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The raising steps method. Applications to the L r Hodge theory in a compact riemannian manifold.

Eric Amar

To cite this version:

Eric Amar. The raising steps method. Applications to the

Lr

Hodge theory in a compact riemannian

manifold.. 2017. �hal-01158323v2�

(2)

The raising steps method. Application to the L r Hodge theory in a compact riemannian manifold.

Eric Amar

Abstract

Let

X

be a complete metric space and

a domain in

X.

The Raising Steps Method allows to get from local results on solutions

u

of a linear equation

Du=ω

global ones in

Ω.

It was introduced in [1] to get good estimates on solutions of

∂¯

equation in domains in a Stein manifold.

As a simple application we shall get a strong

Lr

Hodge decomposition theorem for

p−forms

in a compact riemannian manifold without boundary, and then we retrieve this known result by an entirely different and simpler method.

1 Introduction.

This work proposes a way for passing from local to global: the raising steps method, RSM for short. I introduce it precisely to get L

r

− L

s

estimates for solutions of the ∂ ¯ equation in Stein manifold in [1]. See also [2] to have result in case of intersection of domains in Stein manifolds.

The aim is to generalise it to the case where the ∂ ¯ operator is replaced by an abstract linear operator D acting on a domain in a complete metric space.

We shall deal with the following situation: we have a complete metric space X admitting partitions of unity (see condition (ii) below) and a measure µ.

A domain of X will be a connected open set Ω of X, relatively compact.

We are interested on solutions u of a linear equation Du = ω, in a domain Ω. Precisely fix a threshold s > 1. Suppose you have a global solution u on Ω of Du = ω with estimates L

s

(Ω) → L

s

(Ω). It may happen that we have a constrain:

∃K ⊂ L

s

(Ω) with s

the conjugate exponent for s, such that ∀u :: Du ∈ L

s

(Ω), ∀h ∈ K, hDu, hi = 0.

In this case, in order for a solution of Du = ω to exist, we need to have ω ⊥ K. With no constrain, we take K = {0}.

Very often this threshold will be s = 2, since Hilbert spaces are usually more tractable.

Now suppose that we have, for 1 ≤ r ≤ s, local solutions u on U ∩ Ω of Du = ω with estimates L

r

(Ω) → L

t

(U ∩ Ω) with a strict increase of the regularity, for instance 1

t = 1

r − τ, τ > 0 for any r ≤ s, then the raising steps method gives a global solution v of Dv = ω which is essentially in L

t

(Ω) if we start with a data ω in L

r

(Ω).

In particular we prove:

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Theorem 1.1. (Raising steps theorem) Under the assumptions above, there is a positive constant c such that for 1 ≤ r ≤ s, if ω ∈ L

r

(Ω), ω ⊥ K, there is a u ∈ L

t

(Ω) with 1

t = 1

r − τ, such that Du = ω + ˜ ω, with ω ˜ ∈ L

s

(Ω), ω ˜ ⊥ K and control of the norms.

From this theorem and the fact that there is a global solution v ∈ L

s

(Ω) to Dv = ¯ ω we get that u − v is the global solution we are searching for.

To illustrate the method, we shall apply it for the Poisson equation associated to the Hodge Laplacian in a compact riemannian boundary-less manifold (M, g).

On (M, g) we can define Sobolev spaces W

k,r

(Ω) (see [8]) and if M is compact these spaces are in fact independent of the metric. Moreover the Sobolev embeddings are true in this case and a chart diffeomorphism makes a correspondence between Sobolev spaces in R

n

and Sobolev spaces in M.

Let d be the exterior derivative on M and d

its adjoint; we define the Hodge laplacian acting from p differential forms to p differential forms to be: ∆ := dd

+ d

d. Because M is compact, we have that ∆ is self adjoint. The Poisson equation on M is, for a given p-form ω on M, to find a p-form u on M such that ∆u = ω.

Let H

p

be the set of p-harmonic forms in M, i.e. h ∈ H

p

⇐⇒ h ∈ C

p

(M ), ∆h = 0. We have:

∀h ∈ H

p

, h∆u, hi = hu, ∆hi = 0

hence, in order to solve ∆u = ω, we need to have ω ⊥ H

p

.

We derive, from a solution of the Poisson equation we get by use of the RSM, a L

rp

Hodge decomposition for p differential forms on M.

We shall use, for the local results, the classical ones. Let B be a ball in R

n

, then:

∀γ ∈ L

r

(B), ∃u ∈ W

2,r

(B) :: ∆

R

u = γ, kv

0

k

W2,r(B)

≤ Ckγk

Lr(B)

.

