
DP,-
y
(dp)
and
G
=
LP(R+1,
dp)
hen
II(1
ITp,7 i
l'
P
II
L-(SK,d )
<
C2K(E
II
;IIG)üP
.
7

7
(ii)
I
E
DP
,,
y
(dp)
and
G
=
LP(R+
+1
,
dp)
hen
II(1
:IMw7 jl
p
)
1
'
p
ilLs(SK,d )
<c
2~s
(EII
;II
G)"P
.
VECTOR-VALUED
INEQUALITIES
WITH
WEIGHTS

199
P oo
.
Gi en
K
>_
0,
we
decompose
each
unc ion
=
'
+
"
whe e
= XB
K
, "= - 'and
I
¡y¡
+
u
>
2(Ixl
+
)
hen
Iyl
+u
<
Iy¡
+u+
4
<
Iy1+u+2 +Iy¡+u-2Ixi
<
2(Ix-y¡+ +
u)
.
The e o e
i
(x,
)
E
SK
we
ha e
ITil,7
"
(x> )l
=
1
/,y¡

K7
(x
-
y,
,
u) (y,u)dp(y,u)I
+u>zK+
1
>2(Ixl+ )
lyl+u>2K+
1
lyl+u>2K+
1
BK
=
{(x,
)
:

Ixi
+
<
2K+I}
.
1
.
(y,
u)

dp(y,
u)
(Iy¡+u)
n-7
<
Cn
( Rn+1
l
.
(y,
u)
I
P
(y,
u)dp(y,
u)
(y,U)1-P

)
1/p
<
(¡y¡
+
u)(n-7)P'
dl~(y,
u

.II
~~G
Thus

sup

(E
m
.,7 i(x, )IP)1/P
<
C(57
II jIIc)1/P
and
hen,
(a, )ESK
7

.7
II(1ITp,7 ii1/PIILs(SK,d )
<
CV(5K)1/s(1
:
II
;IIG)1/P
<
<
C2ns
(1
:
li
;IIG)
1
/P
.
On
he
o he
hand,
as s
<
nn7,
we
use
Co la 's
inequali y
(see
[GC,
R
200

L
.
M
.
FERNÁNDEZ-CABRERA,
J
.
L
.
TORREA
de
F,
V
.2
.8])
and
Theo em
(3
.16)
o
ge
he e o e
11
(1
:
I
Tw,y
. j
ip)
1
/plI
LI
(SK,d )
j
<CV(SK

n
)1/s_

II(Y
:ITw,y
. jlp)1/pIILn,ny
(Rn+1,d )
GC2
K
s
["-(n-y)]
¡¡(Y
:
1
i
j
=C2
K
s

(~
L j
(y,
u)
I
p)1/p
dp(y,
u)
~B
K

2K(n
-
y)
j
<C2
K
s

L j
(y,
u)
I
p (y,
u)dM(y,
u)
j
l-p,
(y,
u

11/p

n
+u+
1))-ydu
(Y,u)J

<
C2
K!'
(1
II
. iliG)1/p
.
This
comple es
he
p oo
o
(i)
.
In
o de
o
p o e
(ii),
we
obse e
ha
i
(x,
)
E SK,
hen
(x,
)
E
QK
whe e
QK
is
he
cube
QK=
{y
E
R
n
"

y
=
(Y1,
.
.
.,
yn),
I
NI
:~
2K,

i
=
1,
. .
.,
n},
1
IQxIn-1
~GlK
I
l
~(y,u)Idl~(y,
u)
1/p
,
<
C
n
II
.
II
G
2K(7-n)
(

(
y
u)1-p
dp(y,
u)
I
B/x
1/p'
<
C
n
II
IIG
s11P
CR(y
-n
)p'

(y,
u)1-p
dM(y,
u)
I

<
R>1
I
.J+U<R
<
CnII,
IIG-
Now
he
es
o
he
p oo
ollows
as in
(i)
.
P oo
o
Theo em
(2
.16)
:
I
E
Dp,y(dp),
hen,
by
he
las
P opo-
si ion,
inequali y
(1
.2)
is
sa is ied
o
T
p
,, .
y
wi h
AK=
SK,
G=
Lp( dp),
F
=
R,
cK
=
2K'
.
The e o e
by
Theo em
(1
.1)
he e
exis s
u
sa is ying
(2
.15)
o
T
p
,,
y
.
Mo eo e u
is
such
ha
IIu
-I
XSK
IIL-
1
(AK,d )
<
(aK12
sn
)p
VECTOR-VALUED
INEQUALITIES
WITH
WEIGHTS

