Publicacions
Ma emá iques,
Vol
35
(1991),
299-321
.
REMARKS
ON
KATO'S
SQUARE-ROOT
PROBLEM
JEAN-LIN
JOURNÉ
0
.
In oduc ion
Le
T
be
a
maximal
acc e i e
ope a o
on
a
Hilbe
space
I
-
l,
wi h
domain
V
.
The e
is
a
well-de ined
squa e- oo
o T,
T'I'
which
sa is ies
he
equa ion
T
1
/
2
U
=
1
A
-1/2
(T
-
i
-
A)
-1
Tu
dñ,
o
o
all
u
in V[1]
.
When
T
is
sec o ial,
Ka o
conside s
he
p ehilbe ian
s uc u e
on
V
de ined
by
Suppo ed
by
he
N
.S
.F
.
<u, >y
=
<Tu, >+<u,T >+<u, >,
whe e
<
.
, .
>
deno es
he
scala
p oduc
on
'H
.
Ka o
conjec u ed
ha
he
domain
o£
T1/2
was
he
comple ion
o
V
in 'H
o
he
p ehilbe ian
s uc u e
de ined
by
<
.
,
.
>
y
.
This
conjec u e
was
disp o ed
by
A
.
McIn osh
[2]
.
He
obse ed
ha
he
ailu e
o
his
conjec u e
was
connec ed
wi h
he
ailu e
o
he
inequali y
11
I
AI
B
-
BI
AI
11

<

C

11
AB
-
BA
li,
o
gene al
sel -adjoin
ope a o s,
¡Al
being
(A')'/'
.
Since
he
boundedness
o
he
i s
commu a o
o
Calde ón
[3]
is
a
( ue)
special
case
o
he
abo e
( alse)
gene al
inequali y,
McIn osh
sugges ed
ha
Ka o's conjec u e
migh
be
ue
when
T
is
a
di e en ial
ope a o
on
R'
o
he
o m
di
AV,
whe e
A
is
a
ma ix- alued
unc ion
such
ha ,
o
some
S
>
0
and
o
all
x in
R"
and
1
in
Cn,
Re <
1,
A(x)j
>

>

$
1112
.
I
is
his
special
case
which
is
now
known
as
Ka o's
conjec u e
o ,
mo e
p ecisely,
Ka o's
squa e- oo
p oblem
.
The
connec ion suspec ed
by
McIn osh
u ned
ou
o
be
qui e
signi ican
and
led
simul aneously
o
he
solu ion
o
Ka o's
conjec u e
in
dimension
1
and
o
he
p oo
o
he
L
2
-boundedness
o£
he
Cauchy-ke nel
on
Lipschi z
g aphs
[4],
conjec u ed
by
Calde ón
and
p o ed
by
himsel
in
he
case o£
small
Lipschi z
cons an s
[5]
.
300

J .-L
.
JOURNÉ
A
simple
escaling
on
A
shows
ha
i is
enough
o
conside
he case
whe e
~~
A-I
Iloo
<
1
.
In
dimension
la ge
han
1,
Ka o's
conjec u e has
been
sol ed
when
A
is
a
small pe u ba ion
o he
iden i y,
ha
is
i
11
A-I
jj
'>
"
<
e n
,
whe e
e n
depends
only
on
he
dimension
n and
decays exponen ially
wi h
n
[6],[7]
.
A
na u al
way
o
a ack
Ka o's
conjec u e
is
he e o e o ind bes possible
lowe
bounds
o e n
.
He e
we
ob ain
explici
lowe
bounds
o e
n
,
decaying
like
n -1
/
2
.
In
pa icula ,
i
n
<
5,
one can
ake
e n o
be
1/3
.
Also
we
answe
a
ques ion
o
A
.
McIn osh
conce ning
he
solu ion o
Ka o's
conjec u e
in
dimension
1
.
Rescaling
A
so
ha ,
o
all
x
in
Rn,
Re
A(x)
>
I,
one
may
assume
ha
¡¡A
-1
-Ijj,,
<
1
.
In
dimension
1,
he solu ion
is
ob ained
by
expanding
an
ope a o
depending
on
A
in
a
se ies
o
ope a o s
depending
mul ilinea ly
on
he
new
a iable
B
=
I
-
A
-1
,
hen
by
showing
ha
hese
ope a o s
a e
bounded
on
L
2
and
ha
he
no ms
can
be
summed
when
11
B
~j
'>
"
<
1
.
In
dimension
la ge
han
1,
a simila
p ocedu e
does
no
seem
o
b ing
he
same
kind
o
simpli ica ion
and
hence,
i
was
sugges ed
by
A
.
McIn osh,
as a
i s
s ep o
unde s and
he
highe -dimensional
case,
o
ep o e
Ka o's
conjec u e
in
dimension
1
wi hou
using
he
a iable
B
.
This
is
wha we
do
a
he
end
o
his
pape
.
The
con en
o he
sec ions
o
his
pape
a e as
ollows
.
In he
i s
we
in oduce
some
no a ions
and
ecall
some
o mulas
.
In
he
second
we
ecall
some
basic ac s
abou
Ca leson-measu es
.
In
he
hi d,
we
s a e
a
heo em
abou
mul i-linea
ope a o s,
which
eadily
implies
he
imp o emen
on
he
o de
o
magni ude
o e n
,
and
whose
p oo
we
ou line
.
In
he ou h
and
i h
we
p o e
some
echnical
es ima es
on
ke nels
o
ope a o s
and
in he six h
we
conclude
he
p oo
o
he
a o emen ioned
heo em
.
In
he
se en h
we show
how
o
use
a ious p ope ies
o
he ope a o s
a ising
in
Ka o's
p oblem
o
imp o e
he
lowe
bound
o e,
gi en
by
ou
heo em
.
In
he
eigh h
we
gi e a
di ec
p oo
o
Ka o's
conjec u e
in
dimension
1
.
1
.
No a ions
andFo mulas
The
ollowing
is
pa ly
bo owed om
[7]
.
I
we make
he
change
o a iable
A
=
-2
,
we
see
ha
he
o mula
gi ing
T'I'
can
be
ew i en
as
T'/
2
=2

