Finiteness of pi1 and geometri inequalities in almost positive Ri i urvature

Description

Finiteness of pi1 and geometri inequalities in almost positive Ri i urvature Erwann AUBRY ? Abstra t We show that omplete n-manifolds whose part of Ri i urva- ture less than a positive number is small in Lp norm (for p > n/2) have bounded diameter and nite fundamental group. On the on- trary, omplete metri s with small Ln/2-norm of the same part of the Ri i urvature are dense in the set of metri s of any ompa t dierentiable manifold. Keywords: Ri i urvature, omparison theorems, fundamental group 1 Introdu tion A lassi al problem in Riemannian geometry is to nd topolog- i al, geometri al or analyti al ne essary onditions for the exis- ten e on a manifold of a Riemannian metri satisfying a given set of urvature bounds. For instan e, S. Myers showed that a omplete n-manifold with Ric≥k(n?1) (where k>0) is ompa t (the diameter is bounded by π√ k ) and has nite π1, whereas, on the ontrary, J. Lohkamp showed in [11? that on every n-manifold with n≥3 there exists a metri with negative Ri i urvature. This paper is devoted to the study of the Riemannian manifolds satis- fying only an Lp-pin hing on the negative lower part of their Ri i urvature tensors.

• eitherm has small


• stronger ur


• π1


• nite


• volume


• urvature bounded


• r0 ? π


• riemannian metri

Subjects

• Documents
• Education
• Volume

Informations

• Published by: pefav
• Reads: 9
• Language: English

Exrait

π1
∗
n
pL p>n/2
n/2L
n Ric≥k(n−1) k>0
π√ π1k
n
n≥3
pL
Ric(x) = inf Ric (X,X)/g(X,X)x
X∈T Mx
x ∈ M
f (x)=max(−f(x),0) f−
∗ ◦
dense
in
the
)
an
the
set
v
of
with
metrics
w
of
ture
an
UBR
y
theorem,

er
dieren
and
tiable
tensors.
manifold.
um
Keyw

ords
w
:
ositiv

is

FNRS
ature,


to
theorems,
fying
fundamen
the
tal
their
group
b
1
small
In
p
tro
of

lo
A


e
problem
ature
in
inequalities
Riemannian
.
geometry
Bishop's
is
supp
to
a
nd
e
top
This
olog-
dev
ical,
study

manifolds
or
an
analytical
hing

e

part
for

the
diameter
exis-
v
tence
norm
on
er
a
e
manifold
than
of

a
whose
Riemannian
note
metric
est
satisfying
of
a
at
giv
,
en

set
ann
of


almost
ature
,
b
function
ounds.
rst
F
follo
or
yp

P
S.
b
My
Gran
ers
metric
sho
negativ
w

ed
ature.
that
pap
a
is

oted
are
the
-manifold
of
with
Riemannian
ature
satis-

only

nite
the

of
on
part
negativ
same
lo
the
er
of
of
-norm

(where
ature
small
Let
with
ounded
metrics
e
n
ha
20-101469.
(for

in
(the
is
diameter
b
is
n
b
ositiv
ounded
a
b
less
y
a-


,
part
trary
-manifolds
)
de-
and
the
has
w
nite
eigen

alue
the
the
,
tensor
whereas,
that
on
sho
the
and

W
trary
Y
,
A
J.
Erw
Lohk

amp
e
sho
p
w
in
ed

in
for
[11
arbitrary
℄
and
that
Our
on
result
ev
the
ery
wing
On
t
-manifold
e
with
of
group.
artially
tal
orted
fundamen
y
there
Swiss
exists
t
Finiteness
)
is
1nn(M ,g) p>
2Z
p
ρ = Ric−(n−1) gp −
M
9 1
n 10 10Volg≤VolS (1+ρ )(1+C(p,n)ρ ).p p
Ric ≥
n−1 π1
p > n/2
V > 0 ǫ > 0
V
nρ ≤ǫ
2
n(M ,g) p>n/2
ρp 1≤ M π1VolM C(p,n)
1ρp 10Diam(M,g)≤π× 1+C(p,n) .
VolM
π1
∞L
ρ /VolMp
k> 0 R pkρ ρ = Ric−k(n−1)p p M −
nC(p,n) C(p,n,k) VolS
nVolS π√n π n
2 kk
k≤ 0
1S
π1
ρp
there
exists
a
metric
this
applies
large
prop

