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Analysis in weighted spaces : preliminary version
Frank Pacard
To cite this version:
Frank Pacard. Analysis in weighted spaces : preliminary version. 3rd cycle. Téhéran (Iran), 2006,
pp.75. �cel-00392164�
Analysis in weighted spaces : preliminary version
Contents
1 Weighted L2 analysis on a punctured ball 5
1.1 A simple model problem . . . 5
1.2 Analysis in weighted spaces in the punctured unit ball . . . 9
1.3 The spectrum of the Laplacian on the unit sphere . . . 10
1.4 Indicial roots . . . 11
1.5 A crucial a priori estimate . . . 11
2 Weighted L2 analysis on a punctured manifold 21 2.1 The Laplace-Beltrami operator in normal geodesic coordinates . . . 21
2.2 Two global results . . . 22
2.3 The kernel of the operator Aδ . . . 26
2.4 The range of the operator Aδ . . . 27
2.5 Fredholm properties for Aδ . . . 29
2.6 The deficiency space . . . 30
2.6.1 The kernel of Aδ revisited : . . . 37
2.6.2 The deficiency space : . . . 37
3 Weighted C2,α analysis on a punctured manifold 39 3.1 From weighted Lebesgue spaces to weighted H¨older spaces . . . 39
3.2 An example . . . 48
4 Analysis on ALE spaces 51 4.1 Asymptotically Locally Euclidean spaces . . . 51
4.2 An example from conformal geometry . . . 53
5 Mean curvature of hypersurfaces 57 5.1 The mean curvature . . . 57
5.2 Jacobi operator and Jacobi fields . . . 59 3
4 CONTENTS 6 Minimal hypersurfaces with catenoidal ends 63 6.1 The n-catenoid . . . 63 6.2 Unmarked space of minimal hypersurfaces with catenoidal ends . . . 65 6.3 The marked space of minimal hypersurfaces with catenoidal ends . . . 67 7 Analysis on manifolds with cylindrical ends 69 7.1 Manifolds with cylindrical ends . . . 69 7.2 Manifolds with periodic-cylindrical ends . . . 71
Chapter 1
Weighted L
2
analysis on a
punctured ball
1.1
A simple model problem
Let BR (resp. B¯R) denote the open (closed) ball of radius R > 0 in Rn and BR∗ = BR− {0}
(resp. ¯BR∗) denote the corresponding punctured ball. Given ν ∈ R and a function
f : B1∗⊂ Rn −→ R satisfying
k|x|−νf k L∞(B
1)≤ 1 we would like to study the solvability of the equation
( |x|2∆u = f in B∗ 1
u = 0 on ∂B1
(1.1)
A solution of this equation is understood in the sense of distributions, namely u is a solution of (1.1) if u ∈ L1(B
1− ¯BR), for all R ∈ (0, 1) and if
Z B1 u ∆v dx = Z B1 f v |x|−2dx
for all C∞ functions v with compact support in ¯B1∗. We claim that :
6 CHAPTER 1. WEIGHTED L2 ANALYSIS ON A PUNCTURED BALL
Proposition 1.1.1. Assume that n ≥ 3 and ν ∈ (2 − n, 0). Then there exists a constant c = c(n, ν) > 0 and for all f ∈ L∞loc(B1∗) there exists u a solution of (1.1) which satisfies
k|x|−νukL∞(B 1)≤ 1
The proof of this result is a simple consequence of the maximum principle. First, recall the expression of the Euclidean Laplacian in polar coordinates
∆ = ∂r2+n − 1 r ∂r+
1 r2∆Sn−1
Using this expression we get at once
|x|2∆|x|ν= −ν (2 − n − ν) |x|ν
away from the origin. Now, if ν ∈ (2 − n, 0) (this is where we use the fact that n ≥ 3 !), we observe that the constant
cn,ν:= γ (2 − n − ν) > 0
The existence of a solution of (1.1) can then be obtained arguing as follows : Given R ∈ (0, 1/2), we first solve the problem
( |x|2∆u
R = f in B1− ¯BR
uR = 0 on ∂B1∪ ∂BR
(1.2)
Since f ∈ L∞(B1− ¯BR), the existence of a solution uR ∈ W2,p(B1− ¯BR) for any p ∈ (1, ∞)
follows from the following classical result :
Proposition 1.1.2 ([?], Theorem 9.15). Given p ∈ (1, ∞) and Ω a smooth bounded domain of Rn, if g ∈ Lp(Ω) then there exists a unique solution of
( ∆v = g in Ω
v = 0 on ∂Ω
which belongs to W2,p(Ω) ∩ W1,p 0 (Ω).
In our case Ω = B1− ¯BRand
f ∈ L∞(B1− ¯BR) ⊂ Lp(B1− ¯BR),
for all p ∈ (1, ∞), and hence
uR∈ W2,p(B1− ¯BR) ∩ W 1,p
0 (B1− ¯BR).
1.1. A SIMPLE MODEL PROBLEM 7 Proposition 1.1.3 ([?], Theorem 7.26). If α = 1 −n
p, then
W2,p(Ω) ⊂ C1,α( ¯Ω) provided Ω is a smooth bounded domain of Rn.
The maximum principle also implies that |uR(x)| ≤
1 cn,ν
|x|ν (1.3)
for all x ∈ ¯B1− BR. Indeed, observe that the function
w(x) = 1 cn,ν
|x|ν− uR(x)
is positive on ∂B1∪ ∂BR. Moreover
∆w ≤ 0
in ¯B1− BR. Therefore one can apply the maximum principle
Proposition 1.1.4 ([?], Theorem 8.1). Assume that v ∈ W1,2(Ω) satisfies ∆v ≤ 0 in some
smooth bounded domain Ω ⊂ Rn. Then
inf
Ω v ≥ inf∂Ω(min(v, 0))
This result applies to the function w in B1− ¯BR. We conclude that w ≥ 0 and hence
uR≤
1 cn,ν
|x|ν.
Applying the same reasoning to −uR we obtain the desired inequality. Observe that, in the case
where uRis C2, one can simply invoke the classical maximum principle ([?], Theorem 3.1).
Now, we would like to pass to the limit, as R tends to 0. To this aim, we use the following estimates for solutions of (1.2)
Proposition 1.1.5 ([?], Theorem 9.13). Given a smooth bounded domain Ω ⊂ Rn whose bound-ary has two disjoint components T1and T2, Ω0⊂⊂ Ω ∪ T1 and p ∈ (1, ∞). There exists a constant
c = c(n, p, Ω, Ω0) > 0 such that, if g ∈ Lp(Ω) and v ∈ W2,p(Ω), satisfy ( ∆v = g in Ω
v = 0 on T1
then
kvkW2,p(Ω0)≤ c kvkLp(Ω)+ kgkLp(Ω)
8 CHAPTER 1. WEIGHTED L2 ANALYSIS ON A PUNCTURED BALL
Using this result with Ω = B1− ¯BR, T1= ∂B1and Ω0 = B1− ¯B2R, together with the a priori
bound (1.3), we conclude that, for all R ∈ (0, 1/2) there exists a constant c = c(n, ν, R) > 0 such that
kuR0kW2,p(B1− ¯B2R)≤ c
for all R0∈ (0, R). It is now enough to apply the Sobolev Imbedding Theorem Proposition 1.1.6 ([?], Theorem 7.26). The imbedding
W1,p(Ω) −→ C0,α( ¯Ω)
is compact provided 0 < α < 1 −np and Ω is a smooth bounded domain.
It is now easy to use these two results together with a standard diagonal argument to show that there exists a sequence (Ri)itending to 0 such that the sequence of functions uRi converges to some continuous function u on compacts of ¯B1∗. Obviously u will be a solution of (1.1) and, passing to the limit in (1.3), will satisfy
cn,νk|x|−νukL∞(B
1)≤ 1 (1.4)
We have thus obtained a solution of (1.1) satisfying (1.4), provided ν ∈ (2 − n, 0). This completes the proof of Proposition 1.1.1.
Exercise 1.1.1. Given points x1, . . . , xm∈ Rn, weights parameters µ, ν1, . . . , νm∈ R, we define
two positive smooth functions
g : Rn− {x1, . . . , xn} −→ R and h : Rn− {x1, . . . , xn} −→ R
such that :
(i) For each i = 1, . . . , m, g(x) = |x − xi|νi and h(x) = |x − xi|νi−2 in a neighborhood of the
point xi.
(ii) g(x) = |x|µ and h(x) = |x|µ
away from a compact subset of Rn.
Show that, provided n ≥ 3 and µ, ν1, . . . , νm∈ (2 − n, 0), given a function
f : Rn− {x1, . . . , xn} −→ R
satisfying
|f | ≤ h it is possible to find a solution of the equation
γ2∆u = f which satisfies
1.2. ANALYSIS IN WEIGHTED SPACES IN THE PUNCTURED UNIT BALL 9
1.2
Analysis in weighted spaces in the punctured unit ball
Given δ ∈ R we define the spaceL2δ(B1∗) := |x|δ+1L2(B1)
This space is endowed with the norm kukL2 δ(B ∗ 1):= Z B1 |u|2|x|−2δ−2dx 1/2
It is easy to check that Lemma 1.2.1. The space (L2
δ(B ∗
1), k · kL2
δ(B1∗)) is a Banach space. Exercise 1.2.1. Provide a proof of Lemma 1.2.1.