T

d
.
0
2
T +
I
Le
Dj
_-
i
jai
,
D
=
V
and
D*
=
(D
I ,
. .
.,
D
n
) .
We
deno e
by {A}
he
ope a o
o
poin wise
mul iplica ion
by
he
ma ix
A
.
Then
T
can
be
w i en
as
D*{A}D
.
Le
U
=
A-
I
.
Then
T
=
D*{U}D
+,n,
.
Le
P
=
(
2
A+
I)-1
.
Then
(
2
T
+
I)
-1
can
be
w i en
as
j
:(-1)'P,
[ 2D*{U}DP ]~,
j
;1
0
as
long
as
11
U
jj,,,< 1
.
And
in
his
case,
Le
Q
=
D
*
P
and
R
be
he
ma ix
o
ope a o s
whose
en ies
a e
Dó'
.
Obse e
ha
2
DD*P
=
R(I
-
P
) .
The e o e he
Sobole
space
W1,2
will
be
in
he
domain
o
T
1
/
2
i
he
se ies
o
ope a o s
Q
*
[{U}R(I -
P )]j
d
j>-
o
is
con e gen
in
L
2
-ope a o
no m
.
Tha
each
summand
is
bounded
wi h
a
no m
domina ed
by
C,j
11
U
j1i
is
known
[6],[7]
.
Obse e
ha
i
A
is
bounded
and
s ic ly
acc e i e,
11
AA-
I
11,,~<
1,
o
small
enough
.
Hence
one
can always
educe
o
he
case
whe e
11
U
11
<
,~<
1
.
The e o e
Ka o's conjec u e
will
be
sol ed
i
one can
show
ha ,
o
all
e
>
0,
each
summand
is
bounded
wi h
a
no m
a
mos
C,,(1
+
e)j
when
11
U
jj"
.<
1
.
One
can
also
do
a
powe
se ies
expansion
in
he
a iable
V=
I -
A-1
.
S a ing
om
we
ob ain
KATO'S
SQUARE-ROOT
PROBLEM

30
1
-
=L
.
.(-1)jp
[
2
D{U}D*P
]
j
D*{A}D
=
j
:(-1)jD*P
[
2
{U}DD*P
]
{A}D
.
j_>0
(
2
D*{A}D
+
I)
D*
=
D*{A}(I
+
2
DD*)
-
D*{A}
(I
-
{A}-1)
(2)

T
D
*
R
[{V
}R ]'
d
j30
0

=
D*{A}
(I
-
{V}(I
+
2
DD*)
-1
)(I
+
2DD*),
(
2
D*{A}D
+
I)
-1
D*{A}
=
D*(I
+
2
DD*)
-1
(I
-
{V}(I
+
2
DD
*)
-
1
)
.
Le
R
=
(I
+
2
DD*)
-1
.
Then
i
11
V
li~~<
1,
one
ob ains
ha
W1,2
will
be
he
domain
o
T
1/2
i
he
se ies
o
ope a o s
is
con e gen
.
I is
easy
o
check
ha
D*R
=
Qi
and
ha
R
=
I
-R+RP,
.
Hence
he e
is
a
e y
close
esemblance
be ween
he
wo
expansions
(1)
and
(2)
.
In
dimension
1
howe e ,
R
=
I and
R
=
P
,
so
ha
he
expansion
in
302

J .-L
.
JOURNÉ
he
a iable
V
is
much
easie
o
handle
han
in
he
expansion
in
he
a iable
U
.
By
mul iplying
A
by a
la ge
numbe
so ha
Re
A >
I,
one can
educe
o
he
case
whe e
II
V
II<,
.<
1
.
The e o e
one can
y
o
p o e
Ka o's
conjec u e
by
showing
ha
o
all e
>
0,
he
summands
in
he
se ies (2)
a e
domina ed
by
C
E
(1
+
e)j
II
V
IIi
in
ope a o
no m
.
This
is
how
i
was
done
o iginally
in
dimension
1 [4]
.
In
highe
dimension
i is
no
clea
ha
one
o
he
se ies
(1)
o (2)
has
an
ad an age
o e
he
o he
.
One
can
ask
in
pa icula
i
heie
is
a
simple
ela ion
be ween
hei adii
o
con e gen e
.
In
dimension
1,
i is
easy
o see
a
p io i
ha
hey
a e
he
same
.
Indeed,
a
unc ion
a
is
such
ha
II
a
-1
II
.<
,
Ao
<
1, i
and
onl y
i
II
a-1
-1/(1-
ió)
II<,
,<
A0/(1-
Aó)
.
The e o e
i
Ka o's
conjec u e
is
ue
when
II
a
-
1
II~<
o
<
1,
i
is
also
ue
when
II
(1
-
A2
)a`
-
1
II,,,,
A
o
,
and
also
when
II
a-1
-
1
II,<
A
o
a e
escaling
.
I is
a
li le
su p ising,
bu
easy
o
see,
ha
his
emains
ue
in
highe
dimension
.
I is
a
consequence
o he
ollowing
ac
.
I
one
pu s
on
GL
n
(C)
he
dis an e
induced
by
he
ope a o -no m
on Cn,
d(A,
B)

e,C
uéu=1
II
Uj
-
V
I
II,
hen
in e sion
does
no
in
gene al
map
halls
o
o he
balls
.
Howe e
i
does
map
a ball
cen e ed
a ound
a
mul iple
o
he
iden i y o
a
ball
o
he
same
na u e,
jus as
in
C*
.
This
is
wha
he
nex
lemma
exp esses
.
Lemma
1
.
Le 0
<
Ao
<
1
and
A
in
GL,(C)
.
Then
IIA-III<ao
~
II(1-Aó)A
-
'-III<ao
.
P oo
. .
Le
w
be
a uni
ec o
in
C'
.
We
wan
o
show
II
(1-
aó)A
-
'w
-w
II
,<
A
o
.
Making
he
change
o a iables
=
A-1w,
i
is
enough
o
show
ha
II
(
1
-
Aó)
-
A
jj,<
ao
11
A
ll
Bu
and
A
a e jus
wo
ec o s x
and
y
such
ha
II
y
-
x
II<
- o
II
x
II
.
The e o e
we
can
check
his
inequali y
in
C
le ing
x
=
1
and
y
=
1
+
z
wi h
IzJ
<
A
o
.
A e
di ision
by
Iyi(1
-
ió),
his
is
equi alen
o
I(1
+
z)
-'
-
(1
-
~`ó)-'I