F
tually
t
dense
olume
amongs
pinc
the
trary
length
w
spaces
pro
for
(see
the
a
Gromo
y
v-Hausdor
v

it
familly
a
of
v
Riemannian
a
manifolds
w
of
w
v
it
olume
form
spaces.
Gromo
and
it
with
a
length
w
the
to
on
the

y
v-Hausdor

(see
p
prop
v
osition

9.2).
its
Our
same

Theorem
result
readily
is
for
the
since
follo

wing
Theorems
m
replace
y
y
ers's
of
t
osition
yp
amongs
e
that
theorem.
es
Theorem
generalize
1.2

L
with
et
b
Gromo
and
the
is,
for
w,
family
y

tal
a
implied
b
in
e
on
a
ature,

the
omplete
t
manifold
to
and
that
form
has
h
an
whic
small
nite
ersal
with
satises
.
hing.
If
an
manifolds
an

argumen
for
ws
only

applies
b
so
e
and
is
assumes
ev-
theorem
er,
Bishop
optimal.
the
and
of
w
ersion

v
Bishop
,
the
then
ma

and
is
While

y
omp
length
act
v-Hausdor
with
y
nite
.
The
space
and
vious
and
es
volume
of
nite
3)
of

is
form
then
nite
nite
yp
is
manifold
If
the
.
whic
and
there
manifold
up
omplete
no
that
no
v
ert
nite
of
with
fundamen
manifolds

olic
er
erb
b
yp
purely
h
tegral

hing
a
the


(for
and
ones
is

main
e
oin
b
of
et

L
pro
1.1
e
some
if
A
manifold
few
the

y
ts
and
are
,
in
then
order:
univ
1)


er
h
the
a
pinc
diameter
2)
b
or
ound
y
w
y
as
,
obtained
renormalization
in
t
[14
sho
℄
that
under
e
stronger
replace

that
v
y
ature
sho
assumptions
w
but
optimal
the
1.1
niteness
for
of
ery
the
the
and
ev
manifold
Ho
w
in
as
1.2
a
1.1

vided
(see
e
also
not
[18]).
is
As
theorem,

b
in

[14
implies
℄
joran
if
our
Riemannian
,
On
also
-niteness
the
b
9.1).
obtained
b
w
prop
only
spaces
that
the
,
dense
readily
and
to
b
the
is
univ
spaces
ersal
The

-Euclidean
v
mak
er
ob
(ev
that
en
do
if
not
it
to
is
set

.
that
The
is
pro
not
of
the
small
same
h
for
a
in
v
tegral
h
pinc
er-
hings.
olic
That
sho
is
that
the

reason
the
wh
small
b

ounds
e
on
if
the
e

assume
olume)
ature
tranfer
is

(or
Theorem
2ρp
VolM
1ρ Sp
ρp
p = 1 n = 2 π1
p =n/2 n≥ 3
n(M ,g) n
n≥ 3
(g ) M gm
ρ (g )n/2 m
→ 0
Volgm
n−1
∞L
∞L
π n1
n−1
π1
the

olume
v
v
olume
the
nite
[8
and
b
,
h
ology
on
top
natural
the
has
theorem
on
is
of
still
een
v
presen
alid
the
(
the
innite
ertinen
with
w
-
o
niteness
an
ob
use
viously
the
follo
e
ws
Sob
from
of
the
℄
Gauss-Bonnet
sition
theorem),
wn
but
Gallot
in
requiring

℄
manifold
e
a
the
get
y
e
b
w
and
and
the
small
e
a
w
with
ture,
no
ariation
generalization

of
in
the
rems

T
results
b
v
ha
alid
to
under
theorem
p
fails
oin
℄
t
(see
wise
w
lo
ts
w
tegral
er
one
b
ound
ound
y
on
er
the
the

h

manifolds
ature
but


b
b
e
ound
exp
whereas
ected,

as
y
sho
b
ws
ounded
the
e
follo

wing
Since
theorem,
not
Theorem
an
1.3
b
L

et
e


pro
ula

of
By
h
.
to
nite
pro
b
ers
e
t
any
and

hniques,
omp
a
act
on
R

iemannian
e
and
elop
-manifold
generaliza-
(
My
olume
the
v
form
nite
℄
).
℄
Ther
til
e
pap
exists
prop
a
only
se
b

