Publicacions
Ma emá iques,
Vol
35
(1991),
155-168
.
POINTWISE
SMOOTHNESS,
TWO-MICROLOCALIZATION
AND
WAVELET
COEFFICIENTS
S
.
JAFFARD
In
his
pape ,
we
shall
compa e
h ee
no ions
o
poin wise
smoo hness
:
he
usual
de ini ion,
J
.M
.
Bony's
wo-mic olocal
spaces
Cxós,,
and
he
co espond-
ing
de ini ion
on
he
wa ele
coe icien s
.
The
pu pose
is
mainly
o
show
ha
hese
wo-mic olocal
spaces
p o ide
"good
subs i u es"
o he
poin wise
Hdlde
egula i y
condi ion
;
hey
can
be
e y
p ecisely
compa ed
wi h
his
condi ion,
hey
ha e
mo e
unc ional
p ope ies,
and
can
be
cha ac e ized
by
condi ions
on
he
wa ele
coe icien s
.
We
also gi e
applica ions
o
hese
p ope ies
.
In
Pa
2
some
esul s
on
he
mic olocal
spaces
con ained
in
[B2]
will
be
ecalled
.
The-
o ems
3 and 4
a e also
essen ially
con ained
in [B2]
.
The
s a ing
poin
o his
pape
was
a
no e
([J1])
he
au ho
had
w i en
on
a
compa ison
be ween
he
Hdlde
c i e ion
o egula i y a
a
gi en poin
xoand
a
co esponding
p ope y
de ined
on
he
wa ele
coe iicien s
.
Some
easy
p oc s
a e
omi ed
o ab idged
and
can
be
ound
in
[J2]
.
1
.
Poin wise
smoo hnessand
wo-mic olocaliza ion
Le
s
be
a
s ic ly
posi i e
eal
numbe
.
Le
us
ecall
he
usual
de ini ion
o he
Hdlde
c i e ion
a
xo
.
A
unc ion
belongs
o
C"
i
h ee
exis s
a
polynomial
P(x)
o
deg ee
equal
o he
in eg al
pa
o
s
such
ha
(x)
=
P(x)
+
O(Ix
-
xo
js)
.
The
ollowing
p ope ies
a e
classical
.
I
belongs
o
C2
o
,
no hing
is
implied
on
he de i a i es o
.
In
dimension
1,
he
p imi i e
o
belongs
o
C'+'
.
In
dimension
la ge
han
1,
he
ac ional
in eg a ion
o
o de
1
maps
C',
in o
C9+1
.
x
o
The
wo-mic olocaliza ion
o J
.M
.
Bony
consis s
in
eplacing
he
p eceding
no ion
by
ano he
one
which
allows
o
de i a e
and
in eg a e
.
The
classical
pseudo-di
e en ial
ope a o s
will
ope a e
on
hese spaces,
which
will
be
spaces
o
empe a e
dis ibu ions
.
These
spaces
a e
de ined
by
condi ions
on
he
Li lewood-Paley
decomposi-
ion
.
Le
us
ecall
i s
de ini ion
.
Le
0
be
a
unc ion
in
he
Schwa z
class
such
ha
B(1)
=
1
i
111
<
1/2
and
B(1)
=
0
i
111
>
1
.
156

S
.
JAFFARD
Le
hen
Le
Sj
be
he "low-pass
il e ",
which,
a e
a
Fou ie
ans o m,
is
a
mul ipli-
ca ion
by
0(2
-
j
j)
.
De ine
Aj
=
Sj+1
-S
j
.
Thus
The
Fou ie
ans o m
o
Oj(u)
is
suppo ed
by
he
se
The
wo-mic olocal
spaces
can
now
be
de ined
.
De ini ion
1
.
Le
s
and
s'
be
wo
eal
numbe s
;
C"'
is
he
Banach
space
o dis ibu ions
such
ha
(2)