>,0(j
-
ió)-'
This,
in u n, ollows
om
he
ac
ha
he
in e sion
in
C* maps
B(1,
A
o )
o
B((1
-
Aó)-1,
A
0
(1
-
Aó)-1)
.
This
p o es
Lemma
1
.
Since
he
in e sion
is
an
in olu ion
on
GL

(C),
i
ollows
ha a
ball
cen e ed
a ound
a
mul iple
o
he
iden i y
is
mapped
exac ly
on o
a
simila
ball
.
F om
he
p e ious
lemma
we
see ha ,
i
A
is
bounded
and
ReA>
S
>
0,
min
II
AA-
I
II,,
.=
min
II
AA
-
1-
I
II

.
a>o

a>o
KATO's
SQUARE-ROOT
PROBLEM

30
3
I
ollows
ha
i
he
se ies
(1)
con e ges
when
11
U
II,
> ,
>
<
A
o
<
1,
o
some
Ao
>
0,
hen
W
l,z
is
he
domain
o
T
1
1
2
when
mine,>o
II
AA-
I
II,,~<
Ao,
and
he e o e
when mina
>
o
II
AA
-1
-
I
II,,
.<
Ao
.
So
he
se ies
(2)
mus
con e ge
when
11
V
II
< ,
.<
A
o
.
And
con e sely
.
Ha ing
obse ed
ha
he
adii
o
con e gence
o
(1)
and
(2)
a e
he
same,
we
would
like
o
eco e
his ac di ec ly
om
he
s udy
o
he
mul i-linea
e ms
o
(1)
and
(2),
and
o
show
ha
his
adius
is
1
.
As
we
shall
see,
he
me hod
we
use
gi es
wo
dis inc
alues
o
he
adii
o
con e gence
o
(1)
and
(2),
and
his
shows
ha
i is
no
op imal
.
2
.
P elimina ies
on
Ca leson
Measu es
Le
p
deno e
he
ope a o
o
con olu ion
wi h
he
Poisson-ke nel
.
Then
a
Ca leson-measu e
M on
he
hal -space
R++
1
i
s
a
measu e
o
which
one
has he
es ima e
(3)

IR-+1
Ipj(X)1
2
d1,(X,
)
'<
C(,1)
II
II2
A
necessa y
and
su ñcien
condi ion
o
a
measu e
p
o
ha e
his
p ope y
is
he
exis ence
o
a
cons an
C
>
0
such
ha
o
all
cubes
Q
in
R
n,
(4)

11(Q
x
[0,
si)
S
Cm,
whe e
IQI
and 6
deno e
he
Lebesgue
measu e
and
he
side-leng h
o
Q
.
Le
us deno e
by
c
P
he
bes
cons an
in
(4)
.
Then,
o
some
absolu e
cons an
C,
C(p)
<,
Cc,
.
Also,
i
we
eplace
he
Poisson-app oxima ion
by
some
o he
app oxima ion
o
he
iden i y (p ) >o
sa is ying
app op ia e
es ima es,
hen
he
bes
cons an
in (3)
will
p esumably
change
.
Howe e ,
i
one
looks
a
he
size
o
he
di e ence
%
JR++1
(pj
-
pj)(,)jzdp(,,
)
we
see ha
i
depends
only
on a
cons an
h,,
which
we
de ine o
be
he
bes
cons an
in
he
inequali y
1,(Q
x
[6/2,61)
<,
CIQ1
.
O
cou se
h,,
<
c,
.
The
poin
is
ha
in
gene al,
and
in
wo king
on
Ka o's
p oblem
in
pa icula ,
one
ies
o
es ima e
c
P
o
measu es
o
which
one
al eady
has
a
good
con ol
o
h,,
.
Lemma
2
.
Le
p
and
p
be as
aboye
.
Then, o
some
C
>
0,
J
Rn+
,I(p
-p )(x)1
2
dp(x, )
<
Ch,,
11
II2

30
4

J
.-L
.
JOURNÉ
P oo
:
Obse e
ha
he
ope a o s
Tk
de ined
o
k
E
Z
by
<
9,
Tk
>=

[(P
-
P )(x)]
[(p,9
-
P 9)(x)]
dp(x,
),
R
.^
X
12k ,2k+11
sa is y
he
assump ions
o he
Co la -S ein
Lemma
wi h
a
cons an
depending
only
on
h,,
.
This
p o es
Lemma
2
.
F om
Lemma
2
we
see ha
2
1/2
[Jx
.^+1
P (x)1
d/ (x, )1

[Cocwl2+C1h1,l2]
11
112,
whe e Co
is
independen
o
he
app oxima ion
o
he
iden i y
bu C1,
o
cou se,
is
no
.
Lemma
3
.
The
cons an
C
o
can
be
chosen
equal
lo
2
in
all
dimensions
.
P oo
. .
By
Lemma
2
we
know
ha
we
can
choose
P
as
we
like
o
es ima e
he
bes
Co
.
By
he
same
a gumen ,
we
can
eplace
P"
(x)
by
S
(x)
=
mQ( x)
(x),
whe e
Q( , x)
is
he
dyadic cube
con aining
x o
size
2
k
,
wi h
2k-1
<
<
2
k
.
I
is
easy
o
see
ha
IS (x)l'dp(x, )~
1/2
<
cN'
IIm>ó
IS m0,12
and
i is
a
classical
ma ingale-inequali y
ha
l
maxiS (x)j