olev
e
ere
of
an

trol
omplete

R
y
iemannian
a
metrics
the
,
one
ology
Y
top
lo
innite
ound
on
olume
with
balls
that
h

otheses
onver

ges
almost
to

manifold
not
in
in
the
for
Gr
w
omov-Hausdor
on

ould
e

and
w

set
that
with
a
b
get
w
to
small
℄
but
[8

of
b
A.2
ab
example
v
the
or
dify

mo
ature
tly
ounded).
sligh
w
also
do

assume
Since
t.
1941
lo
sev
er
eral
ound
generalizations

of
a-
My
w
ers's

theorem
the
app
v
eared,
form
under
for
roughly
length
three
geo
dieren
whic
t
is
kinds

of
ol
h
the
yp
of
othesis:
My
a)
theo-
some
of
in
yp
tegrals
a)
of
b).
the
ec

whic

need
ature
priori
along
ounds
minimizing
some
geo
olev

ts,
are
v

b
trolled
dev
([1],
ed
℄
get
℄
tions
[10
the
℄
ers
[12
when
℄

b)
ariation
the
ula

(see

[14
ature
[7
is
[16
almost
Un
b
this
ounded
t
b
er
elo
our
w
o-
b
8.1),
y
t
e
o
W
ounds
nite).
Sob
but

not
w
al-
kno
lo
under
w
in
ed

to
of
tak

e
ature:
v
b
alues
S.
under
requiring
a
b
giv
on
en
diameter
negativ
℄
e
b
n
D.
um
ang
b
a
er
w
([7],
b
[19
on
℄
v
℄
of
[18
small
℄
[20


the
extra
In
yp
is
are
.
(and
w
for
er
with

nonnegativ


ature
ature,
b
are
ound
p
of
t

our
b)
text:
is

replaced
lo
b
er
y
ound
b
v
ounds
w
on
b
other
the
Riemannian
nalit
in
of
v
e
arian
the
ts
of
(for
-manifolds
example

the
ature
v
ounded
olume
elo
b
b
4)
as
b
as
elo
nite
w
not
or
ounded
the
of
diameter
ounded
lo
3n(M ,g)
T
T x
B(x,R )⊃T ⊃B(x,R ) R ≥R >πT 0 T 0
Z 1h i 1 p πp 1002ǫ =R Ric−(n−1) ≤B(p,n) 1−T −VolT R0T
1n 20Diam(M ,g)≤π 1+C(p,n)ǫ M ⊂T
n
2R −π n0
M
T R >π0R p1 Ric−(n−1) 0 R π0VolT T −

VolB x,π /VolB(x,R )0
p1 L Ric−(n−1) 0
−
B(y,r)⊂B(x,R ) VolB(y,r)/VolB(x,R )0 0
r
B(x,R )0
x
M
M
π1
n(M ,g)
1 1˜λ λ λ1 1 1
n1 1 (M ,g)
1ρpn 1 n pλ (M ,g) =λ (M ,g)≥n× 1−C(p,n) ,1 1
VolM 1ρp1 n p˜λ (M ,g)≥ 2(n−1)× 1−C(p,n) .1 VolM
pro
v
e
lemma
elop

eigenvalue
1.4,
of
w
et
e
a
sho
is
w
ed
that
also
e
esp
w
in
er,

v
whic

estimates
ersal
olev
univ
v
the
1.4.
of
and
ature
ortan

of

d
the
tly
trol
of

our
to
based
able
ounds
e
y
b
for
go
o
es
e
to
univ
to
of
when
the
the
ature
and
the
othesis
Prop
-norm
t)
of
it
yp
rst
h
on
extra
and
unnatural
the
these
ws
oid
b
v
the
a
of
o
v
o
Lemma
b
tend
tends
tration
to
no
e
v
and
infered
that
v
for
elop
an
pro
y
1.4.
a
w
manifold
-niteness,
(not
a
ne
in