We define the unbounded operator Aδ by
Aδ: L2δ(B∗1) −→ L2δ(B1∗)
u 7−→ |x|2∆u
The domain of this operator is the set of functions u ∈ L2
δ(B1∗) such that Aδu = f ∈ L2δ(B1∗) in
the sense of distributions : This means that u ∈ W2,2(B
1− ¯BR), for all R ∈ (0, 1/2) and
Z B1 u ∆v dx = Z B1 f v |x|−2dx
for all C∞ functions v with compact support in B1∗.
We start with some properties of Aδ which are inherited from the corresponding classical
properties for elliptic operators.
Proposition 1.2.1. Assume that δ ∈ R is fixed. There exists a constant c = c(n, δ) > 0 such that for all u, f ∈ L2
δ(B1∗) satisfying |x|2∆u = f in B1∗ we have
k∇ukL2 δ−1(B ∗ 1/2)+ k∇ 2uk L2 δ−2(B ∗ 1/2)≤ c (kf kL2δ(B ∗ 1)+ kukL2δ(B ∗ 1))
The proof of this result follows from the :
Proposition 1.2.2 ([?], Theorem 9.11). Given a smooth bounded domain Ω ⊂ Rn, Ω0 ⊂⊂ Ω and p ∈ (1, ∞). There exists a constant c = c(n, p, Ω, Ω0) > 0 such that, if g ∈ Lp(Ω) and
v ∈ W2,p(Ω), satisfy
∆v = g in Ω then
10 CHAPTER 1. WEIGHTED L2 ANALYSIS ON A PUNCTURED BALL
The proof of the Proposition 1.2.1 goes as follows : Given R ∈ (0, 1/2) we define the functions v(x) := u(R x) and g(x) := f (R x)
Obviously, we have
|x|2∆v = g
in B2− ¯B1/2. We can then apply the result of Proposition 1.2.2 with Ω = B2 − ¯B1/2 and
Ω0= B3/2− ¯B1, we conclude that kvk2W2,2(B 3/2− ¯B1)≤ c kvk2L2(B 2− ¯B1/2)+ kgk 2 L2(B 2− ¯B1/2)
Performing the change of variables backward, we conclude that R2−nk∇uk2 L2(B3R/2− ¯BR)+ R 4−nk∇2uk2 L2(B3R/2− ¯BR)≤ c R−nkuk2 L2(B2R− ¯B R/2)+ R −nkf k2 L2(B2R− ¯B R/2)
It remains to multiply this inequality by Rn−2−2δ, choose R = 1 3(
2 3)
i
, for i ∈ N and sum the result over i. We obtain
k∇uk2 L2 δ−1(B ∗ 1/2)+ k∇ 2uk2 L2 δ−2(B ∗ 1/2)≤ c kuk2 L2 δ(B ∗ 1)+ kf k 2 L2 δ(B ∗ 1)
This completes the proof of the result.
1.3
The spectrum of the Laplacian on the unit sphere
We recall some well known facts about the spectrum of the Laplacian on the unit sphere. Proposition 1.3.1 ([?], Theorem ??). The eigenvalues of −∆Sn−1 are given byλj = j (n − 2 + j)
where j ∈ N. The corresponding eigenspace will be denoted by Ej and the corresponding
eigen-functions are the restrictions to Sn−1
of the homogeneous harmonic polynomials on Rn.
One easy computation is the following : If P is a homogeneous harmonic polynomial of degree j, then P (x) = |x|jP (x/|x|) and hence
r ∂rP = j P
Using the expression of the Laplacian in polar coordinates, we find that r2∆P = j (n − 2 + j) P + ∆Sn−1P
1.4. INDICIAL ROOTS 11 Since P is assumed to be harmonic, when restricted to the unit sphere this equality leads to
∆Sn−1P = −j (n − 2 + j) P
This at least shows that the restrictions to Sn−1 of the homogeneous harmonic polynomials of
degree j on Rn belong to E j.
Exercise 1.3.1. What is the dimension of the j-th eigenspace Ej ?
1.4
Indicial roots
We setδj:=
n − 2 2 + j
Definition 1.4.1. The indicial roots of ∆ at the origin are the real numbers given by νj±:= 2 − n
2 ± δj for j ∈ N.
The indicial roots are related to the asymptotic behavior of the solutions of the homogeneous problem ∆u = 0 in Rn− {0}. Indeed, a simple computation shows that
∆(|x|ν±j φ) = 0
if φ ∈ Ej.
1.5
A crucial a priori estimate
We now want to prove the key result which explains the importance of the parameters δj in the
study of the operator |x|2∆ when defined between weighted L2-spaces. This is the purpose of
the :
Proposition 1.5.1. Assume that δ 6= ±δj for j ∈ N. Then there exists a constant c = c(n, δ) > 0
such that, for all u, f ∈ L2δ(B∗1) satisfying
|x|2∆u = f in B1∗, we have kukL2 δ(B ∗ 1)≤ c kf kL2 δ(B ∗ 1)+ kukL2(B1− ¯B1/2)
12 CHAPTER 1. WEIGHTED L2 ANALYSIS ON A PUNCTURED BALL
Observe that this result states that we can control the weighted L2-norm of u in terms of the
weighted L2-norm of f and some information about the function u away from the origin.
To prove the result let us perform the eigenfunction decomposition of both u and f . We write x = r θ where r = |x| and θ = x/|x| ∈ Sn−1 and we decompose
u(r, θ) =X j≥0 uj(r, θ) and f (r, θ) = X j fj(r, θ)
where, for each j ≥ 0, the functions uj(r, · ) and fj(r, · ) belong to Ej. In particular ∆Sn−1uj = −λjuj and ∆Sn−1fj= −λjfj, wherever this makes sense.
Observe that Z B1 |u|2|x|−2δ−2dx =X j≥0 Z B1 |uj|2|x|−2δ−2dx = X j≥0 Z 1 0 kujk2L2(Sn−1)r n−3−2δdr and Z B1 |f |2|x|−2δ−2dx =X j≥0 Z B1 |fj|2|x|−2δ−2dx = X j≥0 Z 1 0 kfjk2L2(Sn−1)r n−3−2δ dr where k · kL2(Sn−1)is the L2(Sn−1) norm. In addition, the functions uj and fj satisfy
|x|2∆u
j = fj (1.5)
in the sense of distributions in B∗1. Indeed, making use of Z B1 u ∆v dx = Z B1 f v |x|−2dx
with test functions of the form v(r, θ) = h(r) φ(θ) where φ ∈ Ej and h is a smooth function with
compact support in (0, 1), we find that uj is a Ej-valued function solution of (1.5). Using the
decomposition of the Laplacian in polar coordinates, we also find that
r2∂r2uj+ (n − 1) r ∂ruj− λjuj = fj (1.6)
in the sense of distribution. Moreover Z 1 0 kujk2L2(Sn−1)r n−3−2δdr < ∞ and Z 1 0 kfjk2L2(Sn−1)r n−3−2δdr < ∞
The Sobolev Imbedding Theorem will help us justifying most of the forthcoming computation : Proposition 1.5.2 ([?], Theorem 7.26). If α = 1 −np then
W2,p(Ω) ⊂ C1,α( ¯Ω) provided Ω is a smooth bounded domain.
1.5. A CRUCIAL A PRIORI ESTIMATE 13
Observe that uj ∈ W2,2((r, 1)) for all r ∈ (0, 1) and hence we find that uj ∈ C 1,1/2 loc ((0, 1]).
Also, using the result of Proposition 1.2.1, we conclude that Z B1 |∂ruj| |x|−2δdx < ∞ and Z B1 |∂2ruj| |x|2−2δdx < ∞ (1.7)
Let j0 denote the least index in N such that
|δ| < δj0 (1.8)
The proof of Proposition 1.5.1 is now decomposed into two parts.
Part 1 : The case where |δ| < δj. Let χ be a cutoff function equal to 1 in B1/2 and
equal to 0 outside B1, let us further assume that χ is radial. We multiply the equation (1.6) by
χ2r−2δ−2u
j and integrate over B1. We obtain using polar coordinates
Z B1 χ2r−2δuj∂r(rn−1∂ruj) dr dθ − λj Z B1 χ2u2jrn−3−2δdr dθ = Z B1 χ2ujfjrn−3−2δdr dθ
where dθ denotes the volume form on Sn−1 and hence the Euclidean volume form is given by dx = rn−1dr dθ.
We integrate the first integral by parts to get Z B1 χ2|∂r(χ uj)|2r−2δdx + (λj+ δ (n − 2 − 2δ)) Z B1 χ2u2jr−2−2δdx = Z B1 (δ ∂r(χ2) − r |∂rχ|2) r−1−2δu2jdx − Z B1 χ2ujfjr−2−2δdx (1.9)
Even is formally, this computation is correct, some care is needed to justify the integration by parts at 0. Let us explain how the integration by parts is performed : We write
−χ2r−2δu
j∂r(rn−1∂ruj) = rn−1−2δ|∂r(χ uj)|2+ δ (n − 2 − 2δ) χ2rn−3−2δu2j
+ (δ ∂r(χ2) − r |∂χ|2) rn−2−2δu2j
− ∂r χ2rn−1−2δuj∂ruj+ δ χ2rn−2−2δu2j
For all R ∈ (0, 1), we integrate this equality over [R, 1] × Sn−1with respect to the measure dr dθ
to get − Z B1− ¯BR χ2r−2δ+1−nuj∂r(rn−1∂ruj) dx = Z B1− ¯BR r−2δ|∂r(χ uj)|2dx + δ (n − 2 − 2δ) Z B1− ¯BR χ2u2jr−2−2δdx + Z B1− ¯BR (δ ∂r(χ2) − r |∂χ|2) r−1−2δu2jdx + Z ∂BR χ2r−2δ r uj∂ruj+ δ r−1u2j rn−1dθ (1.10)
14 CHAPTER 1. WEIGHTED L2 ANALYSIS ON A PUNCTURED BALL
Now, use the fact that, thanks to (1.7) Z 1 0 Z ∂Br r−2δ |uj∂ruj| + r−1u2j r n−1dθ r−1dr < ∞
to show that, for a sequence of Ri tending to 0 we have
lim Ri→0 Z ∂BR i χ2r−2δ uj∂ruj+ δ r−1u2j r n−1dθ = 0
We now use this sequence of radii and pass to the limit in (1.10) to get − Z B1 χ2r−2δ+1−nuj∂r(rn−1∂ruj) dx = Z B1 r−2δ|∂r(χ uj)|2dx + δ (n − 2 − 2δ) Z B1 χ2u2jr−2−2δdx + Z B1 (δ ∂r(χ2) − r |∂χ|2) r−1−2δu2jdx
All subsequent integrations by parts can be justified using similar arguments, we shall leave the details to the reader.
