-
Univers
-
Ebooks
-
Livres audio
-
Presse
-
Podcasts
-
BD
-
Documents
-
- Cours
- Révisions
- Ressources pédagogiques
- Sciences de l’éducation
- Manuels scolaires
- Langues
- Travaux de classe
- Annales de BEP
- Etudes supérieures
- Maternelle et primaire
- Fiches de lecture
- Orientation scolaire
- Méthodologie
- Corrigés de devoir
- Annales d’examens et concours
- Annales du bac
- Annales du brevet
- Rapports de stage
La lecture à portée de main
Dispersive Estimate for the Wave Equation
19
pages
English
Documents
Écrit par
Fabrice Planchon
Publié par
pefav
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
Découvre YouScribe en t'inscrivant gratuitement
Découvre YouScribe en t'inscrivant gratuitement
19
pages
English
Ebook
Obtenez un accès à la bibliothèque pour le consulter en ligne En savoir plus
- references cited there
- wave equation
- dispersive estimate
- physical space via legendre
- multiplication operator
- inverse-square potential
- then obtain
- following dispersive
- bessel function
Disp
ersiv
e
a
um
constan
Estimate
;
for
D'Alem
the
are
W
=
a
n
v
b
e
in
Equation
inequalities
with
tro
the
u
In
f
v
(1.1)
erse-Square
n
P
equation
oten
scaling
tial
erturbation
F
tial.
abrice
in
Planc
a
hon
v
John
follo
G.
8
Stalk
x
er
;
A.
@
Shadi
g
T
@
ah
b
vildar-Zadeh
is
12/27/01
in
Abstract
p
W
degree
e
a
pro
This
v
cannot
e
of
that
v
spherically
imp
symmetric
ersiv
solutions
the
of
e
the
general
Cauc
1
h
Consider
y
linear
prob-
e
lem
:
for
a
the
2
linear
u
w
)
a
x
v
u
e
)
equation
x
with
n
the
t
in
the
v
in
erse-square
and
p
real
oten
er.
tial
in
sat-
from
isfy
tial
a
homogeneous
mo
and
died
same
disp
as
ersiv
ertian
e
that
inequalit
ds
y
e
that
the
b
p
ounds
particular,
the
of
L
a
1
t.
norm
disp
of
e
the
for
solution
w
in
v
terms
equation
of
in
certain
una
Beso
oidable.
v
In
norms
duction
of
the
the
wing
data,
w
with
v
a
equation
factor
<
that
n
deca
+
ys
j
in
j
t
u
for
0
p
(0
ositiv
x
e
=
p
(
oten
)
tials.
t
When
(0
the
x
p
=
oten
(
tial
)
is
where
negativ
=
e
2
w
e
is
sho
D'Alem
w
ertian
that
R
the
+1
deca
a
y
a
is
n
split
b
b
The
et
terest
w
this
een
comes
t
the
and
oten
r
term
,
eing
and
of
the
-2
estimate
therefore
blo
the
ws
w
up
y
at
the
r
b
=
term.
0.
implies
W
p
e
metho
also
alone
pro
b
vide
used
a
studying
coun
eect
terexample
this
sho
oten
wing
In
that
the
the
alue
use
the
of
t
Beso
is
v
ortan
norms
1In
[8
]
in
1,
the
w
e
e
x
sho
1
w
y
ed
for
that
w
in
an
the
t
radial
(
case,
_
i.e.
oten
when
the
the
ab
data
ho
{
e
and
tials
th
ws
us
L
the
(
solution
)
{
all
are
This
radially
k
symmetric,
e
the
(1.2)
solution
holds
to
tial,
(1.1)
it
satises
also
generalized
.
spacetime
estimate
Stric
,
hartz
erse-square
estimates
is
as
the
long
died
as
n
a
(
>
n
(
=
n
4
2)
)
2
g
=
4.
w
In
to
this
one
pap
norms
er
disp
w
the
e
compactly
con
has
tin
e
ue
that
the
of
study
of
of
scale-in
the
,
radial
pro
case
estimate
b
t)
y
implies
pro
ortan
ving
that
a
a
disp
in
ersiv
the
e
negativ
estimate,
blo
i.e.
en
a
negativ
deca
in
y-in-time
e,
estimate
v
for
estimate
the
at
L
o
1
k
norm
)
of
(
the
solution.
n
W
e
n
note
k
that
R
for
k
non-radial
1
data,
L
the
n
same
false
estimate
,
can
sho
b
app
e
pap
pro
ws
v
replace
en
k
in
_
the
in
same
e
w
F
a
a
y
with
for
orted
eac
Beals
h
wn
comp
holds.
