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DISCRETE
AND
CONTINUOUS
W
[20
whic
W
ebsite:
p
h
and
ttp://AIMsciences.org
heat
D
op
YNAMICAL
deriv
SYSTEMS
of
V
p
olume
the
,
x
Num
therein)
b
of
er
estimate
0
y
,
time
Xxxx
.
2001
y-in-time
pp.
um
000
erator.
{
manifolds
000
the
L
studied
p
tum
ESTIMA
un
TES
describ
F
linear
OR
es
THE
for
W
t
A
oin
VE
the
EQUA
norm
TION
a
WITH
for
THE
w
INVERSE-SQUARE
the
POTENTIAL
L
F
ation.
abrice
p
Planchon
in
Lab
e
oratoire
also
d'Analyse
odinger
Num


v
erique,
of
URA
therein),
CNRS
[10
189
.
Univ
prop
ersit
ell-kno

detail
e
the
Pierre
e
et
are
Marie
First
Curie,
es
175
norm
rue
the
Chev
the
aleret,
Secondly
75252
disp
P
a
aris,
of
F
of
rance
n
John
of
G.
deca
St
o
alker

Departmen
terp
t
can
of
L
Mathematics
t
Princeton
terms
Univ
of
ersit
Mathematics
y
ds
,
equation,
Princeton
estimates,
NJ
ell
08544
study
A.
a
Shadi
on
T
W
ahvild
that
ar-Zadeh
Sc
Departmen
ws
t
op
of
a
Mathematics,
2
Rutgers,
b
The
the
State
bustion
Univ
and
ersit
in
y
hanics
of
and
New
ectiv
Jersey
particular,
110
sp
F
of
relingh
are
uysen
These
Road,
in
Piscata
the
w
section.
a
of
y
a
NJ
with
08854
tial,
Abstract.
t
W
basic
e
the
pro
h
v
b
e
L
that
the
Stric
es
hartz-t
at
yp
terms
e
quan
L
time
p
the
estimates
wise,
hold
e
for
giv
solutions
ound
of
1
the
solution
lin-
in
ear
L
w
an
a
b
v
ativ
e
data,
equation
t
with
in
the
t
in
estimates
v
homogeneous
erse
=
square
y
p
b
oten
them
tial,
a
under
for
the
norm
additional
at
assumption
1
that
1
the
the
Cauc
0
h
certain
y
er
data
ct
are
Key
spherically
phr
symmetric.
v
The
v
estimates
tial,
are
gation
then
w
applied
as
to
the
pro
of
v
w
e
v
global
equation
w
conic
ell-p
[4].
osedness
e
in
note
the
the
critical
and
norm
hr
for
o
a
for
nonlinear
elliptic
w
erator
a
+
v
j
e
j
equation.
ha
1.
e
In
een
tro
in
duction.
theory
Consider
com
the
(see
follo
]
wing
references
linear
and
w
quan
a
mec
v
(see
e
]
equation
references
8
resp
<
ely
:
In

the
n
usual
u
ectral
+
erties
a
this
j
erator
x
w
j
wn.
2
are
u
ed
=
more
h
at
(
end
x;
this
t
In
)
case
u
the
(0
w
;
v
x
equation
)
no
=
oten
f
there
(
three
x
yp
)
of
@
estimates:
t
is
u
energy
(0
whic
;
giv
x
a
)
ound
=
the
g
2
(
of
x
rst
)
ativ
(1.1)
of
where
solution

time
n
in
=
of
@
same
2
tit
t
at

zero.
n
,
is
p
the
t
D'Alem
or
b
ersiv
ertian
estimate,
in
h
R
es
n
b
+1
for
and
L
a
norm
is
the
a
at
real
t
n
terms
um
the
b
1
er.
of
The
appropriate
in
um
terest
er
in
deriv
this
es
equation
the
comes
with
from
constan
the
that
p
ys
oten
t
tial
These
term
w
b
classical
eing
are
homogeneous
the
of
case
degree
u
-2
0,
and
b
therefore
in
scaling
olating
the
et
same
een
w
one
a
obtain
y
deca
as
estimate
the
the
D'Alem
p
b
of
ertian
solution
term.
time
Suc
for
h
<
a
<
p
in
oten
of
tial
dual
arises
p
in
norm
the
a
problem
n
of
b
the
1991
stabilit
Subje
y
Classic
of
35L05,35L15.
certain
wor
singular
and
stationary
ases.
solutions
a
of
e
nonlinear
in
w
erse-square
a
oten
v
space-time
e
conju-
equations,
op
as
12
F
ABRICE
w
either
wise
PLANCHON,
g
JOHN
to
G.
to
ST
the
ALKER,
w
A.
is
SHADI
in
T
of
AHVILD
place
AR-ZADEH
+
of
H
deriv
L
ativ
-
es
]).
of
in
the
for
data
co
(since
concerned
this
b
could
L
b
ers,
e
m
fractional
the
the
u
appropriate

norms
k
are
the
Beso
oth
v-space
to
norms,
rapidly
see
in
[15
no
,
h
pp.
W
50{60]).
.
The
hniques
third
1
kind
In
of
a
estimate
x
(or
the
family
explicitly
of

estimates)
for
for
e
the
non-radial
free
y
w
e
a
it
v
e
e
t
equation
n
is
L
a
+
b
;
ound
;
for
(1.1).
the
equation
L
v
p
are
norm
p
in
[22
sp
[2],
ac
of
etime
a
of
from
the
b
solution
conjugation
u
solution
.
ving
In
to
the
of
case
approac
of
example.
zero
t
Cauc
extreme
h
the
y
dieren
data
w
for
al
example,
tion,
this
end
estimate
.
has
sp
the
the
form
can
k
us
u
and
k
x
L
Corollaries
p
In
(
consider
R
the
n
of
+1
ceeding
)
w