ISo(u)(x)I
<_
C(1
+
1x)-9/
and
0(1)
=
0(1/ 2
)
-
B(1)
.
I=So+Do+Di
+
. . .
2'-1
<_
111
<
2j+1
.
láj(u)(x)j
:5
C2
-
j 9
(1
+
j2jxj)-9'
.
The
space
Cxo
9
is
hen ob ained
h ough
a
simple
ansla ion
.
I s'
=
0,
he
space
hus
ob ained
is
he
global
HSlde
space
C9(Rn)
.
The
e ec
o
s' is
o
accen ua e
ei he
he
ole
played
by x
o
,
when
s'
<
0,
o he
beha io
a
in ini y,
when
s'
>
0
.
A
ew
o he
ema ks
will
gi e
a
be e
unde s anding
o
hese condi ions
.
De ine
uj
=
Oj(u)
and
Uj(x)
=
uj(2
-
jx)
.
Then,
he
Fou ie
ans o m
o
Uj
is
a dis ibu ion ca ied
by
he
se
1/2
<
111
<
2,
and
(3)
implies
ha
(
4
)

l
Uj(u)(x)I
<C2
- j'(1
+
Ix1)"
.
Such
an
es ima e
is
s able
unde
de i a ion
and
ac ional
in eg a ion,
mo e
gene ally
unde
he
ac ion
o he
ope a o s
(I
-
0)
9
/ 2
o
(-0)
.9
/
2
,
s
E R,
because
hese
ope a o s
a e
Fou ie
mul iplie s
which,
once
es ic ed
o
1/2
<_
111
<_
2,
coincide
wi h
a
unc ion
o he
Schwa z
class
.
Coming
back
o he
"space
a iable",
we
ge
(-0)9~2Uj
=
K
9
*
Uj
whe e
K
s
belongs
o
he
Schwa z
class
.
This
con olu ion ope a o
p ese es
he
polynomial
inc ease o decay, as
i
appea s
in
(4)
.
So
ha
he
ollowing
equi alence
holds
(5)

u
E
C','

áu
~

E C'
-1
"
'
o
1
<
j
<
n
.
zo

09x
j

xo

-

-
In
he
ollowing
pa ,
we
shall
in es iga e
he
na u e
o
he
elemen s
o
Cxó",
whe he
hey
a e
(e en ually
smoo h)
unc ions
o dis ibu ions
.
We
claim
ha
he
elemen s
o
Ció
'
a e
(in
gene al) dis ibu ions
.
To
sho en
he p oo ,
suppose
n
=
1,
x
o
=
0
and
0
<
s
<
1
.
De ine
whe e
he
Fou ie
ans o m
o
~
belongs
o
he
Schwa z
class,
anishes ou side
[-1/2,1/2],
and
is
equal
o
1
on
[-1/4,1/4]
.
So
ha ,
o
any
N
>
1,
De ine
hen
Then
Le
j
0
be such
ha
hen
I
(x)
-
(0
)I
<
By
de ini ion,
Hence
POINTWISE
SMOOTHNESS
AND
WAVELET
COEFFICIENTS

157
2
.
The
elemen so
Ció"
8(x)
=
0
*
IxI',
B(x)
=
IX],,
+
Q(IXI-N)
.
(x)
=
1
:
2-i80(2ix)e'2'
z
=
1
:
uj(x)
.
0

0
uj(x)
=
2
-
J
9
u(2
1
x)
and
Iu(x)I
<
C(1
+
IxI)',
so
ha
belongs
o
Có'
-
' .
The
es ic ion
o
o
any
in e al
]b/2,
8[,
S
>
0,
is
a
dis ibu ion,
because,
i
5/2
<
x
<
S,
00
(x)
=
IX
I'
1
:
e
d21x
+
O(1)
.
0
We
claim
ha
he
elemen s
o
Cs,~",
when
s'
<
-s,
a e
"hones
unc ions"
.
In
o de
o
p o e
i ,
we
shall
suppose
ha
0
<
s
<
1,
x0
=
0,
and
ob ain
ha
I
(x)
-
(0)I
<
Cixi'
when
0
<
IxI
<
1
.
2 -
(jo+1)
<
IxI
<
2-jo
I
SOAX)
-
SO (0)I
+