<,
2ll ll2
.
>o

z
Tol
e he
wi h Le n na
2
applied
wi h
S
ins ead
o
p
,
hese
wo
inequali ies
eadily
imply
Le n na
3
.
A
ob ious
bu
usc ul
ema k
is
ha ,
in
he
inequali y
/2
[JR^+i
ll íe (x)ll
'
dl (x, )~

1< (
2
cl~
/2
+C1hl
,
/ 2
)ll ll2
.
we
can
eplace
dIÁ(x,
)
by
dp(x,
u ),
whe e
0
<
u
<
1,
and
eplace
P
by
p,,,
o so ne
w>
1
.
KATOS
SQUARE-ROOT
PROBLEM

30
5
3
.
A
Mul ilinea
Es ima e
Le
(Ai)iEN
be
a
amily
o
ma ix- alued
unc ions
sa is ying
IIAjii
,
,~
S
1
.
Le
(Kj)iEN
be
a
sequence
o
con olu ion
ope a o s
mapping
C
d
- alued
unc-
ions
o
C
1
- alued
unc ions
.
We
assume
ha
he
symbols
u(Ki),
which
a e
ma ix- alued,
a e o
he
o m
((Sk, ( )/IIIZ))
whe e
he
Skj's
a e
homogeneous
polynomials
o
deg ee
2
.
Le
(MiEN
be a
sequence
o
numbe s
.
Fo
>
0and
i
E
N
we
de ine
he
ope a o
Ki
=
AiI
- -
Ki(I
-
P
)
.
We
assume
ha
he
symbols
o
he
K¡,
's
and
he
K¡'s,
which
a e
ma ix- alued,
a e
con ac ions
on
Cd,
o
all
1
E
R
d
.
Theo em
1
.
Fo
all
e
>
0
he e
exisis
C
E
>
0
such
¡ha¡,
o
all
F
E
LC,
(R
d
),
[Ja~112
d /
1IQj{Aj}Ki,
. .
.
K
._l, {An}xn,iF
2-J1
2
<
CE(1+2~+ =)nJIFI12
.
The
p oo
which
we
shall
ou line ollows as
usual
om
Ca leson-measu es
es ima es
.
The
imp o emen
o e
[6]
and
[7J
comes
om
Lemma
3
and
om
he
ac
ha
o
he
Ca leson-measu es
ha en e in o he p oo ,
i is
e y
easy
o
ob ain
agood
con ol o
h,,
.
I
(Fí) >o
is
a
amily
o
ec o - alued
o
scala - alued
L
2
- unc ions
we
shall
de ine
I
I
IF
111
by
IIIF III
=
lJo[100
IIF ll
z-l
1/
2
.
Ske ch
o
P oo
.
Fi s
we
educe
(9)
o
a
simila
es ima e
whe e
K,,
is
eplaced
by
P
.
To do
his
we
domina e
by
IIIQ {Aj}K,,
IIIQ {A1}Ki,
. . .
K
._i, {A
.}P Á'
.FIII
. .
.
K

_l, {An}K
., Flll
+

IIIQ {A~
}h1,
...
Kn_1,
{An}(~nI
+
hn)FIII
Fo
all
n
>
0,
X
n
,
Y
n
and i
n
deno e
he
sup
o
hese
3
quad a ic
exp essions
when
IIFII2
<
1
.
Since
{An}(AnI
-}-
K
n
)
is
a
con ac ion,
Y,,
<
X,,_1
.
Hence
(10)

X
n
1<
Y

+
X
._1
So
i
Y,,
g ows
a
mos
like
(1
+
2
-}-
e)n o
all
s
>
0,
hen so does
X
n
.
30
6

J .-L
.
JOURNÉ
To
es ima e
IIIQ {A1}K1,
.
. .
K,l, {A
n
}P
FIII
he
classical
hing
o
do
is
o
decompose
he exp ession
in
he
no m
as a
sum
o
he
ype
{Q {A1}K1,
. .
.
{An_1}Kn_1, An}P F+
e o
e m,
whe e
he
e o
is
o he
o m
L
F
wi h
L
l
=
0
.
The
main e m
is
es ima ed
using
a
Ca leson-measu e
es ima e
which
i sel
ollows
om an
L
2
-es ima e
a
he
o de
n
-
1
.
As we
shall
now
see,
his
p ocedu e
can
be
imp o ed
o yield
be e
cons an s
.
Le
(w ) >o
be
some
adial
smoo h
app oxima ion
o
he
iden i y,
such
ha
w
1
is
non-nega i e
and
suppo ed
in
he
uni
ball
.
Now
le
=
w
*w
.
Obse e
ha con olu ion
wi h
w
o
is
a
con ac ion
on
L
2
o
L°°
.
Now
we
domina e
and
IIIQ {
A
l}Ki,
. . .
Kn_1, {An}P FIII
by
he
sum
o
he
h ee
ollowing
e ms
(
11
)

IIIQ {Al}Ki,
.
.
.
Kn_1, {An}(P1
-
2n )FIII,
(12)
111
Q {A1}Kl,
.
. .
K
n
_
1
,
{A
n
} 2n F-
{Q {A1}K1,
.
. .
K
n_
1
,
¡
A
n
} 2n F1

,
(13)

111
{Q {A1}K,,,
...
Kn_1, An} 2n Fl
The,
i s
e m
is
less
han
III(P
-
2

)FIII,
which
is
domina ed
by
n
II1(Pi
-
!)FIII
+ ,
III( 2k
-
zk-l ) lll
k=1
and
llena
:
by
C(1
+
n)IIFII2
.
The
second
e co
!s
also
o
he
YI>c
IIIL FIII
wi h
L
l
=
0
o
all
>
0,
and
can
be
es i na ed
di ec ly
wi hou
using induc ion
on n
.
The
co esponding
es ima e
g ows
slowe
han
exponen ially
.
To
es i na e
(13)
one
has
o
es ima e
e
!
,
and
h
! ,
o
he
Ca leson-measu e
The
cons an
h
,
can
be
es a na ed
wi hou
induc ion
and
g ows
slowe
han
exponen ially
.
To
es i na e
c
!
,
we
choose
a
cabe
Q
o
side-leng h
6
and we wan
o
es ima e
jQ /2^
{A}K1,
/2
n
. .
.
K
n
_
1
,
/2n
An(x)
2
d dx
I
.EQ, í
KATO'S
SQUARE-ROOT
PROBLEM