Riemannian
essarily
of

a
omplete)
h
which
of

our
ontains
w
a
generalizations
subset
hnero
satisfying
v
the
1.5
quotien
denote
t
,
Then
and
the
to
fol

lowing
o

L
onditions:
the
1.
of
is

is
-forms
star-shap
to
e
.
d

at
a
a
pro
p
v
oint
wing
(se
lo
e

denition
).
2).
(and
is
b
uniformly
tends
b
e
ounded

b
measure
elo
(and
w
mak
b
of
y
required
a
olume
p
then
ositiv
b
e
the

olume
function
dev
of
ed
2.
the
.
of
These
lemma
t
T
w
sho
o
the
opp

osite
w
b

eha
star-shap
viours
domain
of
the
the
ersal


tration
er
of
radius
the
ball
measure
tain
in
whic
for
satises
some
assumptions
.
lemma
3.
Under
pinc

the
assumptions,
that
e
prev
get
en
of
t

the

manifold
Bishop-Gromo
from
theorems.
ha
osition
ving
L
p
us
oin
by
ts
that
to
t
o
imp
far
is
a
,
w
r
also
e
T
the
from
nonzer
and
eigenvalue
.
the
T

o
functions,
pro
rst
v
on
e
-forms
theorem
on
1.1,

w

e
of

get
a
order
go
that
o
Then:
d
sho

small
osition
sucien
of
y
on

in

to
with
star-shap
e
ed
follo
subsets
diameter
and
-sphere
sho
an
w
sum
that
The
either
Remark.
Sob

has
ersion
small
of
.
diameter
rst
ound,
L
to
lemma
dev
1.4
when
apply
to
to
a
at
hnique
least
on
one

of
estimates
these
1.4
subsets.
h
The
e
b
use
ound
b
on
on
the
v
hing
olume
or
a
et
y
T
4np= n≥ 3
2
ρ10 p 1η = ≤ x ∈ MVolM C(p,n)
0≤r≤R
1 1
2p−nVol S(x,R) 2p−1 Vol S(x,r) 2p−1n−1 n−1 2 2p−1− ≤η (R−r) ,
L (R) L (r)1−η 1−η
VolB(x,r) A (r)1
≥ (1−η) ,
VolB(x,R) A (R)1
L (t) A (t)k k
n 1t (S , g)
k

2
Vol S(x,R)≤ 1+η L (R)n−1 1−η

VolB(x,R)≤ 1+η A (R).1
Ric ≥ (n−1)
Vol S(x,.)n−1
L1
r
π
n
ρp ≤C(p,n) p>n/2VolM
n≥ 3 p =n/2

Ric− (n−1)
−
cosr(n−1) r
sinr
π (M) 81
p =n/2

the
to
dev
whic
of
trast
of

w
In
the
also:
in
,
a
our
b
assumptions
pro
do
5
not
b
yield
and
an
on
upp
tly
er
the
b
of
ound
y
on
lemmas
the
ecomes
quotien
This
t
o
e
pro
henc
giv
,
niteness
in
prop
adius
a
r
of
of
w
l)
3,
al
lemma
b
of
esp.
tal
(r
it
e
o
spher
tegral
for
(see
all
less
p
.
ossible
the
v
of
alues
in
of
b

b
b
geo
ecause
diameter
the
1.1
diameter
6.
of
pro
our
sho
manifolds
the

,
b

e
surv
greater
prop
than
v
o
ed
.
need
This
In
results
e
are
lemma
similar
impro
to

the
℄
results
is
obtained
our
ge
theorem
a
vides
.
from
and
e
[14

℄
further
under
for
stronger
and

the
ature
of
assumptions.
adapting
Theorem
with
1.2
result
and

prop

osition
this
1.6
is
imply
4
that
get
the
from
set
e
of
w
of
olume
-
balls.
manifolds
of
satisfying
v
volume
of
the
1.2
for
in
stands
7
)
to
esp.
of
(r
that
e
w
wher

have:
of
we
1.5.
,
e
adii
9
,
the
for
brief
a
ey
r
the
l
erties
al
the
and
olume
l
star-shap
,
domains
is
e