These non trivial estimates are coming from Gilbarg and Trudinger [6, Theorem 9.9, p. 230] and the constant C = C(n, r) depends only on n and r.

Then, together with the R.S.M., we get a solution of the Poisson equation for the Hodge laplacian.

Theorem 1.2. Let M be a compact, C

Riemannian manifold without boundary. For any r, 1 ≤ r ≤ n/2, if g is a p-form in L

rp

(M) ∩ H

p

there is a p-form v ∈ L

t

(M) such that ∆v = g and kv k

Lt

p(M)

≤ ckgk

Lr

p(M)

with 1 t = 1

r − 2 n .

Moreover for any r > 1 and any p form u solution of ∆u = g, we have u in W

p2,r

(M).

From this result we deduce the L

r

Hodge decomposition of p-forms.

Theorem 1.3. Let (M, g) be a compact riemannian manifold without boundary. We have the strong L

r

Hodge decomposition:

∀r, 1 ≤ r < ∞, L

rp

(M ) = H

rp

⊕ Im∆(W

p2,r

(M )) = H

pr

⊕ Imd(W

p1,r

(M )) ⊕ Imd

(W

p1,r

(M )).

The case r = 2 of this decomposition which gives us s = 2 as a threshold, goes to Morrey in 1966 and essentially all results in the L

2

case we use here are coming from the basic work of Morrey [10].

This decomposition is an already known result of C. Scott [12] but proved here by an entirely

different method. Critical to Scott’s proof is a nice L

r

Gaffney’s inequality which he proved and

used to get the L

r

Hodge decomposition, the same way than Morrey [10] did with the L

2

Gaffney’s

inequality [5] to get the L

2

Hodge decomposition.

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In the case of a compact manifold with boundary, G. Schwarz [11] proved also a L

r

Gaffney’s inequality to get the L

r

Hodge decomposition in that case, then he deduced of it a global L

r

solution for the equation ∆u = ω.

In the nice book by F.W. Warner [15], the author proved directly, without the use of Gaffney’s inequality, a global L

2

solution for the equation ∆u = ω in the case of a compact manifold without boundary. He deduced from it the L

2

Hodge decomposition.

Here we use the RSM plus the global L

2

solution for the equation ∆u = ω given by Warner [15], to get a global L

r

solution for the equation ∆u = ω and then recover the L

r

Hodge decomposition, without any Gaffney’s inequalities. Hence we get a completely different proof of the known L

r

Hodge decomposition.

Many important applications of the L

2

Hodge decomposition in cohomology theory and algebraic geometry are in the book by C. Voisin [14].

So it may be interesting to have a short proof of this important Hodge decomposition in the L

r

case.

Finally, in the last section we prove, by use of the "double manifold" technique:

Theorem 1.4. Let Ω be a domain in the smooth complete riemannian manifold M and ω ∈ L

rp

(Ω), then there is a p-form u ∈ W

p2,r

(Ω), such that ∆u = ω and kuk

W2,r

p (Ω)

≤ c(Ω)kωk

Lr p(Ω)

. And we make a short incursion in the domain of manifold with boundary:

Corollary 1.5. Let M be a smooth compact riemannian manifold with smooth boundary ∂M. Let ω ∈ L

rp

(M ). There is a p-form u ∈ W

p2,r

(M ), such that ∆u = ω and kuk

W2,r

p (M)

≤ ckωk

Lr p(M)

. Schwarz [11, Theorem 3.4.10, p. 137] proved a better theorem: you can prescribe the values of u on the boundary. But again the proof here is much lighter.

2 The Raising Steps Method.

We shall deal with the following situation: we have a complete metric space X admitting partitions of unity (see condition (ii) below) and a positive σ-finite measure µ.

2.1 Assumptions on the linear operator D.

We shall denote E

p

(X) the set of C

p

valued fonctions on X. This means that ω ∈ E

p

(X) ⇐⇒

ω(x) = (ω

1

(x), ..., ω

p

(x)). We put a punctual norm on ω ∈ E

p

(X), |ω(x)|

2

:= P

p

j=1

j

(x)|

2

and if U is an open set in X, we consider the Lebesgue space L

rp

(U ):

ω ∈ L

rp

(U) ⇐⇒ kωk

rLr p(U)

:=

Z

U

|ω(x)|

r

dµ(x) < ∞.

The space L

2p

(U ) is a Hilbert space with the scalar product hω, ω

i := R

U

P

p

j=1

ω

j

(x)¯ ω

j

(x) dµ(x).