30
7
We
shall
see
ha
because
o
he
ac o 2n,
his
is
essen ially
domina ed by an
exp ession
o
he
o m
up
o e o
e ms,
wi h
e
n
's
such ha
11°
°
1(
1
+E

)
<
oo
.
O
cou se,
he e
he
no m
o
A
n
has
o
be
aken
in he
Hilbe -Schmid
sense
and
his
is
wha
in oduces
a
ac o
-,íd
-
.
In
conclusion
one
ob ains
an
inequali y o
he
o m
o
lix
-
y¡¡
>
.
X
n-1II
An
11i
2
((1+e,)Q)
,
Y
n
<2X
n
_
1
((1
+
En)
-líd)
+
e o
e ms
.
Combining
his
wi h
(10),
one
ob ains
(14)

X
n
<,
(1
+
2V
d)Xn_1
+
e o
e ms,
which
implies
he
heo em,
modulo
app op ia e
con ol
o he e o
e ms
.
4
.
Technical P elimina ies
An
ing edien
in
mos
subsequen
es ima es
is
he
ollowing
.
Lemma
4
.

The
ke nels
o
he ope a o s
Ki,
sa is y
¡he
inequali y
llhi,
(x
-
y)¡¡
This
lemma
ollows
easily
om
he
asymp o ic
p ope ies
o
he
Fou ie
ans-
o m
o
1/(1
+
2),
which
decays exponen ially
as well as
i s
de i a i es
.
We
omi
he
de ails
..
La e
in
he
p oo s,
we
shall
no
need
he
ull
o ce o
he
exponen ial
ac o
.
A
polynomial
ac o
o
su icien ly
high
deg ee,
depending
on
he
dimension,
would
be
enough
.
Such
decay,
howe e ,
can
only
come
om
su icien
smoo h-
ness o he
symbol
o K¡,
which
is
i
+
( 2111
112
/1
+
2ll1ll2
)
x
Requi ing
ha
he
symbol
o
K
;
be
egula
enough,
and
in
a scale-in a ian
way,
o ces
110
2
u(Ki)
which
is
homogeneous
o
deg ee
2,
o
be
a
polynomial
.
This
jus i ies
ou
assump ion
on
a(Kj)
a
leas
in
la ge
dimension
.
A
consequence
o
Lemma
4
is
he
ollowing
.
Lemma
5
.
Le¡
and
g
be
wo
L
2
- unc ions
such
ha
d
(supp(
),
supp(g))
=
b
>
0
.
Then
i
<
b/n,
(15)

<g,K1
{A1}
...
{An_1}Kn
>

Cn11 J12119112e
2n
31
4
Theo em
2
.
The
adius o
con e gence
o
¡he
Ka o
unc ional
is
a
leas
a
-1
,
whe e
a
is
¡he
la ges
posi i e
oo o
¡he
equa ion
(31)
Be o e
ske ching
he
p oo
o
his
heo em,
le
us
indica e
ha
app oxima e
nume ical
alues
o
a-1
in
dimensions
2,
3, 4,
and
5
a e
espec i ely
.474, .416,
.376,
and
.347
.
Ske ch o
p oo
.
.
Fo
simplici y
we
shall
p oceed
as
i
he
ke nels o
(I
-
P )R
and
Q
we e
suppo ed
in
{Ilx
-
y¡¡
<
}
.
This
o
cou se
is
no
ue,
bu , as
he
p oo
o
Lemma
7
shows,
his
is
ue o
all
p ac ical
pu poses
.
The
ac
ha
R
de ines
p o emen
o (10)
:
(32)
To
see
his
we
jus
need
o
obse e ha
o
a
unc ion
F
in
L
z,
JIP112
- -
11(I
-
R)F112
=
JIF112
.
Hence
X
n
<
maxaz+
,
z=
1
AX
n
_1
-}-
MY,,,
which
is
exac ly
(32)
.
By
he
posi i i y
o
R
we
can
w i e
i
as
~,
whe e
S
is
a
con ac ion
.
Since
K
=
I
-
(I
-
P
)R
=
I -
(I
-
P
)
+
S
we
can
ew i e
i
as
I
ollows
ha
1

(33)

Z
n
<
2
(1
(1+C2)(Zn-1
+
YQXn_1),
+
negligible
e o
e ms
.
No e
ha ,
by
(32),
(34)
Combining
(30)
and
(34)
we
ob ain
J
.-L
.
JOURNÉ
CX-21
Xz-1=XId
.
an
o hogonal
p ojec ion
yie1ds
he
ollowing
¡ni-
X
n
<

Yaz
+Xn_
1
.
ZP +2(I-(I-P,)S)
.
n
Xn
<
C
E
Y2
k=1
1

n-1
(35)

Z
n
<
2
Zn-1
+

E
Z~

+
negligible
e o
e ms
.
j=1
1)
Le
us
igno e
he
e o
e ms
in
(35)
.
Le
C
>
0
and
/a
>
1
be
such
ha
o
1.
<
n,
Zj
<
CQj
.
Then
Z
n
<
CQn
i
9
is
such
ha
a
n
i
1
p
n-1
+
~ld-

pn-1
,
2

1_

KATOS
SQUARE-ROOT
PROBLEM

31
5
ha
is,
i
i
>
a
.
This
implies
Theo em
2
modulo
he
handling
o
he
e o
e ms,
which
is
i ial
.
Le
us
men ion
ha
in
wo king
wi h
he
se ies
(1)
ins ead
o (2)
one
does
no
seem
o
be
able o
imp o e
(10) in o (34)
.
Hence
one
ob ains
signi ican ly
wo se
es ima es
o
X
n
o (1)
han
o
(2),
while, as
he
nex
sec ion
will
sugges ,
one
should
expec
a
disc epancy
g owing
slowe
han
exponen ially
.
8
.
A
di ec
p oo
o
Ka o's
conjec u e
in
dimension
1
We
wish
o
p o e
he
ollowing
:
Theo em
3
.
Fo
all
E
>
0
he e
exis s
a
con,s an
C
E
such
ha
o
all
n