onen
ould
t
sho
in
similar
the
the
spherical
Because
harmonics
ecial
expansion
p
of
equation
the
arian
solution,
y
but
th
w
enough
e
e
do
v
not
h
kno
v
w
t
if
b
it
deca
is
is
p
to
ossible
ev
to
h
re-sum
false
these
0:
to
sho
obtain
that,
the
solution
estimate
a
in
with
the
in
general
oten
case.
up
It
0
is
the
w
oth.
ell-kno
p
wn
app
that
applications
in
o
the
e
free
to
case
a
a
ersiv
=
h
0
blo
the
when
follo
is
wing
dd,
disp
estimate
ersiv
u
e
t
in-
k
equalit
1
y
R
holds
)
j
C
u
(
(
1)
x;
2
t
k
)
)
j
+1
f
C
L
t
(
n
n
1
+
2
(
n
k
4
f
k
k
1
_
R
W
)
n
is
+1
for
2
n
;
as
1
e
+
w
k
an
g
endix
k
this
_
er.
W
sho
n
that
1
cannot
2
the
;
W
1
;p
with
;
H
(1.2)
;p
where
the
_
ersiv
W
inequalit
k
.
;p
or
(
w
R
v
n
equation
)
a
is
supp
the
p
homogeneous
tial,
Sob
[1]
olev
sho
space
that
on
still
R
W
n
w
whic
lik
h
to
is
w
dened
a
to
estimate
b
for
e
solution
the
(1.1).
closure
of
of
sp
C
form
1
the
0
oten
(
this
R
is
n
v
)
t
with
an
resp
a
ect
and
to
us
the
is
norm
to
k
v
u
the
k
o
_
e
W
(whic
k
is
;p
scale-in
(
arian
R
for
n
=
)
i.e.,
=
oundedness
X
the
j
y
It
j
imp
=
t
k
note
k
w
@
er,
suc
u
an
k
is
L
if
p
<
(
It
R
as
n
wn
)
[8]
:
generically
When
the
n
of
is
w
ev
v
en,
equation
the
a
_
e
W
v
k
p
;
tial
1
ws
norms
at
on
=
the
ev
righ
if
t
data
need
smo
to
Since
b
e
e
oten
replaced
do
with
ear
the
nonlinear
Beso
of
v
ab
norms
v
_
w
B
w
k
t
;
pro
1
e
1
mo
,
disp
see
e
[10]
whic
for
allo
details.
for
Ev
wup
en
2the
origin.
Let
the
dr
tation
.
:=
consider
n
terior
2
on
2
R
and
wth
for
a
sp
>
Legendre
f
2
follo
let
R
=
:=
1
p
or
2
2
the
+
space
a:
The
of
follo
using
wing
R
is
x
the
.
main
distinction
result
the
of
n
this
f
pap
r
er:
<
THEOREM
exp
1.1.
L
L
of
et
separate
n
exterior
the
3
e
,
of
el
and
of
.
dene
use
d
ph
as
tegration
ab
erator.
ove,
radially
and
function
assume
=
F
6
e
=
er
n
practice
1
een
2
in
;
L
n
L
3
space
2
h
if
(
n
1
is
n
even,
if
and
f
exp
6
tial
=
en
n
.
3
pro
2
w
;
w
n
t
4
,
2
and
if
r
n
one.
is
case
o
the
dd.
the
L
in
et
the
u
and
b
decomp
e
op
the
ar
r
the
adial
w
solution
represen
of
solution
(1.1)
-
Univers
-
Ebooks
-
Livres audio
-
Presse
-
Podcasts
-
BD
-
Documents
-
Jeunesse
-
Littérature
-
Ressources professionnelles
-
Santé et bien-être
-
Savoirs
-
Education
-
Loisirs et hobbies
-
Art, musique et cinéma
-
Actualité et débat de société
-
Jeunesse
-
Littérature
-
Ressources professionnelles
-
Santé et bien-être
-
Savoirs
-
Education
-
Loisirs et hobbies
-
Art, musique et cinéma
-
Actualité et débat de société
-
Actualités
-
Lifestyle
-
Presse jeunesse
-
Presse professionnelle
-
Pratique
-
Presse sportive
-
Presse internationale
-
Culture & Médias
-
Action et Aventures
-
Science-fiction et Fantasy
-
Société
-
Jeunesse
-
Littérature
-
Ressources professionnelles
-
Santé et bien-être
-
Savoirs
-
Education
-
Loisirs et hobbies
-
Art, musique et cinéma
-
Actualité et débat de société
- Cours
- Révisions
- Ressources pédagogiques
- Sciences de l’éducation
- Manuels scolaires
- Langues
- Travaux de classe
- Annales de BEP
- Etudes supérieures
- Maternelle et primaire
- Fiches de lecture
- Orientation scolaire
- Méthodologie
- Corrigés de devoir
- Annales d’examens et concours
- Annales du bac
- Annales du brevet
- Rapports de stage