subsequen
for
.
the

Gromo
w
v-Hausdor
establish


W
(see
e
3.1),
sho
ving
w
similar
in
lemma
the
[14
last
and

h
that
fundamen
this
for
prop
pro
ert
of
y
1.2:
is
pro
false
a
in
ound
the
ab

v
al
b
for
a
then,
in
and
of
If
e
1.6
details),
osition
℄
Prop
4.1
obtain:
5.1
,
on
ev
part
en
than
for
ofs
the
the
p
By
oin
when
ted
false
Gromo
b
v-Hausdor
of

mean
This
ature

geo
is
spheres
organised
radius
as
.
follo
lemma
ws.
used
F

or
and
our
to
pro
some
of
ounds
of
ab
theorem
v
1.2,
and
w
elo
e
on
need
v
to
of
impro

v
The
e
ofs
the
the
estimates
and
on
olume
v
ounds
olume
theorems
established
and
in
are
[14
en
℄

(see

also
is
[8
oted
℄
the
[20
of
℄
the
and
of
[15
w
℄
e
for
,
other

similar
to
estimates
pro
and
of

osition
hnics).
Finally

w
2
discuss
is

dev
the
oted
last
to
in
In
[15
℄
5x∈M Ux
x U \{0 }x x
∗ n−1 n−1(r,v)∈R ×S S+ x x
x vg
∗ω = exp v =θ(r,v)drdv dv drgx
n−1 ∗S R θx +
∗ n−1R ×S \U 0x+
(r,v) U \{0} h(r,v)x
∂exp (rv)x ∂r
x r h Ux
∂θ(t,v) =h(t,v)θ(t,v)∂r
′s (r)kk h = (n−1)k s (r)k
n(S ,g ) nkk
k
p
sinh( |k|r)
ps (r) = k< 0, s (r) =r k = 0,k k
|k|
 √
sin( kr) π √ √r≤
k ks (r) = k> 0.k
π √0 r> ,
k
π√U U ∩B(0, ) k > 0 ψ = (h −h)x x k k −k
n−1u S I =]0,r(u)[ux
t (t,u)∈U t → ψ (t,u)x k
π√I ∩]0, [u k
(
1) lim ψ (t,u) = 0,k
+t→0
2ψ∂ψ 2ψ hk k k k2) + + ≤ρ ,k∂r n−1 n−1

ψ ρ = Ric−k(n−1)k k −
g(∇△f,∇f) =
1 2 2△|∇f| +|Ddf| +Ric(∇f,∇f) x dx2
wright
for
the
y

w
when
satise
of
the
es
er
asur

me
e
W
e
and
to
).
satises:
at
his
ectors
the
.
(wher
al
of
anonic
spheres
Henceforth,


e
the
map
e
of
ar
all
R
℄
iemannian
B.
and
the
me
V
asur
mean
e
ature
if
ential
wher
right
,
wher
e
spher
and
exterior
set
ell
when
the
if
the
we
the
extend
denote
v
e
to
p
normal
set
of
pap
when
and
set
329).
the

is
extracted)
(where
ois
On
vitation
ordinates
during

2003-2004.
(
ula
resp
and
.
dened
on
This
olar
r
p
this
their
quality
with
by
by
left
of
of
.
at
F

or
of
usual,
Let
as
Pr
if
the
where,
wn
)
form
all
.
),
b
we
at
set
an
ature
domain

onen

,
with
and

iden


simply
oin
-dimensional,
this
w
e
(
,
.
real
F
or
ollo
it
wing
F
[15
p.
℄
[17
w
(
e
is
will
and
use:
Colb
Lemma
for
2.1
in
L
to
et
UNINE
ould
the
b
ear
e
2
an
olume
element
form
of
satises
lik
and
e
on
to
is
.
function
thank
.
and
adius
S.
of
Gallot
e
space
dier
del
ine
mo
is
the
d
on
the
function
and
onding
derivatives

and
the
and
the
e
interval
d
of
enter
values
e
for
the

)
that
Notation.
for
normal
his
).
friendly
oof.
supp
Apply
ort
w
during
kno
m
Bo
y
hner
.
ula
The
(for
function
W
PhD
denote
thesis
y
[2
inje
e
℄
(from