1,
and
al
ELe
(R),
1
<
i
<
n,
and
E
LZ(R),
(39)
Fo
all
u
E]0,1],
we
shall
p o e
(36)

QL

({aá}(I-PL))

<
CE(1+e)n

~laj1j
.JI II2
.
The
p oo
we
shall
gi e
clea ly
yields
mul ilinea
es ima es
.
Howe e we
shall
wo k
wi h
one
single
bounded
unc ion
a
o
no m
1,
and we
shall
deno e
{a}
by a
.
This
symbol
will
s and
o
{a})
o
any
powe
j,
o
ins an e
in
(43)
and
(44)
.
P oo
..

The
p oo
o
Theo em
3
elies
on
a
i ial
ex ension
o
one
iden i y
o
[4],
namely
:
(37)

Q aP
=
P {P a}Q
+
{Q a}P
-
Q {Q a}Q
.
Le
S
i
be such
ha
P
S
L
=
P
a
o
some a
>
0
.
Then
Q
S
=
DP
S,
=
DP
«
=ó
Qa
This,
oge he
wi h
(37)
and
a escaling
immedia ely
gi es
:
Lemma
9
.
Fo
all
u
>
0
and
a
>
0,
(38)

Qu aPa
=
u
Pu {Pu a}Qa
+
{Quia}Pa
-
IXQu {Qu a}Qa
"
11
IQu (a(I
-
P »n
aP
I
<
C(1
+
E)n
~I
IJ2
.
This
clea ly
implies
(36)
.
The
eason
o
in oducing
he
pa ame e
u
will
become
appa en
du ing he
p oo
.
Le
p
be
an
in ege possibly
equal
o
l
.
We
w i e
n
=
qp+
=
q+(p
-
1)q+
=
q+s,
wi h
<
p
.
Le
a
>
1
and
j
=
a9
.
Finally,
o
-y
>
1,
we
se
Qé=
P -Py
.
31 6

J
.-L
.
JOURNÉ
We
wish
o
educe
he
s udy
o
111
Q
(a(I
-
P »n
aP
111
o
he
s udy
o
8+1
(40)
I I
Qu a(I
-
P« )a(I
-
Pa, )a
. .
.
a(I
-P
.
q
-
)
(a(I
- Pp
))

aPp

,
whe e
we
shall
be
able
o
ake
ad an age
o
he
ac o

in
Lemma
9
.
To
do
his
we
eplace
each
P
by
he
co esponding
P-
y
,
whe e
-y
is
some
app op ia e
powe
o
a
.
S a ing
his
p ocess
om
he
igh
and
using
Minkowsky's
inequali y
we
domina e
he
le
hand
side o (39)
by
he
sum
o (40)
and
he
ollowing
exp essions
.
(41)
(42)
Qu
(a(I
-
P ))
n
aQQ
Qu
(a(I
-
p,»
n
-
1
a&aPp
¡Qu
(a(I
-
P ))
n-2
aOAa(I
-
Pp )aPp
Qu
(a(I
-
P
))
-1
aoA
(a(I
-
Pp ))
9
aPp
Qu
(a(I
-
P ))
9-2
aQ
9-1
(a(I
-
pR »s+1aPp
II
s+1
Qu aQ
a(I
-
Pa2 )a
. .
a(I
-
P,,,,-l )
(a(I
-
PP )
/
aPp
To
s udy
(40)
we
expand
each (I
-
Py )
and
eg oup
he
esul ing
2n
e ms
aeco ding
o
he
loca ion
o
he
i s
P
y
which
appea s
when
going
om
he
le
.
A
new
applica ion
o
Minkowski's
inequali y
hen
shows
ha (40)
is
less
han
he
sum
o he
ollowing
exp essions
:
(43)
(44)
IQu aP« a(I
-
P,,2,)a
. .
a(I
-
P
a
,~,,)
(a(I
s+i
Qu aP
.,,
,
(a(I
-
Pp )
/
aPp
~
Qu aPp
(a(I
-
Pp
)l
9
aPp
111
111
Q« aPp aPp
Q- aPp ]
11
KATO'S
SQUARE-ROOT
PROBLEM

31
7
An
applica ion
o
Lemma
9
pe mi s
o
decompose
each
o
hese
exp essions
in
he
sum
o h ee
e ms,
he
wo
las
o
which
can
be
handled
ia
he
usual
Ca leson-measu e a gumen ,
hanks
o
he
ollowing
lemma
.
Lemma
10
.
Leí
0
<
wo
. .
.
<
w,
Then
he
ke nel

I'
(x,
y) o
P,,,oa(I-P
.,)a(I-P,)
...
(I-P,,,

_1)aP
.

is
domina ed
by
C
(w

/wo)i12
(l+n)3
(_
. )1,+,,,,1,2
.
~ ,
zThe
same
is
ue
i
P,,
o
is
eplaced by
Q,,,
o
.
We
de e
he
p oo
o his
lemma
un il
he
end
o
he
sec ion
.
We
in oduce
some
no a ions
o bes
cons an s
in
quad a ic
es ima es,
o
which
we
shall
ob ain
es ima es
by
induc ion
.
These
cons an s
depend
on n
which,
o he
ime
being,
is
ixed
.
Recall
ha
u E]0,1]
.
The
i s
cons an
DQ
s
is
he
bes
cons an
in
he
inequali y
(
45
)

(
40
)
-
C11
11
2
.
(46)