We shall now make use of the following Hardy type inequality Lemma 1.5.1. The following inequality holds
(n − 2 − 2δ)2 Z Rn r−2−2δu2dx ≤ 4 Z Rn r−2δ|∂ru|2dx
provided the integral on the left hand side is finite.
Using this Lemma together with (1.9) we conclude that δ2j− δ2 Z B1 χ2u2j|x|−2−2δdx ≤ Z B1 χ2|fj| |uj| |x|−2−2δdx + c Z B1−B1/2 u2jdx
where the constant c = c(n, δ) > 0 does not depend on j. Now, we set
η := δj2 0− δ
2> 0
This is where it is important that δ 6= ±δj. Using Cauchy-Schwarz inequality together with the
inequality 2 a b ≤ η a2+ η−1b2 we get 2(δ2j− δ2) − η Z B1 χ2u2j|x|−2δ−2dx ≤ η−1 Z B1 χ2fj2|x|−2−2δdx + 2 c Z B1−B1/2 u2jdx. (1.11)
1.5. A CRUCIAL A PRIORI ESTIMATE 15 Observe that, for j ≥ j0,
2(δ2j− δ2) − η ≥ 2(δ2 j0− δ
2) − η = η.
We can sum all the inequalities (1.11) over j ≥ j0 to conclude that
η Z B1 χ2u˜2|x|−2δ−2dx ≤ η−1 Z B1 χ2f˜2|x|−2−2δdx + 2 c Z B1−B1 2 ˜ u2dx. (1.12)
where we have set
˜ u := X j≥j0 uj and f :=˜ X j≥j0 fj
Proof of Lemma 1.5.1: We now provide a proof of the Hardy type inequality we have used. Assume that n − 2 6= 2 δ and also that
Z
Rn
|∂ru|2|x|−2δdx < ∞
since otherwise there is nothing to prove. Then, start with the identity (n − 2 − 2δ) Z ∞ 0 v2rn−3−2δdr = Z ∞ 0 v2∂r(rn−2−2δ)dr = −2 Z ∞ 0 v ∂rv rn−2−2δdr
where the last equality follows from an integration by parts. Use Cauchy-Schwarz inequality to conclude that (n − 2 − 2δ)2 Z ∞ 0 v2rn−3−2δdr ≤ 4 Z ∞ 0 |∂rv|2rn−2−2δdr
The inequality in Lemma 1.5.1 follows from the integration of this inequality over Sn−1. Observe
that, in order to justify the integration by parts, it is enough to assume thatR∞
0 v
2|x|−2δ−2dx
converges.
Part 2 : The case where |δ| > δjand δj 6= 0. It remains to estimate uj, for j = 0, . . . , j0−1.
Here we simply use the fact that we have an explicit expression for uj in terms of fj. In order to
simplify the discussion, we first assume that δj6= 0. Then, we define ˜uj by
˜ uj(r, ·) = 1 2δj r2−n2 +δj Z r ∗ tn−42 −δjfj(t, · ) dt − r 2−n 2 −δj Z r ∗ tn−42 +δjfj(t, · ) dt
where ∗ has to be chosen according to the position of δ with respect to ±δj. In fact (see below)
we will choose ∗ = 0 when δ > δj and ∗ = 1 when δ < δj. It is easy to check that
|x|2∆˜u j= ˜fj
16 CHAPTER 1. WEIGHTED L2 ANALYSIS ON A PUNCTURED BALL
Basic strategy : Consider a quantity of the form u(r) = r2−n2 ±δj
Z r
∗
tn−42 ∓δjf (t) dt
where we assume that
Z 1
0
f2(r) rn−3−2δdr < ∞
It is a simple exercise to compute, using an integration by parts that Z 1 R u2(r) rn−3−2δdr = 1 2(±δj− δ) u(1)2− Rn−2−2δu(R)2− 2 Z 1 R f (r) u(r) rn−3−2δdr (1.13) This is where, once again, it is important that δ 6= ±δj.
A simple application of Cauchy-Schwarz inequality, yields |u(r)|2≤ r2δ+2−n 2|δ ± δj| Z 1 0 f2(t) tn−3−2δdt ,
provided we choose ∗ = 0 when δ > δj and ∗ = 1 when δ < −δj. This is where the choice of ∗ is
crucial.
Plugging this information in (1.13) and using Cauchy-Schwarz inequality, immediately implies that Z 1 R u2rn−3−2δdr ≤ 1 2|δ ± δj|2 Z 1 0 f2(r) rn−3−2δdr + 1 |δ ± δj| Z 1 0 u2(r) rn−3−2δdr 1/2 Z 1 0 f2(r) rn−3−2δdr 1/2
It is a simple exercise to check that this implies that Z 1 R u2rn−3−2δdr ≤ c Z 1 0 f2(r) rn−3−2δdr
for some constant c = c(δ, n, j) > 0.
Using this result, and passing to the limit as R tends to 0, we conclude that Z B1 ˜ u2j|x|−2δ−2dx ≤ c Z B1 fj2|x|−2δ−2dx (1.14)
for some constant c = c(δ, n, j) > 0.
It remains to evaluate the difference between the the functions uj and ˜uj. Since
|x|2∆(u
1.5. A CRUCIAL A PRIORI ESTIMATE 17 we find that
uj− ˜uj = r
2−n
2 +δjφ + r2−n2 −δjψ
where φ, ψ ∈ Ej. Remembering that uj− ˜uj ∈ L2δ(B1∗) we find, in the case where δ > δj, that the
only possibility is φ = ψ = 0. Therefore, in this case the proof is already complete since (1.14) provides the desired estimate. When δ < −δj, it is very likely that φ and ψ are not equal to 0.
In this case, we evaluate
kφkL2(Sn−1)+ kψkL2(Sn−1)≤ c k˜uj− ujkL2(B 1− ¯B1/2)
for some constant c = c(δ, j, n) > 0. To obtain this estimate without much work observe that the space of functions
{r2−n2 +δjφ + r 2−n
2 −δjψ : φ, ψ ∈ Ej} is finite dimensional and that we have two (equivalent) norms on it. Namely
N1(r 2−n 2 +δjφ + r2−n2 −δjψ) := kφk L2(Sn−1)+ kψkL2(Sn−1) and N2(r 2−n 2 +δjφ + r2−n2 −δjψ) := kr2−n2 +δjφ + r2−n2 −δjψk L2(B 1− ¯B1/2).
Observe that we have implicitly used the fact that δj6= 0 and hence the functions r −→ r
2−n 2 +δj and r −→ r2−n2 −δj are linearly independent.
Granted this estimate, we conclude that kujkL2 δ(B1∗)≤ c kfjkL2 δ(B1∗)+ kujkL2(B1−B1/2)
This completes the proof of the result when all δj 6= 0. Collecting this estimates together with
(1.12) this completes the proof of the Proposition 1.5.1 when δj 6= 0, for all j ∈ N.
Part 3 : The case where |δ| > δj = 0. We now turn to the case where δj = 0. This case
happens when n = 2 and j = 0. The equation satisfied by u0reads
r2∂r2u0+ r ∂ru0= f0
This time, the explicit formula we will use is ˜ u0(r) := Z r ∗ s−1 Z s ∗ t−1f0(t) dt ds
where ∗ will be chosen appropriately, namely ∗ = 0 when δ > 0 and ∗ = 1 when δ < 0. Again, one can check directly that |x|2∆˜u
0= f0.
To start with use the strategy developed above to prove that k∂ru˜0kL2 δ−1(B ∗ 1)≤ c kf0kL2δ(B ∗ 1)
18 CHAPTER 1. WEIGHTED L2 ANALYSIS ON A PUNCTURED BALL
We leave the details to the reader. Once this is done, use again the above strategy to show that Z 1 R ˜ u20r−2δ−1dr ≤ 1 2|δ|2 Z 1 0 f02(r) r−2δ−1dr + 1 |δ| Z 1 0 ˜ u20(r) r−2δ−1dr 1/2 Z 1 0 |∂ ˜u0|2(r) r−2δ+1dr 1/2
Collecting these two estimates, we conclude that k˜u0k2L2 δ(B ∗ 1)≤ c kf0k2L2 δ(B ∗ 1)+ kf0kL 2 δ(B1∗)k˜u0kL2δ(B1∗)
from which it follows that
k˜u0kL2
δ(B∗1)≤ c kf0kL2δ(B1∗) Once this estimate has been obtained, we observe that
u0− ˜u0= α + β log r
When δ > 0, α = β = 0 since u0− ˜u0∈ L2δ(B∗1) and when δ < 0 we can argue as what has been
already done when δj 6= 0 to obtain
|α| + |β| ≤ c k˜u0− u0kL2(B 1− ¯B1/2)
for some constant c = c(n, δ) > 0. Collecting all the estimate, we conclude that ku0kL2 δ(B1∗)≤ c kf0kL2 δ(B∗1)+ ku0kL2(B1−B1/2) (1.15) This completes the proof in all cases.