201
wi h
u
=
(P
) and
~aP
<
+oo,
hen
U(X)
1-a

°°
n-7)P~
d (x,
)
<
~
2-K(--7)p'
(aK12
'-)P(a-1)
,
R
n+1
(1
+
+
~x~)(

K=O
bu
as
(2)'
<
p'
we
ha e
-(n
--
y)p'
+
sp(a
-
1)
<
0
.
The e o e
i is
enough
o
choose
aK
=
2-K,
wi h
E
small
enough
and
hen
u«
E
DP,7 wi h
cx
-
0
-
1
.
p
This
inishes
he
p oo
o
(i)
.
The
p oo
o
he
su ciency
o
condi ion
DP
,
, y
(d¡z)
in
(2,16)
is
ob ained
in a
simila
way
using
(3
.18)
(ii)
.
Fo
he
necessi y
obse e
ha
o
any
ball
B
BC
{(x, )
:
Mw,7 (x, )
>
IBI~-1
hen
(2
.15)
o
T
=
M,,,7
implies ha
he e o e
o
=
XB
1-
p
we
ge
he
esul
.
1
(y,
u)
I
dw(y,
u)
},
Á
u(x, )dp(x, )
<
<
C
(Á
I
(y,
u)
I
dp(y,
u»
-P
I
BI
(!-1)P
R
~
+1
(y,
u)
I
P
(y, u)di (y,
u),
P oo
o
Theo em
(2
.17)
:
Since
T
N
,,
.
y
is
essen ially
sel -adjoin ,
a
simple
duali y
a gumen shows
ha
he
pai
(u
(x,
),
(x, ))
sa is ies
(2
.15)
o
he
exponen
p
i
and
only
i
he
pai
( (x,
)
1-
P
,
,
u(x,
)
1-P
)
sa is ies
he
some
inequali y
wi h
exponen
p'
.
Thus
(i) is
ac ually
equi alen
o
(2
.16)
(i)
.
The
necessi y
o
(ii) is
ob ained
as in
(2
.16)
(ii)
.
Fo
he
su iciency
we
conside
he
Q°°- alued
ope a o
Tw,7
(x,
)
=
{
X
QT
(
x
'
)
IQ
I1-7/n
. QT
(y,
u)dp,(y,
u»IER
whe e
Q
is
he
cube
cen e ed
a
o igine
and
wi h
side
leng h
.
I
is
clea
ha
M
m
,,
(x,
)

=

JIT
N
,,
y
(x,
)
I1E_
.

The e o e
(u
(x,
),
(x, ))
sa is ies
(2
.15)
o
M
N
,,,
y
i
and
only
i
sa is ies
(3
.19)
a
pai
L
.+1
IIT~,7
(x,
)
IIé-u(x,
)d (x,
)
<
C
IRn+1
I
(x,
)
I
P (x, )dp(x,
)
.
208

L
.
M
.
FERNÁNDEZ-CABRERA,
J
.
L
.
TORREA
[R de
F,
R,
T]
RUBIO
DE
FRANCIA,
J
.
L
.,
RUIZ,F
.
J
.
AND
TORREA,
J
.
L
.,
Calde ón-Zygmund
heo y
o
ec o
alued
unc ions,
Ad
.
i
n
Ma h
.
62
(1986),
7-48
.
[R,
T1]
Ruiz,
F
.
J
.
AND
TORREA,
J
.
L
.,
Weig hed and
ec o - alued
inequali ies
o
Po en ial ope a o s,
7' ans
.
Ame
.
Ma h
.
Soc
.
295
(1986),213-232
.
[R,
T2]
Ruiz,
F
.
J
.
AND
ToRREA,
J
.
L
.,
Vec o - alued
Calde on-
Zygmund
heo y
applied
o en
spaces,
Colloquium
Ma h
.
62
(1991),265-277
.
[S]
SAWYER,
E
.,
A
cha ac e iza ion
o
Two
Weigh
No m
inequali ies
o
F ac ional
and
Poisson
in eg als,
73
-
ans
.
Ame
.
Ma h
.
Soc
.
30
8
(1988),533-545
.
Luz
M
.
Fe nández-Cab e a
:
Escuela
Uni e si a ia
de
Es adís ica
Uni e sidad
Complu ense
de
Mad id
Mad id
SPAIN
Rebu
el
4
de
Se emb e
de
1992
José
L
.
To ea
:
Depa amen o
de
Ma emá icas
Uni e sidad
Au ónoma
de
Mad id
28049
Mad id
SPAIN