1
:

I
u
j(
x
)
-
uj(0)I
+
E
Iuj(x)I
+
E
Iuj(o)I
.
0<j<jo

j>ju
j>jo
Iuj(x)I
<C2
-
j"(1
+2jix1)"
.
1
V
uj(x)I
<
C2j('
-
s)(1
+
2j1xI)-9,
.
So
ha ,
i
0
<
j
<
j0,
Iuj(x)-uj(0)I
:5
Cl
2
' ( '
-J)
IxI
158

S
.
JAFFARD
and
he
o al
con ibu ion
o
hese
e ms
is
C2j(1
-9
),
which
is
equi alen
o
CIxl9
.
The
se ie
E
Iuj(x)I
is
bounded by
C
E
2-j9(2jjxj)-",
which
is
equi alen
7>70

7>70
o
C2
-1
and
he
same
es ima e
holds
o
E
Iuj(0)I
.
Since,
by
Be ns ein's
7>7o
inequali y,
I
S
o
(x)
-
S
o
(0)
I
<
C
o
Ix1,
he
esul
is
p o ed
.
One
can
also
easily
check
ha ,
i s
>
0
and
s'
+
s
>
0,
hen
CxoC" ,-
"
isinclu
de
dinC
x
o
.
In
he
nex
pa ,
we
shall
examine
he
egula i y o he
elemen s
o
Cxó
s
a
x
o
.
and
Chis
esul
is
op imal
.
3
.
A
compa ison
be ween
Ciá
s
and
Cx
o
Le
s
>
0,
we
saw
ha
he
elemen s
o
Cxó
9,
e en
es ic ed
o
R'
-
{xo}
a e in
gene al
"wild
dis ibu ions"
o
which
(1)
canno
hold
.
Though,
we
shall
p o e
he
ollowing
esul
.
Theo em
1
.
Le s
and
l
be s ic ly
posi i e
numbe s,
and
u
an elemen
o
Cid
'
n
CO(R
n
)
.
The e
exisis
a
polynomial
P
o deg ee
less
han s
such
ha ,
i
IX
-
XOI
<
1,
(6)

Iu(x)-P(x)I
:5
CIx-x0I
9
1og
Ix
2
xol
'
Rema k
ha ,
in
his
heo em,
we
a e
looking
o
egula
poin s
in
an
i egula
backg ound,
which
is
mo e
sub le
han
he
usual
app oach
ha
consis s
in
inding
i egula
poin s
in
a
C°°
o analy ical
backg ound
(de e mina ion
o
he
singula
suppo s)
.
This
heo em
can
be
in e p e ed
as a
aube ian
heo em
.
We
ha e
in o -
ma ion
on
he
beha io
o
a e ages
o
( i s
Li lewood-Paley
decomposi ion)
and
a
aube ian
condi ion
o
minimal
global
egula i y,
which
allow
o
ob ain
a
poin wise
esul
.
P oo
o
Theo em
1
:
De ine
j
o
and
j
l
by
2-)0-1
<
I
x
-
xo
I
<
2 -
io
and
Ji
=
s
jo
.
Q
Le
us
es ic
o
he
case 0
<
s
<
1
and
0
<
0
<
s
.
Then
P(x)
=
u(0),
and
7o
u(x)
-
UMI
5
ISOU(x)
-SOu(o)I
+
E
Iu7(x)
-
Uj(0)I+
0
00
E(IUj(x)I
+
IU
;(o)I)
+
57(I
Uj(x)I
+
IUj(0)I)
=
A+
B
+
C
+
D
.
7o

h
POINTWISE
SMOOTHNESS
AND
WAVELET
COEFFICIENTS

159
To
es ima e
A
is
a
s aigh o wa d
consequence
o
Be ns ein's
inequali y
.
As
conce ns
B,
we
use
u E
C¿8'-"1
so
ha
;
whe e
j j(x)j
_<
c(1
+
jxj)
-
.
The
Fou ie
ans o m
o Nj
anishes
ou side
he
se
1/2
<
111
<
2,
so
ha
hence
ui(x)
=
2-19 ij(21x)
l
pj(x)
-
j(
0)1
_<
cixi
i
IxI
<
1
;
l
uj(x)
-
uj(0)j
<
c2jixi2-j9
.
Adding up
hese
inequali ies,
we
ge ei he
B
<
cixi
9
i
s
<
1,
o ,
i s
=
1,
B
<
cixijo
<
c'ixllog
I2I,