Exercise 1.5.1. Observe that there is another formula we could have used for ˜u0, namely
˜ u0(r) = log r Z r ∗ t−1f0(t) dt − Z r ∗ t−1 log t f0(t) dt.
Prove the estimate (1.15) starting from this formula.
Exercise 1.5.2. Show that, in the main estimate in the statement of Proposition 1.5.1, one can replace kukL2(B
1− ¯B1/2)by kukL1(B1− ¯B1/2).
Exercise 1.5.3. Let a : B1∗−→ R be a function which satisfies the bound
|a(x)| ≤ c |x|−2+α
in B∗1, for some α > 0. Show that the result of Proposition 1.5.1 remains true if the operator |x|2∆ is replaced by the operator |x|2(∆ + a).
1.5. A CRUCIAL A PRIORI ESTIMATE 19 Exercise 1.5.4. † Show that the result of Proposition 1.5.1 remains true if the operator |x|2∆
is replaced by the operator |x|2
∆ + d, where d ∈ R is fixed, provided we define δj = < n − 2 2 + j 2 + d !1/2
Chapter 2
Weighted L
2
analysis on a
punctured manifold
2.1
The Laplace-Beltrami operator in normal geodesic
co-ordinates
Given a Riemannian manifold (M, g), the Laplace-Beltrami operator is defined in local coordinates x1, . . . , xn ∆g= X i,j 1 √ detg∂xi p det(g) gij∂xj
where gij are the coefficients of the inverse of the matrix (g ij)i,j.
Recall that, in local coordinates, the volume form on M is given by dvolg=
p
detg dx1. . . dxn In particular, if u is a smooth function of M , we have
Z M u ∆gu dvolg= − Z M gij∂xiu ∂xju dvolg= − Z M gij|∇u|2gdvolg
Using the exponential mapping, we can define normal geodesic coordinates in a neighborhood of a point p ∈ M as follows : first choose an orthonormal basis e1, . . . , em of TpM . Then define
the mapping
F (x1, . . . , xm) := Expp X
i
xiei
!
One can prove that F is a local diffeomorphism from a neighborhood of 0 in (M, g) into a neighborhood of p in M .
22 CHAPTER 2. WEIGHTED L2ANALYSIS ON A PUNCTURED MANIFOLD
Proposition 2.1.1. ([?], Theorem ??) In normal geodesic coordinates, the coefficients of the metric g can be expanded as
gij= δij+ O(|x|2)
The functions O(|x|2) are smooth function which vanish quadratically at the origin. As a
simple consequence of this result, we have the expansion of the Laplace-Beltrami operator in normal geodesic coordinates :
∆g= ∆eucl+ O(|x|2) ∂xi∂xj+ O(|x|) ∂xk (2.1) The operator O(|x|2) ∂
xi∂xj is a second order differential operator whose coefficients are smooth and vanish quadratically at the origin and the operator O(|x|2) ∂
xk is a first order differential operator whose coefficients are smooth and vanish at the origin. This last expansion follows from a direct computation using the formula of ∆gin local coordinates and the result of Proposition 2.1.1.
2.2
Two global results
Using the normal geodesic coordinates, we extend the results of Proposition 1.1.1 and Proposi-tion 1.5.1 in a global setting.
As in the previous section (M, g) is a compact n-dimensional Riemannian manifold without boundary. We choose points p1, . . . , pk∈ M and denote by
M∗:= M − {p1, . . . , pk}
Given R small enough, we define BR(p) ⊂ M (resp. ¯BR(p) ⊂ M ) to be the open (resp. closed)
geodesic ball of radius R centered at p. The corresponding punctured balls are denoted by BR∗(p) and ¯BR∗(p). Finally, we set
MR:= M − ∪jB¯R(pj)
We fix a smooth function
γ : M∗−→ (0, ∞) such that, for all j = 1, . . . , k
γ(p) = dist(p, pj)
in some neighborhood of pj.
Given δ ∈ R we define the space
L2δ(M∗) := γδ+1L2(M ) This space is endowed with the norm
kukL2 δ(M∗):= Z M |u|2γ−2δ−2dvolg 1/2 Again, we have
2.2. TWO GLOBAL RESULTS 23 Lemma 2.2.1. The space (L2
δ(M
∗), k · k L2
δ(M∗)) is a Banach space. We define the unbounded operator Aδ by
Aδ : L2δ(M ∗) −→ L2 δ(M ∗) u 7−→ γ2(∆ gu + a u)
where a is a smooth function on M . The domain D(Aδ) of this operator is the set of functions
u ∈ L2 δ(M
∗) such that A
δu = f ∈ L2δ(M
∗) in the sense of distributions : This means that
u ∈ W2,2(M
R), for all R > 0 small enough and
Z M u (∆gv + a v) dvolg = Z M f v γ−2dvolg
for all C∞ functions v with compact support in M∗. It is easy to check that Lemma 2.2.2. The domain of the operator Aδ is dense in L2δ(M
∗) and the graph of A
δ is closed.
Exercise 2.2.1. Give a proof of Lemma 2.2.2.
The result we have obtain in Proposition 1.2.1 translates immediately into :
Proposition 2.2.1. Assume δ ∈ R is fixed. There exists a constant c = c(n, δ) > 0 such that for all u, f ∈ L2δ(M∗) satisfying γ2(∆gu + a u) = f in M∗ we have
k∇ukL2
δ−1(M∗)+ k∇
2uk L2
δ−2(M∗)≤ c (kf kL2δ(M∗)+ kukL2δ(M∗))
The proof of the result goes as follows : First observe that the result of Proposition 1.1.1 remains true if one changes B1∗ with BR∗. In which case the estimate of Proposition 1.1.1 has to be replaced by k∇ukL2 δ−1(B∗R)+ k∇ 2uk L2 δ−2(B∗R)≤ c kf kL2 δ(B∗R)+ kukL2δ(B∗R) (2.2) if u, f ∈ L2δ(BR∗) satisfy |x|2∆u = f in B∗R. This can be seen easily by performing a simple change v(x) = u(R x) and g(x) = f (R x) so that v and g satisfy |x|2∆ v = g in B∗1, then the estimate follows from the corresponding estimate in Proposition 1.1.1.
Close to the puncture pj we use normal geodesic coordinates so that γ = |x| and write the
equation γ2(∆gu + a u) = f as
|x|2∆
24 CHAPTER 2. WEIGHTED L2ANALYSIS ON A PUNCTURED MANIFOLD
Using the result of Proposition 1.1.1 together with the result of Proposition 2.1.1, we evaluate k|x|2(∆ eucl− ∆g) u − |x|2a ukL2 δ(B∗R)≤ c R 2 kuk L2 δ(M∗)+ k∇ukL2δ−1(M∗)+ k∇ 2uk L2 δ−2(M∗)
for some constant c = c(n, δ) > 0 which does not depend on R > 0, small enough. Next we apply (2.2) to conclude that k∇ukL2 δ−1(B ∗ R)+ k∇ 2uk L2 δ−2(B ∗ R) ≤ c kf kL2 δ(B ∗ R)+ kukL2δ(M∗) + R2(k∇uk L2 δ−1(M∗)+ k∇ 2uk L2 δ−2(M∗))
for some constant c = c(n, δ) > 0 independent of R > 0 small enough. This can also be written as (1 − c R2) (k∇ukL2 δ−1(BR∗)+ k∇ 2uk L2 δ−2(B∗R)) ≤ c kf kL2 δ(BR∗)+ kukL2δ(M∗)
If R > 0 is chosen so that c R2≤ 1/2 we conclude that
k∇ukL2 δ−1(B ∗ R)+ k∇ 2uk L2 δ−2(B ∗ R)≤ 2 c kf kL2 δ(B ∗ R)+ kukL2δ(M∗)
We now use the elliptic estimates provided by
Proposition 2.2.2. ([?], Theorem ??) Assume we are given Ω ⊂ M , Ω0 ⊂⊂ Ω and p ∈ (1, ∞). Then there exists c = c(M, g, Ω, Ω0) > 0 such that, if v ∈ W2,pand g ∈ L2(Ω) satisfy ∆gv = g in
Ω, then
k∇vkLp(Ω0)+ k∇2vkLp(Ω0)≤ c kgkLp(Ω)+ kukLp(Ω) with Ω = MR/2 and Ω0 = MR to show that
k∇ukL2 δ−1(MR)+ k∇ 2uk L2 δ−2(MR)≤ c kf kL2 δ(MR/2)+ kukL2δ(MR/2)
for some constant c = c(n, R) > 0. The estimate then follows from the sum of the two estimates we have obtained.
The following result is a consequence of Proposition 1.5.1.
Proposition 2.2.3. Assume that δ 6= ±δj for j ∈ N. Then there exists a constant c = c(n, δ)
and a compact K in M∗ such that, for all u, f ∈ L2
δ(M∗) satisfying γ2(∆gu + a u) = f in M∗, we have kukL2 δ(M∗)≤ c kf kL2 δ(M∗)+ kukL2(K)
2.2. TWO GLOBAL RESULTS 25 Again this result states that we can control the weighted L2- norm of u in terms of the weighted
L2-norm of f and some information about the function u away from the punctures.