whic
me
h

a
of
part
exp
is
by

we
ontinuous,
tial
right
at
and
w
left
in
dier
tify
entiable
the
everywher
to
e
function
in
ts
I
6|∇d | = 1 Dd(d ) R∇dx x x
⊥x ∇d hx
2 ∂h h ∂ ∂
+ +Ric , ≤ 0
∂r n−1 ∂r ∂r
n(S ,g )kk
h∼ (n−1)/r+o(1)
q.e.d.
x∈M T ⊂ M T
x y ∈ T
x y T
T = exp T T U ⊂x x xx
T Mx
T M x A (r)T
B(x,r)∩T L (r)T
n−1(n−1) (rS )∩U ∩Tx xxR
θ(r,.)dv L (r) = n−1 1l θ(r,v)dv A (r) =T T TxSxRr
L (t)dt θ A LT0
n(S ,g ) θ A Lk k k kk
LT
AT
T (M,g)
LT
AT
LT
α∈]0,1]
Z Zα r α−1L (r) α L (s) θT T
f(r) = − 1l ψT kxn−1 n−1L (r) VolS L (s) θk k k0 Sx
∗ π√R k≤ 0 ]0, [ k> 0+ k
(i) θ(r,v)1lTx
n−1r 1l (r,v) exp r → θ(r,v)1lT Tx x x
]0,r(v)[ [r(v),+∞[
R
1l θ T MT xx
L [0,+∞[T
(i)
b
y
e
in
assume
e
y
b
uous
,
stands
will
Let
manifold
of
and
an
del
that
mo
e
resp
).
ectiv
the
ely
of
.
,
The
ed
regularit
subset
y
,
prop
uous
erties
ounded
of
y
the
on
functions
(if
and
T
ergence
,
and
ed


esgue
denote
used
y
subsequen
,
tly
shap
are
ositiv
summarized
for
in
v
the
where
follo
righ
wing
w
lemma:
W
Lemma
tained
2.2

L
the
et
ery
and
alen
a
the
star-shap
1).
e
of
d
oof.
subset
pro
of
a
,
that
the
.
to
Denition.
onding
the

and
.
v
(i)
e
functions
let
the

is
at
a
star-shap
right
a

Giv
ontinuous,
is
left
on
lower
al
semi-c
there
ontinuous
star-shap
function,
on
(ii)
exists
,
minimizing
Finally
so
is

a
left

semi-con
.

.
infer
e
to
Hessian
is
denoted
ev
the
.
derivative
This
is
oundedness
zero
on
.
subset
(iii)
,
Given
or
on
easily
and
for
equal
get
to
V
the
olume

,
fondamen
Pr
,

the
o
function
v
tal
y
form
w
of
note
the
same
and
In
geo
star-shap

domains:
sphere

of
is

pro
ter
of
on
.
,
olume
w
the
e
W
infer
sa
that
that
satises
b
the
the
follo
of
wing
is

ed
inequation,
hence
This
star-
inequation
subset
b
,
ecomes
en
an
.
equation
at
that
p
Note
e
.
an
on
terv
the
if
mo
all
del
of
spaces
ed
,
,
whic
anishes
h
ane
sure
is
mea-
a
the
,
for
geo
easily
and
giv
is
es
t
inequalit
tin
y
and
2)
lo
of
er
lemma
tin
2.1.
on
Since
.
(see
e
[17
also
℄
from
for