-
x
In
o de
o
es ima e
C,
ema k
ha
luj(x)j
<
c(
2-j9
+
jxj9),
so
ha
C
is
a
mos
O(jxj
9
(jl
-
jo))
=
O(Ixi'
log 2
I
)
.
Because
u
is
in
CQ,
11
uj
jj,,,,<
c2
-
jO,
so
ha
D
is
a
mos
O(2
-
j"Q)
=
00x0,
which
ends
he
p oo
.
The
case
s
>
1 is
le
o he
eade
.
One
easily
checks
ha ,
i
(1)
holds,
hen
u
belongs
o
Czó
s
.

So
ha ,
i
s+s'
>
0,
Cio9
,
C
Ci
o
C
Cx~
s
.
The
necessi y
o
make
he
global
CO
asump ion
and
he op imali y
o
he
loga i hmic
e m
in
he
esul
ha e
been
p o ed by
Y es
Meye
(pe sonal
com-
munica ion),
using
wa ele s
and
will
be
gi en
in
he
nex
sec ion
.
4
.
Wa ele
coe icien s
and
Ció"
spaces
One
o
he
in e es ing
p ope ies
o
he
space
CxO'
is
ha
i
can
be
cha ac-
e ized
by
condi ions
on
he
wa ele
coe icien s
.
The
in ui i e
eason
o
ha
is
because
he
wa ele
coe icien s
o a dis ibu ion a e
gi en
by
a
sampling
( ollowing
Shannon's
ule)
on
he
il e ing
gi en
by
he
Li lewood-Paley
de-
composi ion
.
I is
he e o e
na u al ha
spaces
de ined
by
local
condi ions
on
hei
Li lewood-Paley
decomposi ion
can
be
hus
cha ac e ized
.
We
assume
in
he
ollowing
ha
he
o hono mal
basis o
wa ele s
used
has
enough
egula i y
and
decay
.
We
use he
usual no a ions
1
,
j,k(x)
=
2
nil2
0(2
j
x
-
k),
j
E
Z,
kE
Z
n
.
Then,
he
ollowing
heo em
is
e y easy
o
check
.

16
0

S
.
JAFFARD
Theo em
2
.
A
dis ibu ion
u
belongs o
CxoC
-" ,-
"
i
ando
nl
yi
1
<
u
,Oj,k
>
1
<
C2-(n/2+s)j(1
+
Ik
_
2jxa1)-9'
.
The
o he
wo-mic olocal
spaces
can
also
be
cha ac e ized
by
condi ions
on
he
wa ele
coe icien s
.
Recall
ha
wi h
1
:Icj
12
<
oo
.
Then,
u
belongs
o
H9,9,
i
u
EH9,9,

2
'9(
1
+
2j
ix1)
"
uj JILI<
cj
L
.2's(1
+
2i
i~
-x01)29/¡Cj,k12
<
oo
.
We
now
gi e he
coun e -examples
ha
show
he op imali y
o
Theo em
1
.
Assume
ha
0
is
a
compac ly
suppo ed
wa ele , as
cons uc ed
in [D]
.
One
easily
checks
ha
i is
possible
o
suppose
wl h
Choose
hen
E
;

such
ha
0(0)
7~
0
.
We
i s
p o e
ha
he global
CQ
asump ion
is
needed
in
Theo em
1
.
Le
m
be
a posi i e in ege
and
e,,,
a
eal
numbe
such
ha
2'e