The proof of this second Proposition, also follows from a perturbation argument. First observe that the result of Proposition 1.5.1 remains true if one changes B∗
1 with BR∗. In which case the
estimate of Proposition 1.5.1 has to be replaced by kukL2 δ(B ∗ R)≤ c kf kL2 δ(B ∗ R)+ R −δ−1kuk L2(B R− ¯BR/2) (2.3) if u, f ∈ L2δ(BR∗) satisfy |x|2∆u = f in B∗R. This can be seen easily by performing a simple change v(x) = u(R x) and g(x) = f (R x) so that v and g satisfy |x|2∆ v = g in B∗
1, then the estimate
follows from the corresponding estimate in Proposition 1.5.1.
Close to the puncture pj we use normal geodesic coordinates so that γ = |x| and write the
equation γ2(∆gu + a u) = f as
|x|2∆
euclu = f + |x|2(∆eucl− ∆g) u − |x|2a u
Using the result of Proposition 2.2.1 together with the result of Proposition 2.1.1, we evaluate k|x|2(∆ eucl− ∆g) u − |x|2a ukL2 δ(BR∗)≤ c R 2kuk L2 δ(M∗)
for some constant c = c(n, δ) > 0 which does not depend on R > 0, small enough. Next we apply (2.3) to conclude that kukL2 δ(BR∗)≤ c kf kL2 δ(B∗R)+ R 2kuk L2 δ(M∗)+ R −δ−1kuk L2(BR− ¯B R/2)
for some constant c = c(n, δ) > 0 independent of R > 0 small enough. Adding on both sides kukL2(M R)we conclude that kukL2 δ(M∗)≤ c kf kL2 δ(B ∗ R)+ R 2kuk L2 δ(M∗)+ R −δ−1kuk L2(M R/2)
where c = c(n, δ) > 0 does not depend on R > 0 small enough. In other words (1 − c R2) kukL2 δ(M∗)≤ c kf kL2 δ(B ∗ R)+ R −δ−1kuk L2(MR/2)
It remains to fix R > 0 such that c R2≤ 1/2 and let K = MR/2. this completes the proof of the
result.
Exercise 2.2.2. Show that the result of Proposition 2.2.3 remains true if the function a : M∗−→ R only belongs to L∞loc(M∗) and satisfies the bound
|a| ≤ c γ−2+α
26 CHAPTER 2. WEIGHTED L2ANALYSIS ON A PUNCTURED MANIFOLD
Exercise 2.2.3. † Show that the result of Proposition 2.2.3 remains true if, near any of the pi,
the function a can be decomposed as a = d γ−2+ ˜ai where d ∈ R is a constant and the function
˜
ai satisfies the bound
|˜ai| ≤ c γ−2+αi
in BRi(pi), for some αi> 0 and provided we define
δj = < n − 2 2 + j 2 + c !1/2
Exercise 2.2.4. † Show that the result of Proposition 2.2.3 remains true if, near any of the pi
there exists local coordinates x1, . . . , xn in which the coefficients of the metric can be expanded as gij = δij+ O(|x|β)
and if in addition
∇gij = O(|x|β−1)
for some β > 0.
Exercise 2.2.5. ‡ Extend the result of Proposition 2.2.3 to handle the case where, near any of the pi, the function a can be decomposed as
a = diγ−2+ ˜ai
where di∈ R are constants and the function ˜ai satisfies the bound
|˜ai| ≤ c γ−2+αi
in BRi(pi), for some αi> 0.
2.3
The kernel of the operator A
δThe results of the previous sections will now be used to derive the functional analytic properties of the operator Aδ. We start with the :
Theorem 2.3.1. The kernel of Aδ is finite dimensional.
For the time being, let us assume that δ 6= ±δj. We argue by contradiction and assume that
the result is not true. Then, there would exist a sequence (um)m of elements of L2δ(M∗) which
satisfy Aδum= 0.
Without loss of generality we can assume that the sequence is normalized so that Z
M
2.4. THE RANGE OF THE OPERATOR Aδ 27
and also that
Z
M
umum0γ−2δ−2dvolg= 0. (2.5)
for all m 6= m0. Using the result of Proposition 2.2.3 we obtain kum− um0k
L2
δ(M∗)≤ c ku
m− um0k
L2(K) (2.6)
where c = c(n, δ) > 0 does not depend on m.
Using (2.4) together with the result of Proposition 2.2.1 we conclude that um is bounded in
W1,2(K). Now, we apply Rellich’s compactness result :
Proposition 2.3.1. ([?], Theorem ??) Given a smooth bounded domain Ω ⊂ M , the imbedding W1,2(Ω) −→ L2(Ω)
is compact.
This result allows us to extract some subsequence (which we will still denote by (um)
m) which
converges in L2(K). In particular, the sequence (u
m)mis a Cauchy sequence in L2(K). In view
of (2.6) we see that the sequence (um)m is a Cauchy sequence in L2δ(M∗). This space being a
Banach space, we conclude that this sequence converges in L2
δ(M∗) to some function u.
Clearly, passing to the limit in (2.4) we see that Z
Ω
|u|2γ−2δ−2dvol g= 1
While, passing to the limit m0−→ ∞ in (2.5), we get Z
Ω
umu r−2δ−2dx = 0
and then passing to the limit as m tends to ∞, we conclude that Z
Ω
u2r−2δ−2dx = 0
Clearly a contradiction. This completes the proof when δ 6= ±δj, for all j ∈ N. In order to
complete the proof is all cases it is enough to observe that if u ∈ KerAδ then u ∈ KerAδ0 for all δ0≤ δ. Therefore, one can always reduce to the case where δ06= ±δ
j for all j ∈ N.
2.4
The range of the operator A
δWe pursue our quest of the mapping properties of the operators Aδ by studying the range of this
28 CHAPTER 2. WEIGHTED L2ANALYSIS ON A PUNCTURED MANIFOLD
Theorem 2.4.1. Assume that δ 6= ±δj for j ∈ N. Then the range of Aδ is closed.
Let um, fm∈ L2
δ(M∗) be sequences such that fm:= Aδum converges to f in L2δ(M∗). Since
we now know that Ker Aδ is finite dimensional, it is closed and we can project every each um
onto u ∈ L2δ(M∗) : Z M u v r−2δ−2dx = 0 ∀v ∈ Ker Aδ
the orthogonal complement of Ker Aδ in L2δ(M∗), with respect to the scalar product associated to
the weighted norm. Therefore, without loss of generality, we can assume that umis L2
δ-orthogonal
to Ker Aδ.
Since fm converges in L2δ(M∗), there exists c > 0 such that kfmk
L2
δ(M∗)≤ c. (2.7)
Now, we claim that the sequence (um)
mis bounded in L2δ(M
∗). To prove this claim, we argue
by contradiction and assume that (at least for a subsequence still denoted (um) m) lim m→+∞ku mk L2 δ(M∗)= ∞ We set vm:= u m kumk L2 δ(M∗) and gm:= f m kumk L2 δ(M∗)
so that Aδvm = gm. Applying the result of Proposition 2.2.1, we conclude that the sequence
(vm)
mis bounded in W1,2(K) and hence, using Rellich’s Theorem, we conclude that a subsequence
(still denoted (vm)
m) converges in L2(K). Now the result of Proposition 2.2.3 yields
kvm− vm0kL2 δ(M∗)≤ c kgm− gm0kL2 δ(M∗)+ kv m − vm0kL2(K) . (2.8)
On the right hand side, the sequence (gm)
m tends to 0 in L2δ(M∗) and the sequence (vm)m
converges in L2(K). Therefore, we conclude that (vm)
m is a Cauchy sequence in L2δ(M∗) and
hence converges to v ∈ L2δ(M∗).
To reach a contradiction, we first pass to the limit in the identity Aδvm= gmto get that the
function v is a solution of Aδv = 0 and hence v ∈ Ker Aδ. But by construction kvkL2
δ(M∗) = 1 and also
Z
M
vmv γ−2δ−2dvolg= 0
(since v ∈ Ker Aδ) and, passing to the limit in this last identity we find that kvkL2
δ(M∗)= 0. A contradiction.
Now that the claim is proved, we use the result of Proposition 2.2.1 together with Rellich’s Theorem to extract, form the sequence (um)
msome subsequence which converges to u in L2δ(M∗).
Once more, Proposition 2.2.3 implies that kum− um0kL2 δ(M∗)≤ c kfm− fm0kL2 δ(M∗)+ ku m − um0kL2(K) . (2.9)
2.5. FREDHOLM PROPERTIES FOR Aδ 29
This time, on the right hand side, the sequence (fm)
m converges in L2δ(M
∗) and the sequence
(um)
mconverges in L2(K). Therefore, we conclude that (um)mis a Cauchy sequence in L2δ(M ∗)
and hence converges to u ∈ L2
δ(M∗). Passing to the limit in the identity Aδum= fmwe conclude
that Aδu = f and hence f belongs to the range of Aδ. This completes the proof of the result.
2.5
Fredholm properties for A
δ It will be convenient to identify the dual of L2δ(M
∗) with L2
−δ(M∗). This is done using the scalar
product
hu, vi := Z
M
u v γ−2dvolg (2.10)
Clearly, given v ∈ L2−δ(M∗), we can define Tv∈ L2δ(M∗)
0 by Tv(u) = hu, vi Moreover, we have kTvk(L2 δ(M∗)) 0 = kvkL2 −δ(M∗) Conversely, given T ∈ L2 δ(M∗) 0
there exists a unique v ∈ L2
−δ(M∗) such that hu, vi = T (u) for
all u ∈ L2 δ(M∗).
