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Publié par | profil-zyak-2012 |
Nombre de lectures | 29 |
Langue | English |
Extrait
Remarks
on
Stric
regularit
the
f
hartz
the
estimates
recast
for
L
n
g
ull
)
forms
R
F
2
abrice
f
Planc
led
hon
?
Lab
@
oratoire
f
d'Analyse
2
Num
and
i
erique,
oposition
URA
(
CNRS
j
189,
k
Univ
of
ersit
ab
_
e
r
Pierre
=
et
1
Marie
one
Curie,
2
4
f
place
where
Jussieu
2
BP
2
187,
L
75
transforms.
252
of
P
Let
aris
and
Cedex
)
Abstract
R
W
(4)
e
k
pro
2
v
view
e
an
some
follo
impro
g
v
f
ed
for
Stric
that
hartz
;
estimates
L
for
@
n
2
ull
while
forms,
ull
some
(3)
of
r
whic
2
h
ij
w
g
ere
1
conjectured
1
recen
space:
tly
1
in
if
[7
1
].
h
The
where
results
the
follo
ma
w
in
from
transforms,
com
([2
bining
;
the
2
usual
Q
Stric
;
hartz
R
estimates
j
with
f
div-curl
.
lemma
ij
tec
g
hniques
1
rather
k
than
g
space-time
:
F
the
ourier
suc
transform.
one
In
to
tro
question:
duction
f
Consider
negativ
the
,
bilinear
g
form
s
(1)
>
Q
turns
ij
has
(
f
f
r
;
2
g
2
)
)
=
f
@
g
i
L
f
;
@
for
j
n
g
form
@
has
i
r
g
;
@
g
j
L
f
=
;
Q
where
(
f
;
and
)
g
H
are
;
functions
H
of
denotes
x
Hardy
2
h
R
H
n
if
.
only
Suc
h
h
L
forms
and
app
i
ear
2
in
1
v
R
arious
denote
instances
Riesz
connected
One
with
y
geometric
(3)
PDEs,
term
either
Riesz
elliptic
Pr
([10]
1
and
])
references
f
therein)
g
or
L
h
,
yp
let
erb
ij
olic
f
([17]
g
and
=
references
i
therein).
R
In
g
the
i
later
R
con
g
text,
Then
they
kQ
are
(
called
;
n
)
ull
H
forms
.
(along
f
with
L
other
k
bilinear
k
forms
2
whic
In
h
of
w
pro
e
of
will
h
in
estimate,
tro
is
duce
naturally
later).
the
These
wing
forms
what
ha
out
v
;
e
with
cancellation
e
prop
y
erties.
namely
F
;
or
2
the
H
simple
,
pro
s
duct
0
H
It
older
out
yields
one
(2)
1Pr
oposition
2
Let
1
b
k
f
teractions
;
k
g
R
2
of
_
o
H
the
s
then
,
to
for
)
0
of
<
1
s
<
er
1
23
2
R
.
i
Then
g
(5)
k
k
k
1
(
jrj
g
2
structure
s
as
Q
the
ij
the
(
pro
f
y
;
n
g
one
)
similar
k
T
L
1
k
.
@
k
1
f
1
k
_
(1
H
ma
s
k
1
g
R
k
;
_
f
H
B
s
oth
:
and
Before
an
pro
In
ceeding
b
with
_
the
e
pro
h
of,
frequency:
w
w
e
imp
remark
of
one
ev
has
one
in
and
fact
generic
an
e
ev
,
en
]
b
adv
etter
write
result,
j
as
g
Q
i
ij
(
f
)
;
1
g
ev
)
R
2
.
_
L
B
2
2
"
s;
<
1
v
1
j
.
f
Through
B
what
+
follo
f
ws
g
the
j
reader
i
is
R
assumed
R
to
2
b
s;
e
Notice
familiar
sp
with
Q
Beso
v
v
are
spaces
as
([29]).
<
In
eac
order
four
to
individually
pro
v
v
2
e
.
(5),
left
w
term,
e
with
p
the
erform
a
a
of
parapro
distributions,
duct
in
decomp
without
osition
deca
([1]),
t
writing
Ho
(6)
there
R
instances
i
lo
f
requiremen
R
ij
j
them.
g
of
=
can
(
in
R
a
i
[5,
f
4,
;
references
R
tak
j
tage
g
w
)
R
+
f
(
k
R
j
j
g
=
;
1
R
R
i
g
f
(
)
f
+
k
(
where
R
p
i
W
f
ceed
;
k
R
k
j
g
L
)
;
f
with
k
(
k
F
2
;
s
G
~
)
Assuming
=
2
X
sum
k
high
S
(10)
k
X
2
F
j
2
k
2
G;
;
(
(
F
i
;
;
G
j
)
)
=
R
X
f
j
R
k
g
k
(
0
j
j
;
1
i
)
k
_
F
2
1
k
:
G
b
where
the
ecial
k
of
is
ij
a
the
lo
alue
calization
s
op
irrelev
erator
t
at
long
frequency
s
j
0.
fact
j
h
these
2
terms
k
elong
from
to
a
Beso
Littlew
space
o
B
o
s;
d-P
1
aley
W
decom-
are
p
with
osition,
third
and
whic
S
deals
k
in
=
at
P
same
l
for
<k
simple
duct
l
t
is
o
a
dening
lo
is
w
general
frequencies
ossible
lo
regularit
calization.
and/or
Since
y
f
the
2
w
_
distributions.
H
w
s
er,
is
are
equiv
umerous
alen
where
t
can
to
w
2
these
k
ts,
s
Q
k
is
of
k
More
f
classes
k
sym
L
ols
2
b
=
treated
"
a
k
w
2
y
l
see
2
3,
,
,
w
9
e
and
ha
therein.
v
o
e
e
(recall
an
s
of
<
structure,
0),
e
2
(9)
k
i
s
k
k
R
S
k
g
f
i
k
k
L
R
2
=
f
@
k
(
2
l
f
2
j
(and
k
similar
)
estimates
j
for
1
g
k
,
R
with
~
g
"
;
k
and
=
~
.
k
e
),
pro
from
to
whic
aluate
h
w
e
f
get
j
(7)
k
k
k
S
1
k
k
2
1
R
k
i
k
f
2
k
g
R
L
j
.
g
k
k
2
L
)
1
k
.
"
2
:
2
s
k
1
s
one
y
k
o
~
er
"
frequencies
k
get
;
and
=
then
k
(8)
1
(
k
R
R
i
f
g
;
_
R
1
j
s;
g
1
)
2and
nally
(11)
(
cs,
:
,
f
2
;
and
g
p
)
L
=
y
@
+1
i
and
u
j
stands
@
q
j
that
equation
i
k
2
v
_
(
B
the
2
e
s;
the
1
1
)
:
efore
If
p
s
=
1
n
2
u;
one
,
ma
k
y
k
sligh
can
tly
as
mo
p
dify
are
the
argumen
>
t
in
and
2
write
ma
a
estimates
p
t<