1
is
an
in ege ,
and
E n
-> 0
when
m
-+
oo
.
The
p ecise
alue
o
Em
will
be
gi en
la e
.
Le
a be
such
ha
0
<a<
1
.
The
wa ele
coe icien s
o
he
coun e -example
a e
de ined
by
:
i
2-
<
j
<
2-+1
and
k
=
6,n27,
Cj,k
=
2-j/2Emi
else,
C
j
,
k
=
0
.
Then
de ine
00
x

=
1
:
.
(x)
m=0
n(x)
=
Em

1
:

0(
2j
(
x
-
cm.»
.
2-<j<2-+
1
The
suppo s
o
he
m
a e
disjoin ,
¡ m(x)I
<
C2'em,
and
(0)
=
0
.
Then
is
con inuous
(because
2
m
e"
1
-+
0)
.
Bu
m(em)
=
C2'em,
so
ha
Then
POINTWISE
SMOOTHNESS
AND
WAVELET
COEFFICIENTS

161
limsup
I (x)-
(x°)I
>
limsupC2'E'
-
=+ooVy
>
0
.
xy
-
Hence
is
no
in
Có
o
any
alue
o
y,
al hough
condi ion
(3)
holds
a
0
.
The
ollowing
coun e -example
shows
ha
he
loga i hmic
e m
is
needed
in
(6)
.
Take
he
same
cons uc ion
as
be o e,
bu
wi h
E,,,,
=
2
-
Q
2m
o
a
gi en
/i
>
0
.
I (E-)-
(
0
)I
>C2m
>
C'logjemi
.
m
Hence
he op imali y
o he
loga i hmic
e m
.
I
should
be
no iced ha
o he
condi ions
simila o
condi ion
(7)
can
be
in oduced
in
o de
o
be compa ed
wi h
o he ypes
o
poin wise
egula i y
condi ions
.
Fo
example,
a
compa ison
wi h
poin wise
di e en iabili y
is
gi en
by
he
ollowing
p oposi ion,
he
p oo
o
which
is
simila o he
one
o
Theo em
1
.
P oposi ion
1
.
Leí
be
a
unc ion
di e en iable
a
xo
wi h
wa ele
coe -
cien s
cj,k
.
Leí
A
be he
poin
(k2
-
j,
2
-
j)
i a
he
uppe
hal -plane
.
Then,
he
ollowing
es ima e
holds
whe e
77(A)
<
1
and
l(A)
=
o(1)
whenA
ends
o
(xo,
0)
.
Con e sely,
i
is
in
C
,
(R")
o
a
si icily
posi i e
,l
and
i
he e
exis s
a
posi i e
unc ion
0
de ined
o
posi i e
alues
o 1
such
ha
E
B(j)
<
oo
and
hen
is
di e en iable
a
x
o
.
Icj,kl :5
C~l(A)2-(2+')j(1
+
Ik
-
2'xol)
Icj,kl :5
Cil(A)B(j)2
-cg
+
'>
j(1
+
Ik
-
2'xo1),
5
.
Pseudo-di e en ial
ope a o s
and
wo-mic olocaliza ion
We
shall
now
s udy
he
ac ion
o
gene alized
pseudo-di e en ial
ope a o s
on
he
spaces
Ciós
,
.
The
ope a o s
T
ha
we
shall
conside
will
belong
o
he
algeb as
Op(M7)
(c
.
[DJ],[L]
and
[M2])
de ined
by
condi ions
on
hei
dis ibu ion-ke nel
K(x,
y) as ollows
.
16
2

S
.
JAFFARD
De ine
0
7
o
be
he
class
o
ope a o s
such
ha ,
o
he
main
diagonal,
hei
dis ibu ion-ke nel
K
is
a
unc ion
sa is ying
he
ollowing
es ima es
:
o
any
in ege
a
such
ha
ce
<
y,
I
a
is
he
in ege
such
ha
y
-
1
<a<
y,
C
la°K(x,y)¡-<
Ix-yln+«
.
'Clx
-
x'1
7
-«
l
a«K(x,
y)
-
a
«
K(x,
y)l
<