of
in
olume
b
is
on
de
ery

of
e
that
asing
.
either
yields
on
b
v
of
-dimensional
ma
details),
ev
(if

w
of
e
w
also
tly
)
ontinuous,
right
W
dier
infer
Equiv
entiable
function
from
of
Leb
and
dominated
Since
v
on
theorem
the
7(ii) (i)
AT
L VolM\exp (U ) = 0T xx
(m) 1T ⊂ U m ≥ 1 T = (1− )T ⊂ Tx x x x xm
1T 0 (1− )x m
(m)(m)T M T = exp (T )x x0 x
A = lim A (m) L = lim L (m)T TT T
m→∞ m→∞
(m)(iii) T
f : [a,b]→R
f(x+h)−f(x)x∈[a,b[ limsup ≤ 0+h→0 h
x∈]a,b] liminf −f(x+h)≥f(x)h→0
R
θL A r → 1l ψ (r,v)dvn−1 (m)T T kS T θx x k
I =]0,+∞[ k≤kR Rrπ θ√0 I =]0, [ k > 0 r → n−1 1l (m)ψk k0 S θk x T kx
I fk
(m)r n−1r > 0 S = {v∈S / rv∈T }x(m) xT
r˜L(r+t) (r+t).S θ(r+t,.)dv(m)T
(m) ˜T x L(r+t) ≥ L (m)(r+t)x T
t = 0
(m) (m) ˜ ˜L (r+t)−L (r) L(r+t)−L(r)T Tlim ≤ lim
+ +t→0 t t→0 t
R
∂θ˜L(r+t) = θ(r+t,v)dv = hθrS ∂r
(m)T
˜L hθ ψ θk
rS T(m)T
1(m)T t∈[0, r[
m−1
r(r+t).S T M(m) xT
(n−1)
U \{0 } Sxx x
m rr > r+t S (m)m−1 T
1∂θh = U \{0 }x xθ∂r
(m)
(r+t).S (r)T
Z Z
˜ ˜L(r+t)−L(r)
lim = h1l (m)θdv≤ (ψ +h )1l (m)θdvk kT T
+ n−1 n−1t→0 t S Sx x
of
olume
v
the
ounded
assume
and
y
y
b
v
denote

e
monotone
W
uniformly
.
deni-
let
wing
all
w
Hence
.
).
e
e
the
v
b
pro
let
w
and
no
lemma
e
equalit
W
use
2.3.
e
lemma
ecause
of
remains
(b)
[17
y
b
inequalit
er-
the
set
satises
and
function
oth
the
e
Since
ev
so
or
;
w
on
w
tiable
note
dieren
pro
t
b
righ
tary
uous,
in
tin
is

elongs
is
.
and
to
),

if
uous
on
it
(resp.
e
if
and
on
elo
uous
v
tin
theorem,
semi-con

er
By
w
;
lo
in
left

and
is
uous,
the
tin
image

and
t
ounded
righ
set
is
in
function
.
,
ma
w
,
e
.
obtain,
of
b
b
y
w
dieren
is
tiating
e
this
the
in
the
tegral
lemma:
expression
elemen
of
follo
the
the
,
will
(Note

that
b
and
to
for
W
and
for
As
sho
.
b
,
the
are
is
in
tin
tegrable
on
on
only
the
Hence,
set
and
l
(see
al
℄
for
b
(b)
b
,
w
,
y

ha
h
w

gence
b
v
e
on
false
the
for
.
l
and
and
But,
this
function
is

wh
ter
y
of
w
homothet
e
smo
in
on
tro
y

of
the
the
sets
,
al
so
for
b
(a)
on
onditions
ery
):
b
for
tegers
an
all
y
F

that
two
y
the
e
satises
),
it
since
if
measure
only
Since
and
,
if
tion
asing
y
e
that,
,
e
the
2.2,
for
of
the
of
o
star-shap
T
ha
of

of


e
at
.
ed
denition
w
y
Prop
ws
,
(with
F
if
no
follo
y
de
is
y
function
ert
A
lemma.
2.3
atou
Lemma
(a):
the
F
or
8Z
L (r+t)−L (r)(m) (m)T Tlim + ≤h (r)L (m)(r)+ 1l (m)ψ θ.k kt→0 T T
n−1t Sx
α = 1 (a) Lk
h Lk k
L (r+t) L (r)(m) (m)T T−L (r+t) L (r)k klimsup =
+ tt→0
L (r+t)−L (r)(m) (m)T Tlimsup
+ tL (r)kt→0
h i1 1 1
+ lim L (r +t) −(m)T
+t→0 t L (r+t) L (r)k k
h i1 L (m)(r+t)−L (m)(r)T T
= limsup −h (r)L (m)(r)k Tn−1VolS θ (r) t+k t→0
R
1 θB = n−1 1l (m)ψ dv ǫ > 0n−1 kVolS θS Sx kT
(m) (m)
L (r+t) L (r)T Tt > 0 t∈]0,t [ ≤ +ǫ ǫ L (r+t) L (r)k k
t(B+ǫ)
! ! !α α α−1(m) (m) (m)
L (r) L (r) L (r)T T T+t(B+ǫ) − ≤α η(B+ǫ)
L (r) L (r) L (r)k k k
α−1
(m)
F(r+t)−F(r) L (r)Tlimsup + ≤ α(B+ǫ)t→0 t L (r)k
ǫ> 0 (b) α∈]0,1]
ǫ 0
q.e.d.
ψk
π√k> 0 r→
k
π√k∈R p>n/2 r > 0 r ≤
2 k
k> 0
Zp−1 r
n−12p−1 ppψ (r,v)θ(r,v)≤ (2p−1) ρ (t,v)θ(t,v)dt.k k2p−n 0
wing
lemma
impro
e
easily
e
v
.
es
that
lemma
and
2.2
.
in
o
[15
w
℄
e
and
and
theorem
;
2.1
if
in
b
[14
on
℄
get:
W
e
e
exists
pro
or
vide
inequalit
a
and
p
ev
oin
ws
t
It
wise
,
b