We are interested in solution of a linear equation Du = ω, where D = D

p

is a linear operator acting on E

p

.

In order to have Du = ω, it may happen that we have a constrain: there is a subspace K ⊂ L

r

(Ω) such that

∀h ∈ K, ∀u :: Du ∈ L

r

(Ω), hDu, hi = 0 ⇐⇒ Du ⊥ K.

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The absence of constrain is done by setting K = {0}.

Now on the integer p will be fixed so the explicit mention of the integer p will be often omitted.

We shall make the following hypotheses.

Let Ω be a domain in X. There is a τ ≥ δ with 1 t = 1

r − τ, and a positive constant c

l

such that:

(i) Local Existence with Increasing Regularity (LEIR): for any x ∈ Ω, ¯ there is a ball B :=

B (x, R

x

) such that if ω ∈ L

rp

(B), we can solve Du

x

= ω in B

:= B(x, R

x

/2) with L

rp

(B) − L

tp

(B

) estimates, i.e. ∃u

x

∈ L

t

(B

), Du

x

= ω in B

and ku

x

k

Lt(B)

≤ c

l

kωk

Lr(B)

.

It may append, in the case X is a manifold, that we have a better regularity for the local existence:

(i’) Sobolev regularity: if ω ∈ L

rp

(B), we can solve Du

x

= ω in B

:= B(x, R

x

/2) with L

rp

(B ) − W

pα,r

(B

) estimates, i.e. ∃u

x

∈ W

pα,r

(B

), Du

x

= ω in B

and ku

x

k

Wpα,r(B)

≤ c

l

kωk

Lr(B)

.

By compactness we can cover Ω ¯ by a finite set of balls {B(x

j

, R

j

/2)}

j=1,...,N

of the previous form.

Set B

j

:= B(x

j

, R

j

), B

j

:= B (x

j

, R

j

/2). Set u

j

the local solution of Du

j

= ω with ku

j

k

Lt(Bj)

≤ c

l

kωk

Lr(Bj)

.

(ii) Partition of unity: If {B

j

}

j=1,...,N

is a covering of Ω, ¯ then there is an associated set of functions {χ

j

}

j=1,...,N

such that χ

j

has compact support in B

j

, ∀j = 1, ..., N, 0 ≤ χ

j

(x) ≤ 1 and P

N

j=1

χ

j

(x) = 1 for x ∈ Ω. ¯

(iii) Commutator condition: We set ∆

j

= ∆(χ

j

, u

j

) := χ

j

Du

j

− D(χ

j

u

j

). There is a constant δ > 0 such that, with 1

t = 1

r − δ, we have:

k∆

j

k

Lt(Bj)

≤ c(χ

j

)(kωk

Lr(Bj)

+ ku

j

k

Lr(Bj)

).

(iv) Global resolvability: We can solve Dw = ω globally in Ω with L

s

− L

s

estimates, i.e.

∃c

g

> 0, ∃w s.t. Dw = ω in Ω and kwk

Ls(Ω)

≤ c

g

kωk

Ls(Ω)

, provided that ω ⊥ K.

It may append, in the case X is a manifold, that we have a better regularity for the global existence:

(iv’) Sobolev regularity: We can solve Dw = ω globally in Ω with L

sp

− W

pα,s

estimates, i.e.

∃c

g

> 0, ∃w s.t. Dw = ω in Ω and kwk

Wα,s

p (Ω)

≤ c

g

kωk

Ls(Ω)

, provided that ω ⊥ K.

Theorem 2.1. (Raising steps theorem) Under the assumptions (i), (ii), (iii), (iv) above, there is a positive constant c

f

such that for 1 ≤ r ≤ s, if ω ∈ L

r

(Ω), ω ⊥ K there is a u = u

s

∈ L

t

(Ω) with

1 t = 1

r − τ, such that Du = ω + ˜ ω, with ω ˜ ∈ L

s

(Ω), ω ˜ ⊥ K and control of the norms.

If moreover we have (i’) then u ∈ W

pα,r

(Ω) with control of the norm.

Proof. Let r ≤ s and ω ∈ L

r

(Ω), ω ⊥ K; we start with the covering {B

j

}

j=1,...,N

and the local solution Du

j

= ω with 1

t = 1

r − τ, u

j

∈ L

t

(B

j

) given by hypothesis (i).

If (i’) is true, then we have u

j

∈ W

pα,r

(B

j

), Du

j

= ω.