(x
-yln+y
i
lx-x'I
-< Ix
2
yI
,
lá'K(x,y)-a°K(x,Y,)¡<CI
l
x_yl+7a
i Iy-y'I
<
Ix
2
y¡
,
and
he
ope a o
T
is
such
ha
T(X")
=T
*
(X
a)
=
0
o
a
less
han
o
equal
o
^
y .
The
algeb a
Op(M-
1
)
is
hen
he
union
o
all
he
O^
o
y'
>
y
.
The
usual
pseudo-di e en ial
ope a o s
o
o de
0 a e
he
sum
o
such
an
ope a o
and
o a
egula izing
ope a o
.
Y es
Meye
p o ed
ha
he
ollowing
ca ac e iza ion
holds
(c
.
[M2])
.
P oposi ion 2
.
An
ope a o
T
belongs
o
Op(M
7
)
i
i s
"wa ele
coe -
cien s" de ined
by
c(A,
A')
=<
TOa10a
,
>
saiis y
¡he
ollowing condi ion
:
he e
exisis
y'
>
y,
such
ha
Ie(A,
A')I
<
w
(A, A')
wi h
and
w(A
A')
=
C2-h-j'I(2+7)(

2-i
+
2-j/

)n+-y'
'

2-~
-l-
2
-
j'
+
la
-
A'I
A
=
(k2-i,2-)
)
.
We
shall
now
p o e
he
ollowing
esul
.
Theo em
3
.
I
belongs
o
Cio9,
and
T
belongs
o
Op(M^~)
wi h
hen
T( )
belongs
o
Cx
Ó
"
.
y>
sup(IS
+
S'
I,
.s,
IS'I,
-
n
-
S),
I
we
keep
Theo em
1
in
mind,
his
heo em
can
be
in e p e ed
as ollows
.
The
posi ion
o
he
poin s o
egula i y
o
a
unc ion
is
essen ially
p ese ed
unde
he
ac ion
o
singula
in eg al
ope a o s
such
as
he
Hilbe
ans o m
.
POINTWISE
SMOOTHNESS
AND
WAVELET
COEFFICIENTS

163
P oo
o
Theo em
3
:
Le
B(,1)
=
2-(2'+9)j(1
+
2j
IA
-
xoI)"
.
We
mus
p o e
ha
No ice
ha ,
o
any
s'
,(A,
A')B(, )
<
CB(A')
.
a
(8)

(1+2
j
IA-xol)
-9
'
<(1+2jIA'-xoI)
-
s
,
(1+2jIA-
We
spli
he
sum
Ew(A,A')B(A)
in o
wo
pa s
.
a) I
j
<
j',
hen
and
an
w(A,A')9(A)
<
C

2-(j'-j)(Z+- )
2-(2
+9)x(
1
+2jIA-xoI)
-9a

a
(1+2jIA-
<
C

2-(j'-j)(Z+- )
2-
(z+9)j(
1
+
2,IA'
-
x,1)-s'
b
y
(8)
-

(1+2jIa-~'I)n+7-~9'~
a
<
C2
-
(z+9)j'

2-(j'-~)(7-9)

(
1
+
2jIA'
-
xoI)-9,
-

(1
+
2j
IA
a
We
in oduce
now
he
wo
ollowing
subcases
.
i)I s'<0 hen
(1+2
1.
Ia'-xoI)
-9
'
<-(1+2j'IA'-xoI)"
~w(a,
a')B(a)
<
e(a')
~

2-cj'-jx7
-9)
a

a

(1
+
Ik
-
2'a'Un+,-19'1
<C9(A')i -y>sand~y-Is'j
>0
.
ii)
I s'
>
0
hen
(1
+
2j
IA'
-
xoI)
--"
=
2(j'-j)9'(2(j'-j)
+
2j' IAl
-
xoI)-9'
<
2(j'-D,9'(1
+
2j'
IA'
-
xo
I)
-9
d
~
w(a'
a
)B(A)
<
B(a)
~

2-
(1
+
2j IA
-
a'
I)-+7-191
<CB(,')i y>s+s'andy-Is'I>0
.
b)
I
j
>
j',
hen