We define A∗δ, the adjoint of Aδ
A∗δ : L2δ(M∗)0 −→ L2 δ(M
∗)0
is defined to be an unbounded operator. An element T ∈ L2δ(M∗)0 belongs to D(A∗δ), the domain of A∗δ, if and only if there exists S ∈ L2
δ(M∗)
0
such that T (Aδv) = S(v)
for all v ∈ D(Aδ). We will write A∗δ(T ) = S.
Granted the above identification of L2 δ(M∗)
0
with L2
δ(M∗) it is easy to check that we can
identify A∗δ with A−δ. Indeed, if we write T = Tu and Aδ∗(T ) = Tf, for u, f ∈ L2−δ(M∗), then, by
definition
Tu(Aδv) := hu, Aδvi
and
A∗δ(T )(v) := hf, vi for all v ∈ D(Aδ). Hence, we have
Z
M
u(∆g+ a)v dvolg=
Z
M
30 CHAPTER 2. WEIGHTED L2ANALYSIS ON A PUNCTURED MANIFOLD
for all v ∈ D(Aδ). This in particular implies that γ2(∆g+ a) u = f in the sense of distributions.
Since u, f ∈ L2
−δ(M∗), we conclude that u ∈ D(A−δ) and f = A−δu.
Conversely, if u ∈ D(A−δ), we can write for all v ∈ D(Aδ)
hu, Aδvi = Z M u (∆g+ a)v dvolg = Z M v (∆g+ a)u dvolg = hA−δu, vi
The integrations by parts can be justified since, according to the result of Proposition 2.2.1, we have ∇v ∈ L2
δ−1(M∗), ∇u ∈ L2−δ−1(M∗), ∇2v ∈ L2δ−2(M∗) and ∇2u ∈ L2−δ−2(M∗). Therefore
Tu∈ D(A∗δ) and A∗δ(Tu) = TA−δu.
With these identifications in mind, we can state the Theorem 2.5.1. Assume that δ 6= ±δj for all j ∈ N. Then
Ker Aδ = (Im A−δ)⊥
and
Im Aδ = (Ker A−δ)⊥
The first part is a classical property for unbounded operators with closed graph and dense domain (see Corollary II.17 in [?]). The second result follows from classical results for unbounded operators with dense domains, closed graph and closed range (see Theorem II.18 in [?]).
Observe that, because of our identifications, F⊥ is obtained from F using the scalar product defined in (2.10).
Very useful for us will be the :
Corollary 2.5.1. Assume that δ 6= ±δj for all j ∈ N. Then Aδ is injective if and only if A−δ is
surjective.
2.6
The deficiency space
Even though the previous results seem already a great achievement, since it will provide right inverses for some operators, we will need a more refined result. As usual, this result for operators defined on the punctured manifold M∗ are obtained by perturbing the corresponding results in Euclidean space.
2.6. THE DEFICIENCY SPACE 31 Lemma 2.6.1. Assume that δ 6= ±δj, for j ∈ N. There exist an operator
Gδ : L2δ(B1∗) −→ L 2 δ(B1∗)
and c = c(n, δ) > 0 such that for all f ∈ L2 δ(B
∗
1), the function u := Gδ(f ) is a solution of
|x|2∆ u = f in B1∗ and kukL2 δ(B∗1)+ k∇ukL2δ−1(B1∗)+ k∇ 2uk L2 δ−2(B1∗)≤ c kf kL2δ(B1∗)
At first glance this result looks rather strange wince we are not imposing any boundary data. Nevertheless, some boundary data are hidden in the construction of the operator Gδ. Observe
that we state the existence of Gδ and do not state any uniqueness of this operator !
The proof of the existence of Gδ relies on the eigenfunction decomposition of the function f .
We decompose as usual
f =X
j≥0
fj
where f (r, ·) ∈ Ej for all j ∈ N. Let j0∈ N be the least index for which
|δ| < δj0 We set ˜ f = X j≥j0 fj Clearly ˜f ∈ L2
δ(B1∗) and, for all R ∈ (0, 1/2) one can solve
|x|2∆˜u R = f˜ in B1− ¯BR ˜ uR = 0 on ∂B1∪ ∂BR
The existence of ˜uRfollows from Proposition 1.1.2 and we have the estimate
k˜uRkL2(B1− ¯BR)≤ c k ˜f kL2(B1− ¯BR)
for some constant c = c(n, R) > 0. We claim that there exists a constant c = c(n, δ) > 0 such that
k˜uRkL2
δ(B1− ¯BR)≤ c k ˜f kL2δ(B1− ¯BR) (2.11) Here the norm in L2
δ(B1− ¯BR) is nothing but the restriction of the restriction of the norm in
L2δ(B1∗) to functions which are defined in B1− ¯BR. The proof of the claim follows the Part 1 of
32 CHAPTER 2. WEIGHTED L2ANALYSIS ON A PUNCTURED MANIFOLD
Arguing as in the proof of Proposition 1.2.1 we conclude that there exists a constant c = c(n, δ) > 0 such that
k∇˜uRkL2
δ−1(B1− ¯BR)≤ c k ˜f kL2δ(B1− ¯BR) In particular, given R0 ∈ (0, 1/2), there exists c = c(n, δ, R0) > 0 such that
k˜uRkL2(B1− ¯B0
R)+ k∇˜uRkL2(B1− ¯B0R)≤ c k ˜f kL2δ(B ∗ 1)
for all R ∈ (0, R0). Then using Rellich’s Theorem together with a simple diagonal argument, we conclude that there exists a sequence of radii Ri tending to 0 such that the sequence (˜uRi)i converges in L2(B
1− ¯BR), for all R ∈ (0, 1/2). Passing to the limit in the equation we obtain a
solution ˜u of |x|2∆˜u = f˜ in B∗ 1 ˜ u = 0 on ∂B1 (2.12)
Moreover, passing to the limit in (2.11), we have the estimate k˜ukL2 δ(B ∗ 1)≤ c k ˜f kL2δ(B ∗ 1)
To finish this study observe that the solution of (2.12) which belongs to L2 δ(B
∗
1) is unique. To
see this, argue by contradiction. If the claim were not true there would exists two solutions and taking the difference we would obtain a function ˜w ∈ L2
δ(B ∗ 1) satisfying ( |x|2∆ ˜w = 0 in B∗ 1 ˜ u = 0 on ∂B1
Performing the eigenfunction decomposition of ˜w as ˜ w = X j≥j0 ˜ wj we find that ˜ wj = r 2−n 2 +δjφj+ r 2−n 2 −δjψj where φj, ψj ∈ Ej. Using the fact that ˜wj ∈ L2δ(B
∗
1) we conclude that ψj = 0. Next, using the
fact that ˜wj= 0 on ∂B1, we get φj= 0 and hence ˜w = 0.
Therefore, we can define
Gδ( ˜f ) = ˜u.
It remains to understand the definition of Gδ acting on fj, for j ≤ j0− 1. For the sake of
simplicity, we assume that δj = 0 (When δj = 0, the formula has to be changed according to
what we have already done in Part 3 of the proof of Proposition 1.5.1) and we use an explicit formula Gδ(fj) = 1 2δj r2−n2 +δj Z r ∗ tn−42 −δjf j(t) dt − r 2−n 2 −δj Z r ∗ tn−42 +δjf j(t) dt
2.6. THE DEFICIENCY SPACE 33 where ∗ = 0 if δ > δj and ∗ = 1 if δ < −δj. The estimate follows at once from the arguments
developed in Part 2 of the proof of Proposition 1.5.1. We omit the details. Let us now provide a few applications of this result.
Application # 1 : The first application is concerned with the extension of the previous result to the operator defined on the manifold in a neighborhood of one puncture.
Lemma 2.6.2. Assume that δ 6= ±δj, for j ∈ N. Given pi∈ M one of the punctures, there exists
Ri= R(pi, n, δ) > 0, an operator
G(i)δ : L2δ(BR∗i(pi)) −→ L2δ(B ∗ Ri(pi)) and c = c(n, δ, pi) > 0 such that for all f ∈ L2δ(B
∗ Ri(pi)), the function u := G (i) δ (f ) is a solution of γ2(∆g+ a) u = f in BR∗ i(pi) and kukL2 δ(B∗Ri(pi))+ k∇ukL2δ−1(B∗Ri(pi))+ k∇ 2uk L2 δ−2(BRi∗ (pi))≤ c kf kL2δ(B∗Ri(pi))
This result follows from a simple perturbation argument. First observe that, a scaling argu-ment shows that the result of Lemma 2.6.1 holds when the radius of the ball, which was chosen to be 1, is replaced by R. The corresponding operator will be denoted by Gδ,Rand the estimate
holds with a constant which does not depend on R > 0. We leave this as an exercise. Thanks to the result of Proposition 2.1.1 we can write
kγ2(∆ g− ∆eucl+ a) ukL2 δ(B∗Ri(pi)) ≤ c R 2kuk L2 δ(BRi∗ (pi))+ k∇ukL2δ−1(BRi∗ (pi)) +k∇2ukL2 δ−2(B ∗ Ri(pi))
provided R > 0 is small enough. This implies that kf − Aδ◦ Gδ,Rf kL2 δ(B ∗ Ri(pi)) ≤ c R 2 kf kL2 δ(B ∗ Ri(pi))
for some constant c = c(n, δ) > 0 which does not depend on R. This clearly implies that the operator Aδ ◦ Gδ,R is invertible provided R is fixed small enough, say R = Ri. To obtain the
result, it is enough to define
G(i)δ := Gδ,Ri◦ (Aδ◦ Gδ,Ri). The relevant estimate then follows at once.