ound
e
on
v
.
bining
to
t
whic
w
h,

in
ws,

ha
tend
Comparison
letting
ature
y
deriv
admits
.
a
Let
sharp
get
p
,
olynomial
w
blo

w-up
for
when
that
b
follo
y
assume
an
get:
for
w
y
y
;
vit
these
y
b
W
oth
have:
impro
er,
v
Moreo
emen
mean
ts
the
are
last

w
for
inequalities,
our
e
pro
The
of
of
of
follo
follo
noting
The
v
all
e
that
lemma

theorem
1.2
,
(see
for
the
h
3
pro
of
there
of
,
lemma
has
4.1).
all
Lemma
F
3.1
ativ
L
:
et
ery
Com
9π π√ √k> 0 <r< ,
2 k k
√
4p−n−1 2p−1sin ( kr)ψ (r,v)θ(r,v)k
Zp−1 rn−1 pp≤ (2p−1) ρ (t,v)θ(t,v)dtk2p−n 0
n−1v∈Sx
θ 1l θ s ≥ 0[0,s [ vv
p n/2
n = 2 ψ Lk 1
ρk
1φ C U \{0}x
0
2p−1
r → φ(r,v)ψ (r,v)θ(r,v)k
Iv

∂ 2p−n2p−1 2p−2 2p
(φψ θ)≤ (2p−1)ρ φψ θ− φψ θkk k k∂r n−1
4p−n−1 1∂φ 2p−1
+ h − φψ θk kn−1 φ∂r −
Rr 2p∂θ =hθ≤h θ+ψ θ X = φψ θdtk k∂r 0 k
Z r 1/p 1 2p−n2p−1 p 1−p0≤φψ θ(r)≤ (2p−1) φρ θdt X − Xk k n−10Zh r i2p 1/2p4p−n−1 1∂φ 11−
2p+ h − φθdt X (∗)k
n−1 φ∂r −0
2p−1
limφ(t,v)ψ (t,v)θ(t,v) = 0kt→0
11−
pX
1
2pX
sZ Z1 r r 1/2p2p (n−1)(2p−1)2p p
φψ θdt ≤ φρ θdtk k2p−n0 0
Z 1/2pr 2pn−1 2p−1+(2p−n) ∂φ/∂r
+ h − φθdt .k
2p−n n−1 φ −0
φ(r,v) = 1
hk
Z Zpr r
(2p−1)(n−1)2p p
ψ θdt≤ ρ θdt.k k2p−n0 0
tak
that
olynomial
W
y
and
p
any

e
a
b
obtain
y
e
Indeed
w
we
,
e
y
two
b
).
out
e
Dividing
b
.
everywher
satises:
o
used
ositivit
e
infer:
w
l
where
and
e
at
ativ
div
deriv

its
w
and
pro
,
rst
on
lemma
tiable
taking
dieren
by
t

righ
the
and
e
uous
the
tin
of

if
is

function
.
the
for
2.1,
tegrating,
lemma
over
By
v
.
then
of
erge
d
ounds
o
The
orho
Remark.
b
where
neigh
e
the
e
in
v
ounded
the
b
inequalit
,
of
on
3.1
function
y
e,
used
nonnegativ
(for
a
e
e
e
b
.
Let
then,

ab
oof.
v
Pr
inequalit
).
and
of
p
-norm
y
the
r
y
yield:
b
even
of
,
trol
ve

normal
a
Setting
yields
al
then
get:
h
w

in

e
es
if
a
e
non-negativ
alue
to
hold
tends
These
when
the
in
ine

and
qualities
w
Mor
10