A
C
j
k
notation

oth
u
ws
k
o
L
that
p

0
k
(
(
R

n
f
+1
+
)
L
;
t
p
(
=
)
2(
ds
n
h
+1)
estimate
n
or
1
v
;
a
p
oten
0
L
=
q
2(
kno
n
if
+1)
tial
n
e
+3
[6]),
:
ying
(1.2)
],
(Hence
metho
for
ving
this
presence
particular
oten
p
deduce
there
case
is
oten
no
studying
gain
wining
in
erator,
regularit
es
y
one
for
other,

it
1
;p
but
k
the
appropriate
gain
and
in
is
in
tak
tegrabilit
]
y
the
is
doing
the
fail
maxim
er
um
p
p
1
ossible,
oin
i.e.
needs
the
approac
same
pap
as
are

the
1
symmetric
).
e
This
considering
remark
that
able
on
estimate
and
w
this
as
of
rst
form
obtained
oten
b
op
y
free
Stric
e
hartz
allo
[18
obtain
]
estimate
and
generalizations,
subsequen
a
tly
2
repro

v
and
ed,
statemen
rened
t
and
e
generalized
disp
b
y
y
case,
man
as
y
estimates
others
Before
(see
w
[7
clarify
],
mean
[12
op
],

and
j
references
.
therein).
the
No
(
)
w
of
for
case
the
follo
w
from
a
ab
v
v
e
calculation
equation
k
with
(
a
;
p
)
oten
_
tial,
1
the
R
energy
)
estimate
C
is
kr
still
k
v
2
alid
k
and
k
easy
2
to
Z
obtain
0
if
h
the

p
s
oten
k
tial
2
is

nonincreasing
whic
in
is
t
energy
.
for
In
F
particular,
the
for
a
the
e
case
with
of
smo
the
p
in
tial,
v
arious
erse-square
p
p
L
oten
estimates
tial,
also
one
wn
can
hold
obtain
the
an
oten
energy
is
estimate
nonnegativ
b
([1],
y
],
m
or
ultiplying
deca
the
([9],
equation
[21
in
[5
(1.1)
One
b
d
y
pro
@
estimates
t
the
u
of
and
p
in
tial
tegrating
to
on
them
a
the
h
with
yp
p
erplane
tial
t
y
=
the
const,
tert
to
or
get
op
d
whic
dt
tak
p
the
E
of
(
equation
u;
the
t
pro
)
that

maps
1
k
p
in
2
W
k
;p
h
an
(
range

k
;
p
t
This
)
the
k
h
L
en
2
[21
(
for
R
Since
n
tec
)
for
;
this
where
ypically
the
to
quan
v
tit
the
y
cases
E
=
(
;
u;
,
t
p
)
t
is
estimate
the
a
energy
t
of
h.
the
this
solution
er,
u
e
at
only
time
with
t
spheric
,
ly
E
w
(
v
u;
equa-
t
i.e.
)
solutions
:=
(1.1)
1
dep
2
only
Z
j
R
j
n
t
j
In
@
case,
t
ecause
u
the
j
ecial
2
of
+
p
jr
tial,
x
conjugation
u
erator
j
the
2
case
+
b
a
computed
j
,
x
wing
j
to
2
the
j
p
u
(1.2)
j
its
2
with
dx:
+
An
j
application
j
of
in
Hardy's
of
inequalit
(see
y
3.3
sho
3.9
ws
precise
that
ts).
the
subsequen
ab
pap
o
w
v
will
e
the
energy
ersiv
b
inequalit
ounds
in
the
radial
square
as
of
ell
the
generalizations
_
these
H
to
1
cases.
Sob
pro
olev
further
norm
e
of
ust
u
what
pro
e
vided
b
that
the
n
erator

=
2
n
and
a
a
x
>
2
(
W
n
adopt
2)
standard
2
D
4
for
:
space
In
smo
this
functionsL
p
ESTIMA
TES
b
(1.9)
dimensional
F
e
OR
ove.

se
+
transform,
a
A
j
extensions.
x
which
j
+
2
e
3
an
on
egin
a
are
domain
f

of

or
R
ar
n
ne
.
ctrum
Let
j

The
:=
]
n
of
2
e
2
denition
:
one
(1.3)
.
F
8
or
r

;
2
ep
R
e
,
b
W
is
e

denote
two
b
them
y
<
D
these

e
(
j
R
a
n
spheric
)
ar
the
ctr
space
[10
of
e
smo
w
oth
erator
functions
con

pro
on
A
R
on
n

n
b
f
y
0
radial
g
@
suc
=
h
;
that
@
r
g

we

to

then
extends
d,
to
ly
a
ompletely
smo

oth
describ
function
<
on
is
R
extensions.
n
ab
v
dilation
anishing
half
to
multiplicity
innite
e
order
A
at
p
innit
.
y
K
.
+
W
Y
e
wher
ha
Bessel
v
and
e
or
D
e

d,
(
on
R
omp
n
[14
)
much

out
L
ly
2
erator.
(
ho
R
conjugates
n
the
)
The
if
done
and
the
only
a
if
o

e
>
what
1.
0
Let
0
D
1.
b
these
e
A
the
Cauc
dilation
w
op
in
erator
is
(
:
D
t

)
)(
(
x
u
)
)
=
r

u
(
)
c
r
1
1.1.
x
e
),
e
c
op
>
onditions
0
e
and
e
R
de

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