Application # 2 : Recall that the functions |x|2−n2 ±δjφ
34 CHAPTER 2. WEIGHTED L2ANALYSIS ON A PUNCTURED MANIFOLD
are harmonic in B1∗ provided φ ∈ Ej. Building on the result of the previous application, we
now prove that one can perturb these functions to get, near any puncture pi a solution of the
homogeneous problem associated with the operator γ2(∆
g + a). This is the content of the
following :
Lemma 2.6.3. For all puncture pi ∈ M , given j ∈ N and φ ∈ Ej, there exists Wj,φ± (i) which is
defined in BR∗
i(pi) and which satisfies
γ2(∆g+ a) W ± (i) j,φ = 0 in B∗R i(pi). In addition, Wj,φ± (i)− |x|2−n2 ±δjφ ∈ L2 δ(B ∗ Ri(pi)) for all δ < ±δj+ 2. Finally the mapping
φ ∈ Ej −→ W ± (i) j,φ
is linear.
In this result, Ri is the radius given in Lemma 2.6.2 and x are normal geodesic coordinates
near pi.
The proof of this Lemma uses the following computation which follows at once from Propo-sition 2.1.1 γ2(∆g+ a) |x| 2−n 2 ±δjφ = γ2(∆ g− ∆eucl+ a) |x| 2−n 2 ±δjφ ∈ L2 δ(BR∗i(pi)) for all δ < ±δj+ 2. The result then follows from Lemma 2.6.2.
For each i = 1, . . . , k, we define χ(i)to be a cutoff function which is identically equal to 1 in
BRi/2(pi) and identically equal to 0 in M − B3Ri/4(pi). The main result of this section is :
Proposition 2.6.1. Given δ < δ0, δ, δ0 6= ±δj, for all j ∈ N. Assume that u ∈ L2δ(M∗) and
f ∈ L2δ0(M∗) satisfy
γ2(∆g+ a) u = f
in M∗. Then, there exists v ∈ L2
δ0(M∗) such that u − v ∈ Dδ,δ0 := Span {χ(i)W±(i)
j,φ , : φ ∈ Ej, δ < ±δj < δ0} In addition kvkL2 δ0(M ∗)+ ku − vkDδ,δ0 ≤ c (kf kL2 δ0(M ∗)+ kukL2 δ(M∗)) for some constant c = c(n, δ, δ0) > 0.
2.6. THE DEFICIENCY SPACE 35 The proof of this result relies on the corresponding result for the Laplacian in the punctured unit ball.
Lemma 2.6.4. Given δ < δ0, δ, δ06= ±δj, for all j ∈ N. Assume that u ∈ L2δ(B∗1) and f ∈ L2δ0(B∗1) satisfy
|x|2∆ u = f
in B1∗. Then, there exists v ∈ L2δ0(B1∗) such that
u − v ∈ Dδ,δ0 := Span {|x| 2−n 2 ±jφ, : φ ∈ Ej, δ < ±δj< δ0} In addition kvkL2 δ0(B ∗ 1)+ ku − vkDδ,δ0 ≤ c (kf kL2δ0(B1∗)+ kukL2δ(B∗1)) for some constant c = c(n, δ, δ0) > 0.
To prove the Lemma, we use the result of Lemma 2.6.1 and set ¯v = Gδ0f ∈ L2δ0(B1∗). Therefore
|x|2∆ (u − ¯v) = 0 in B∗1. We have k¯vkL2 δ0(B ∗ 1)≤ c kf kL2δ0(B ∗ 1)
for some constant c = c(n, δ) > 0. We set w = u − ¯v which we decompose as usual w =X
j
wj
where wj(r, ·) ∈ Ej. We fix j0 to be the least index for which
|δ| < δj0 and |δ 0| < δ j0 We define ˜ w = X j≥j0 wj We claim that ˜w ∈ L2
δ0(B1∗) and also that kvkL2
δ0(B ∗
1)≤ c kwkL2(B1− ¯B1/2)
for some constant c = c(n, δ) > 0. The proof of the claim follows the arguments of Part 1 in the proof of Proposition 1.5.1. We omit the details.
Next, observe that, for j = 0, . . . , j0− 1 the function wj is given by
wj = |x| 2−n 2 +δjφ j+ |x| 2−n 2 −δjψ j
36 CHAPTER 2. WEIGHTED L2ANALYSIS ON A PUNCTURED MANIFOLD
for some φj, ψj∈ Ej. Observe that φj = 0 if δj< δ and ψj = 0 if −δj< δ since wj ∈ L2δ(B ∗ 1). It
is easy to see that
kφjkL2(Sn−1)+ kφjkL2(Sn−1)≤ c kwjkL2(B1− ¯B 1/2) for some constant c = c(n, j) > 0.
We set v = ¯v + w + X j=0,...,j0−1, δj>δ0 |x|2−n2 +δjφ j+ X j=0,...,j0−1, −δj>δ0 |x|2−n2 −δjψ j so that u − v = X j=0,...,j0−1, δ<δj<δ0 |x|2−n2 +δjφ j+ X j=0,...,j0−1, δ<−δj<δ0 |x|2−n2 −δjψ j
The estimate follows from collecting the above estimates. This completes the proof of Lemma 2.6.4. We proceed with the proof of Proposition 2.6.1. Choose
˜
δ ≥ inf(δ0, δ + 1)
such that ˜δ 6= ±δj, for all j ∈ N. Using the result of Proposition 2.2.1 we have
k∇ukL2 δ−1(M∗)+ k∇ 2uk L2 δ−2(M∗)≤ c (kf kL 2 δ0(M ∗)+ kukL2 δ(M∗))
Using the decomposition given in Proposition 2.1.1, we conclude that, near any puncture pi, we
have
|x|2∆ u = f − |x|2(δ
g− δeucl+ a) ∈ L2˜δ(BR∗i) We apply the previous result which yields the decomposition
u = v + X
δ<±δj<δ0
|x|2−n2 ±δjφ
where φ ∈ Ej. Next use the result of Lemma 2.6.3 and replace all |x|
2−n
2 ±δjφ by χ(i)W±(i)
j,φ to
get the decomposition
u = v + X δ<±δj<δ0 (|x|2−n2 ±δjφ − χ(i)W±(i) j,φ ) + X δ<±δj<δ0 χ(i)Wj,φ±(i)
Observe that the function ˜ u = v + X δ<±δj<δ0 (|x|2−n2 ±δjφ − χ(i)W±(i) j,φ ) ∈ L 2 ˜ δ(M ∗)
and also that γ2(∆
g+ a) ˜u = ˜f ∈ L2δ0(M∗). If ˜δ = δ0 then the roof is complete. If not, apply the same argument with u replaced by ˜u, f replaced by ˜f and δ replaced by ˜δ and proceed until the gap between δ and δ0 is covered.
2.6. THE DEFICIENCY SPACE 37
2.6.1
The kernel of A
δrevisited :
Thanks to the result of Proposition 2.6.1 we can state the :
Lemma 2.6.5. Fix δ < δ0 such that δ, δ06= ±δj, for j ∈ N. Assume that u ∈ L2δ(M∗) satisfied
γ2(∆gu + a u) = 0
in M∗. Then u ∈ L2δ0(M∗) provided the interval (δ, δ0) does not contain any ±δj, for some j ∈ N.
This Lemma is a direct consequence of the result of Proposition 2.6.1. It essentially states that the kernel of the operator Aδ does not change as δ remains in some interval which does not
contain any ±δj, for j ∈ N.
2.6.2
The deficiency space :
We now define
Definition 2.6.1. Given δ > 0, δ 6= δj, for all j ∈ N, the deficiency space Dδ is defined by
Dδ := Span {χ(i)Wj,φ±(i), : φ ∈ Ej, −δ < ±δj < δ}
Observe that the dimension of Dδ can be computed as follows
dimDδ = 2
X
j,δj<|δ| dim Ej
As a first by product, we obtain
Proposition 2.6.2. Given δ > 0, δ 6= δj, for all j ∈ N. Assume that Aδ is injective. Then the
operator ˜ Aδ: L2δ(M∗) ⊕ Dδ −→ L2δ(M∗) u 7−→ γ2(∆ gu + a u) is surjective and Ker A−δ= Ker ˜Aδ
As a consequence of the previous Proposition, we have the
Corollary 2.6.1. Given δ > 0, δ 6= δj, for all j ∈ N. Assume that Aδ is injective. Then
dim Ker A−δ= codim Im Aδ =
1 2dim Dδ
38 CHAPTER 2. WEIGHTED L2ANALYSIS ON A PUNCTURED MANIFOLD
Under the assumptions of the Corollary, we have
dim Ker A−δ = dim Ker ˜Aδ
and
dim Dδ = dim dim Ker A−δ+ codim Im Aδ
But, by duality, we have dim Ker A−δ= codim Im Aδ. The result then follows at once.
Chapter 3
Weighted C
2,α
analysis on a
punctured manifold
3.1
From weighted Lebesgue spaces to weighted H¨
older
spaces
As far as linear analysis is concerned the results of the previous sections are sufficient. However, we would like to apply them to nonlinear problems for which is will be more convenient to work in the framework of H¨older spaces. The purpose of this section is to explain how the analysis of the previous section can be extended to weighted H¨older spaces.
We begin with the definition of weighted H¨older spaces.
Definition 3.1.1. Given ` ∈ N, α ∈ (0, 1) and δ ∈ R, we define Cδ`,α(M∗) to be the space of
functions u ∈ Cloc`,α(M∗) for which the following norm
kukC`,α δ (M∗):= kukC `,α δ (MR)+ k X i=1 sup ρ∈(0,R) ρn−22 −δku(Exp pi(ρ · ))kC`,α( ¯B2−B1⊂TpiM ) is finite.