1I
Qu
(a(I
-
P )
I
k
aP
111
<,
Cli
112
.
Pp »
s+laPp
The
cons an
E',
independen
o
a,
is
he
bes
cons an
in
he
inequali y
We
wan
o de i e
an
inequali y
o
DQ
s
.
As we
al eady
obse ed,
(40)
is
less
han
he
sum
o
(q+s+1)
quad a ic
gauan i ies,
(q-1)
o
{43)
and
(s+2)
o
31 8

J
.-L
.
JOURNÉ
(44)
.
Each
o
hese
quan i ies
can
be
domina ed
by
h ee
o he s
using
Lemma
9
.
I
will
ollow
ha
D9
,
is
domina ed
by
he
sum
o
3(q+s--1)
numbe s
.
Co esponding
o he
i s
e m
in
Lemma
9
we
ha e
he
ollowing
(q+s+1)
numbe s
:
~Dq-1,s,
-
D
q
-2,s,
...
,
aq
1
D
i
,,
-E
s
,
QE
s
-1,
. . . ,

Eo,
~CO
.
a
2

-
He e
c
o
deno es
he
bes
cons an
in he
inequali y
IIIQ , 111
<
CII
. ll2-
Those numbe s
a e
ob ained by
escaling
and
using
he
ac
ha he
P
's
a e
con ac ions
on
L
2
and
L°°
.
The
wo
las
e ms
in
Lemma
9
can
be
eg ouped
in
a
single
one
.
Lemma
10
and
he
usual
Ca leson-measu e
a gumen
gi e
a
global
es ima e
in
C
1 /
2
(1
-}-
n)
4
.
In
doing
his,
one
uses ha
he
measu e
IQ
u
a1
2
dxd /
is
a
Ca leson-measu e
uni o mly
in
u
.
The
p e ious
ema ks
yield he
ollowing
q-1

s
(
47
)

D
,s
<

.Z
Dq-j,s
+
u
co
+
zs

E
m
+
Caq/
2
(1
+
,n)4
.
j-1
a

p

l
"'=o
The
s a egy
is
o
use
(47) o
show
ha
i
(48)

ER,
<
C1(1
+
E)'
o
some
e
>
0
and
uni o mly
o
m
E
N
and
u
E]0,1],
hen
o
some
e'
<
E,
(49)

Du
,s
<
C2
(q,
s)(
1
+
E')q(1
+
e)9
whe e
C2(q,
s)
has he
o m
C(q
+
s
+
1)4
and
is
independen
o u
E]0,1],
This
will
equi e
an
app op ia e
choice
o
a
and p
.
Then
one
shall
show
ha
o
some e"
<
e,
C
>
0,
and
uni o mly
in u
E]0,1],
(50)

E
uu
m
<
C(1
+
e")"`
.
By
i e a ing
his
p ocedu e
one
can
malee
e
as
small
as
we
wan ,
hus
p o ing
he
heo em
.
Le
us
be mo e
p ecise
.
Le
e
>
0
and
choose
p
such
ha
(51)

21/2 +1

<

1
+
e
<
21/2p-2
.
Then
we
cla,im
ha
i
(48) holds,
hen
(49)
holds
o
any
(52)
1/3
e'
>
((1
+E)2 -2/

-1
.
No ice
ha ,
by
(51),
(1+E2

1/3
p
_)

-
1
<e
.
Also
we
claim
ha
in
(50)
we
can
choose
any
e"
such
ha
(53)
(56)
KATO'S
SQUARE-ROOT
PROBLEIII

319
e"
>
(1
+
e)
1-1
/p(1
-I-
e
')1/p
-
1
.
Since
2
1
/
2
p+
1
i
s
he
only
ixed
poin
o
he
ans o ma ion
x
___>
(

2

)1/3
x2p-2
(55)

a(1
+e')
>
2
.
Obse e
ha ,
by
(55),
j
:i~,
l
(a(1
+
E'))
inc easing
one
jus
needs
i
ollows
by
i e a ion
ha
we
can
make
e
as
close
as
21/2p+1
-
1
as
we
wan
.
Then,
by
inc easing
he
alue
o
p
we
can
ge
e
as
close
o 0 as
we
wan
.
Hence,
o
inish
he
p oo
o
he
heo em,
we
jus
need
o
p o e
wo
claims
and
Lemma
10
.
P oo
o
he
i s
claim
:
We
assume
(48)
and
(51),
and
we
wan
o
conclude
(49) o
any
e'
such
ha
(54)

1+e'
>
(
(1+e)
2 p
-2
)
Le
a
=
(1
+
e)
2
p
-2
(1
+
E')
2
.
No ice
ha ,
by
(54),
In
o de
o
deduce
(49)
om
(48),
we
shall
use
an
induc ion
on
q
based
on
(47)
.
Then
i
will
be
su icien
ha
C
2
be
so ha ,
o
all
0
<
k'
<
q,
k'-1
C2
s k
~
-j)
i
C2
(s,k')(1+E)9(1+S')k
,
E

(
k
.

(1+E)'(1+e')k-
j-1
+Ca
lo
/
2
(s
+
k'
+
1)4
+
Cl
s
k,
1
(1
+
e)'
.
<
1
.
Hence,
since
C2(s,
.)
will
be
(57)

C2
(s,
k
'
)( 1
+
e)"
(1
+
e')k'
>
C
3
ce k
'/
2
(s
-1-
k'
+
1)4
-F-
s
á
k
,
1
(1
+
e)9,
whe e
C3
is
some
cons an
which
emains
bounded
i
eand
e'
s ay
away
om
hei
minimum
alues
and Cl
emains
bounded
.
Le
us
se
C2(s,
k')
=
2C3(S+

32
0

J .-L
.
JOURNÉ
k'
+
1)a
and
check
ha
wi h
his
choice (57)
is
sa is ied
.
Equi alen ly
we
need
ha
o
all
0
<
k'
<
q,
(5g)