For example, the function γ2−n2 +δ ∈ C`,α
δ0 (M∗) if and only if δ ≥ δ0. It also follows directly from this definition that
Cδ`,α(M∗) ⊂ L2δ0(M∗) for all δ > δ0.
Lemma 3.1.1. The space (Cδ`,α(M∗), k · kC`,α δ (M∗)
) is a Banach space.
40 CHAPTER 3. WEIGHTED C2,α ANALYSIS ON A PUNCTURED MANIFOLD
Exercise 3.1.1. Show that the embedding
Cδ`,α(M∗) −→ Cδ`00,α0(M∗) is compact provided `0+ α0< ` + α and δ < δ0.
The last easy observation is that the operator Aδ : C `,α δ (M ∗) −→ C`,α δ (M ∗) u −→ γ2(∆ gu + au)
is well defined and bounded.
The extension of our results to weighted H¨older spaces rely on the following regularity result. Proposition 3.1.1. Assume that δ, δ0∈ R are fixed with δ < δ0. Further assume that the interval
[δ, δ0] does not contain any ±δ
j for j ∈ N. Then, there exists c = c(n, δ, δ0) > 0 such that for all
u, f ∈ L2 δ(B ∗ R(pi)) satisfying γ2(∆g+ a) u = f in M∗, if f ∈ Cδ0,α0 (M∗) then u ∈ C 2,α δ0 (M∗) and kukC2,α δ0 (M ∗)≤ c kf kC0,α δ0 (M ∗)+ kukL2 δ(M∗)
Before we proceed to the proof of this result, let us explain how it can be used.
Application # 1: The first application of the result of Proposition 3.1.1 is concerned with the kernel of the operator Aδ.
Lemma 3.1.2. Assume that δ ∈ R is fixed with δ 6= δj, for j ∈ N. Further assume that
u ∈ L2
δ(M∗) is a solution of
γ2(∆g+ a u) = 0
in M∗. Then u ∈ Cδ2,α(M∗).
In other words, in order to check the injectivity of Aδ, it is enough to check the injectivity of
Aδ, which in practical situation is easier to perform.
Application # 2 : Observe that, if u ∈ L2δ(M∗) is in the kernel of Aδ then u is also in the
kernel of Aδ0 for all δ0 ≤ δ since L2δ(M∗) ⊂ L2δ(M∗). However, it follows from Proposition 3.1.1 the the following is also true :
Lemma 3.1.3. Assume that δ ∈ R is fixed with δ 6= δj, for j ∈ N. Further assume that
u ∈ L2δ(M∗) is in the kernel of Aδ. Then u is also in the kernel of Aδ0 for all δ0 > δ for which [δ, δ0] does not contain any ±δj, for j ∈ N
3.1. FROM WEIGHTED LEBESGUE SPACES TO WEIGHTED H ¨OLDER SPACES 41 Application # 3 : The third application of the result of Proposition 3.1.1 is concerned with the extension of the result of Proposition 2.6.1 to weighted H¨older spaces and this will be useful when dealing with nonlinear differential operators. We have the :
Proposition 3.1.2. Given δ < δ0, δ, δ0 6= ±δj, for all j ∈ N. Assume that u ∈ L2δ(M∗) and
f ∈ Cδ0,α0 (M∗) satisfy
γ2(∆g+ a) u = f
in M∗. Then, there exists v ∈ Cδ2,α0 (M∗) such that
u − v ∈ Dδ,δ0 := Span {χ(i)W±(i)
j,φ , : φ ∈ Ej, δ < ±δj < δ 0} In addition kvkC2,α δ0 (M ∗)+ ku − vkDδ,δ0 ≤ c (kf kC0,α δ0 (M ∗)+ kukL2 δ(M∗)) for some constant c = c(n, δ, δ0) > 0.
There are important by products of this result :
Given δ, δ > δ0, δ 6= δj for all j ∈ N. Assume that Aδ is injective, then, according to the
result of Corollary 2.5.1, the operator A−δis surjective and hence there exists
G−δ: L2−δ(M∗) −→ L2−δ(M∗).
a right inverse for A−δ (i.e. A−δ◦ G−δ= I). In particular, given
f ∈ Cδ0,α(M∗) ⊂ L2−δ(M∗),
the function u := G−δf ∈ L2−δ(M∗) solves
A−δu = f
in M∗. Applying the result of Proposition 3.1.2, we see that there exists v ∈ C2,α
δ (M∗) such that u − v ∈ Dδ:= Span {χ(i)W ±(i) j,φ , : φ ∈ Ej, −δ < ±δj< δ} and in addition kvkC2,α δ (M∗)+ ku − vkDδ≤ c (kf kC 0,α δ (M∗)+ kukL 2 δ(M∗)) for some constant c = c(n, δ) > 0.
If δ ∈ (−δ0, δ0) and if Aδis injective. Then, according to the result of Lemma 3.1.3 the operator
A−δ0 is also injective for all δ0 ∈ (−δ0, δ0). Therefore, according to the result of Corollary 2.5.1 the operator Aδ0 is surjective. This implies that there exists
42 CHAPTER 3. WEIGHTED C2,α ANALYSIS ON A PUNCTURED MANIFOLD
a right inverse for Aδ0. In particular, if −δ0< δ0 < δ < δ0, given f ∈ Cδ0,α(M∗) ⊂ L2δ0(M∗), the function u := Gδ0f ∈ L2δ0(M∗) solves
Aδ0u = f
in M∗. Applying the result of Proposition 3.1.2, we see that u ∈ Cδ2,α(M∗) and in addition kukC2,α δ (M∗) + ku − vkDδ ≤ c (kf kC0,α δ (M∗) + kukL2 δ(M∗)) for some constant c = c(n, δ) > 0. Collecting these result, we have proven the :
Proposition 3.1.3. Given δ > −δ0, δ 6= δj, for all j ∈ N, let us assume that Aδ is injective,
then the operator
˜
Aδ : C`,αδ (M∗) ⊕ Dδ −→ Cδ`,α(M∗)
u −→ γ2(∆
gu + au)
is well defined, bounded and surjective. In addition dim Ker( ˜Aδ) = 12dimDδ.
In particular, under the assumptions of the Proposition, there exists an operator Gδ : Cδ0,α(M∗) −→ Cδ2,α(M∗) ⊕ Dδ.
which is a right inverse for the operator γ2(∆ g+ a).
As a special case, when δ ∈ (δ0, δ0), then Dδ is empty and the above statement simplifies into
the :
Proposition 3.1.4. Given δ ∈ (−δ0, δ0). Let us assume that Aδ is injective, then the operator
Aδ0 is an isomorphism for all δ0∈ (−δ0, δ0).
We now proceed with the proof of Proposition 3.1.1. We start with the :
Lemma 3.1.4. Assume that δ, δ0∈ R are fixed with δ < δ0. There exists c = c(n, δ, δ0) > 0 such
that for all u, f ∈ L2δ(M∗) satisfying
γ2(∆g+ a) u = f in M∗, if f ∈ Cδ0,α0 (M∗) then u ∈ C 2,α δ (M ∗) and kukC2,α δ (M∗)≤ c kf kC0,α δ0 (M ∗)+ kukL2 δ(M∗)
3.1. FROM WEIGHTED LEBESGUE SPACES TO WEIGHTED H ¨OLDER SPACES 43 Proposition 3.1.5. ([?], Theorem ??) Assume that ¯Ω0 ⊂⊂ Ω is fixed. There exists c = c(n, Ω0, Ω) > 0 such that for all u, f ∈ L2(Ω) satisfying
(∆g+ a) u = f
in Ω, if f ∈ C0,α( ¯Ω) then u ∈ C2,α
δ ( ¯Ω0) and
kukC2,α( ¯Ω0)≤ c kf kC0,α( ¯Ω)+ kukL2(Ω)
Close to the punctures, we use normal geodesic coordinates together with (2.1) and write the equation satisfied by u as
|x|2(∆euclu + O(|x|2) ∂xi∂xju + O(|x|) ∂xiu + O(1)u) = f For all r ∈ (0, R) we defined the rescaled functions
ˆ
u(x) = u(R x) and f (x) = f (R x)ˆ so that
|x|2(∆
euclu + O(Rˆ 2) ∂xi∂xju + O(Rˆ 2) ∂xiu + O(Rˆ 2) ˆu) = ˆf
in B2− ¯B1/2. Applying the result of Proposition 3.1.5 with Ω = B2− ¯B1 and Ω0= B3/2− ¯B3/4
we conclude that kˆukC2,α( ¯B 3/2−B3/4)≤ c k ˆf kC0,α( ¯B 2−B1)+ kˆukL2(B2− ¯B1) But we have k ˆf kC0,α( ¯B2−B1)≤ c R 2−n 2 +δ 0 kf kC0,α δ (M∗)≤ c R 2−n 2 +δkf k C0,α δ (M∗) and kˆukL2(B2− ¯B1)≤ c R 2−n 2 +δkukL2 δ(M∗) for some constant c = c(n, δ, δ0) > 0. Therefore, we conclude that
kˆukC2,α( ¯B 3/2−B3/4)≤ c R 2−n 2 +δ kf kC0,α δ0 (M ∗)+ kukL2 δ(M∗)
which by definition of the weighted H¨older norm, implies that kukC2,α δ ( ¯B ∗ 2R)≤ c kf kC0,α δ0 (M ∗)+ kukL2 δ(M ∗)
This completes the proof of the Lemma. The next result we will need reads :