2(s
+
k'
+
1)
4
(1
+
e),(
1
+
e
l
)
k
'
>
a
k
'/
2
(s
+
k'
+
1)4
+
s
a
,
1
(1
+
e)9
.
In
(58),
he
second
e m
o
he
igh
hand
side
is
ob iously
less
han
hal o
he
le
hand
side
.
To
see
ha
his
is
he
same
o
he
i s
e m,
we
i s
obse e
han
when
k'
=
q
i
ollows
om
a
9/2
=
(1+e)(p-1)Q(l+e')a
'
(1+e),(1+e')9
.
To
deduce
i
o
smalle
alues
o
k',
jus
obse e ha
a
l/2
>
(1
+
e')
.
Hence,
(58)
is
p o ed,
om
which
ollow
(57)
and
(56)
.
Hence
(49) ollows
om
(48)
by
induc ion,
using
(56)
and
(47)
.
P oo
o
he
second
claim
:
Wi h
ou
choice
o
C
2
,
(49)
eads
as ollows
:
(59)

D9
,
,
<
C(q
+
s
+
1)
4
(1
+
e)
, (
1
+
e')9
.
We
wan
o
deduce
(50)
om
(48)
and
(59)
,
assuming
ha
e"
sa is ies
(53)
.
The
decomposi ion
o
he
le
hand
side
o (39)
in
(41)+(42)+(40)
implies
ha
Eñ
is
domina ed
by
he
sum
o
(s
+
2)
+
(q
-
1)
+
1
e ms
.
To
handle
he
(s
+
2)
i s
ones,
we
decompose
Qá

P
as
-
2Pu Qu
L
.
By
(48),
his
gi es a
con ibu ion
o c
o
logp
o
he
i s
one
and
Cl
log
/l(1
+
e»,

0
j
<, s,
o
he
(s
+
1)
ollowing
.
By
(59)
one has
con ibu ions
o he
o m
C(q
-
j
+
s
+
1)4
j
log
a
(1
+
e)9
(1
+
e')
9-
j,
1
<
j
<
q
-
1
o
he
e ms
cons i u ing
(42)
and
inally
C2
(q,
s)
(1
+
e)'
(1
+
e')Q
o (40)
.
Summing
hese
es ima es,
and
hen
using
(53)
and
he
ac
ha
(q
+
1)(p
-
1)
>,
s
we
ob ain
Eñ
'<
C(q
+
s
+
1
)
6 (
1
+
e)'(1
+
e')9
<
C
E
,,(1
+
e1L)n
.
This
p o es he second
claim
.
P oo
o
Lemma
10
:
An
ob ious
es ima e
is
l(x,
Y),<¡¡
PWO (x
-
-)¡1211PW

L(
.
-
Y)¡¡
2
=
C1-1(WOWn)-1/2
.
This
is
su icien
when
Ix
-
y¡
<
C(1
+
n)
3/2
Wn
since
in
his
case,
)_1/2

Wn
L/2
)3
Wn
_1
(WOW
n

<
C
~-)

(1
+12
WO

(Wn )
2
+
(x
-y)2
I
IX
-
M
> C(1
+
7L)
,
/
1
W
n
,
we
unca e
P,,,
ol
(x
-
.)
and
P,,, al
(
.
-
y)
on
balls
espec i ely
cen e ed
a
x
a ld
y
and
o
diame e
Ix
-
y
J/4
.
The
a -away
pa s
ha e
L2-
no ins
o
he
a de
o
I
x
-
yj
-s
/2
wo
and IX
-
yI
-s
/2
w
n
espec i ely
.
Hence, hey
will
gi e
con ibu ions
o
a
nos
CW
n Ix
-
y¡-s/2(wo )-1/2
.
I
e nains o con ol
he
con ibu ion
o
he
local
ha s
.
He e
we
use
ha
hei
suppo s
Na e
a
dis ance
Ix-y¡/4
.
As
we
inen ioned
in
he
ema le
ollowing
he
p cxl
o
Lemma
5,
we
can
apply
i
in
ou
si ua ion
and
his
gi es a
con ibu ion
C~nC~z_bI/~W^L(W~Wn)'~/2 -~,
This
concludes he
p oo
o
Lemma
10
and
o
Theo e n
3
.
KATO'S
SQUARE-ROOT
PROBLEM

32
1
Re e ences
1
.

T
.
KATO,
"Pe u ba ion
Theo y
o
Linea
Ope a o s,"
Sp inge -Ve lag,
1966
.
2
.

A
.
MCINTOSH,
On
he
compa abili y
o
A
1
/
2
and
A*
1
/
2
,
P oc
.A
.M
.S
.
32
(1972),430-434
.
3
.

A
.
P
.
CALDERÓN,
Commu a o s
o
Singula
In eg al
Ope a o s,
P oc
.
Na
.
Acad
.
Se¡
.
U
.S .A
.
53
(1965),
1092-1099
.
4
.

R
.
C0IFMAN,
A
.
MCINTOSH
AND
Y
.
MEYER,
L'in ég ale
de
Cauchy
dé ini
un
opé a eu
bo né
su
L
Z
pou
les
cou bes
lipschi ziennes,
Ann
.
o
Ma h
.
116
(1982),
361-388
.
5
.

A
.P
.
CALDERÓN,
Cauchy
in eg als
on
Lipschi z-cu es
and
ela ed
ope -
a o s,
P oc
.
Na
.
Acad
.
Se¡
.
U
.S .A
.
74
(1977),
1324-1327
.
6
.

E
.
FABES,
D
.
JERISON
AND
C
.E
.
KENIG,
Mul ilinea
Li lewood-Paley
es ima es
wi h
applica ions
o
pa ial
di e en ial
equa ions,
P oc
.
Na
.
Acad
.
Se¡
.,
U
.S .A
.
79
(1982),
5746-5750
.
7
.

R
.
C0IFMAN,
D
.G
.
DENG
AND
Y
.
MEYER,
Domaine
de
la
acine ca ée
de
ce ains
opé a eu s
di é en iels
acc é i s,
Ann
.
Ins
.
Fou ie
33
(1983),
123-134
.
Depa men
o
Ma hema ics
P ince on
Uni e si y
Fine
Hall,
Box
708
P ince on
NJ
08544
U
.S
.A
.