Let {χ

j

}

j=1,...,N

be the partition of unity subordinate to {B

j

}

j=1,...,N

given by (ii). Because 0 ≤ χ

j

(x) ≤ 1 we have χ

j

u

j

∈ L

t

(Ω) hence

j

u

j

k

Lt(Ω)

≤ ku

j

k

Lt(Bj)

≤ c

l

kωk

Lr(Ω)

. (2.1) Set v

0

:= P

N

j=1

χ

j

u

j

. Then we have, setting now 1 t

0

= 1

r − τ :

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• v

0

∈ L

t0

(Ω) because χ

j

u

j

∈ L

t0

(Ω) for j = 1, ..., N, and kv

0

k

Lt0(Ω)

≤ ckωk

Lr(Ω)

with c = Nc

l

by (2.1). If (i’) is true, i.e. X is a manifold, then we can choose χ

j

∈ D(B

j

), i.e. in the space of C

functions of compact support in B

j

, hence kχ

j

u

j

k

Wα,r

p (Ω)

≤ cku

j

k

Wα,r

p (Bj)

≤ c

l

kωk

Lr(Ω)

. So

• v

0

∈ W

pα,r

(Ω) with kv

0

k

Wα,r

p (Ω)

≤ ckωk

Lr(Ω)

. We have

• Dv

0

= P

N

j=1

χ

j

Du

j

+ P

N

j=1

∆(χ

j

, u

j

).

Setting ω

1

(x) :=

N

X

j=1

∆(χ

j

, u

j

)(x), we get

∀x ∈ Ω, Dv

0

(x) =

N

X

j=1

χ

j

(x)ω(x) +

N

X

j=1

∆(χ

j

, u

j

)(x) = ω(x) + ω

1

(x) By hypothesis (iii), ∆(χ

j

, u

j

) ∈ L

s0

(B

j

) with 1

s

0

= 1

r − δ, hence ω

1

∈ L

s0

(Ω), with

1

k

Ls0(Ω)

≤ Gkωk

Lr(Ω)

(2.2)

and G = c

sl

N.

The regularity of ω

1

is higher by one step δ > 0 than that of ω. Moreover

∀h ∈ K, hω

1

, hi = hω, hi − hDv

0

, hi = 0 because hω, hi = 0 and hDv

0

, hi = 0.

If s

0

≥ s we notice that ω

1

∈ L

s

(Ω) because L

s0

(Ω) ⊂ L

s

(Ω) for Ω is relatively compact. So we are done by setting u

s

= v

0

, ω ˜ = ω

1

.

If s

0

< s we proceed by induction: we set t

1

such that 1 t

1

= 1 s

0

− τ = 1

r − δ − τ, and we use the same covering {B

j

}

j=1,...,N

and the same partition of unity {χ

j

}

j=1,...,N

and with ω

1

in place of ω, we get ∃v

1

∈ L

t1

(Ω), Dv

1

= ω

1

+ ω

2

and, setting s

1

such that 1

s

1

= 1 s

0

− δ = 1

r − 2δ, we have that the regularity of ω

2

raises of 2 times δ from that of ω.

We still have

∀h ∈ K, hω

2

, hi = hω

1

, hi − hDv

1

, hi = 0

so by induction, after a finite number k of steps, we get a s

k

≥ s. The linear combination u :=

P

k−1

j=0

(−1)

j

v

j

now gives Du = ω + (−1)

k

ω

k

with ω

k

⊥ K. And again we are done by setting

˜

ω = (−1)

k

ω

k

.

If (i’) is true, then we have v

0

∈ W

α,r

(Ω) and v

1

∈ W

α,s0

(Ω), kv

1

k

Wα,s0(Ω)

≤ ckω

1

k

Ls0(Ω)

. Hence by (2.2) we get kv

1

k

Wα,s0(Ω)

≤ cGkωk

Lr(Ω)

.

A fortiori, v

1

∈ W

α,r

(Ω) because r < s

0

with the same control of the norm. Likewise we have

∀j = 1, ..., k, kv

j

k

Wα,r(Ω)

≤ ckωk

Lr(Ω)

, hence the same control of the norm of u.

Corollary 2.2. Under the assumptions of the raising steps theorem and with the global assumption (iv), there is a constant c

f

> 0, such that for r ≤ s, if ω ∈ L

r

(Ω), ω ⊥ K there is a v ∈ L

t

(Ω) with t := min(s, t

0

) and 1

t

0

= 1

r − τ, such that

Dv = ω and v ∈ L

t

(Ω), kvk

Lt(Ω)

≤ ckωk

Lr(Ω)

.

If moreover (i’) and (iv’) are true then we have v ∈ W

α,r

(Ω) ∩ L

t

(Ω) with kv k

Wα,r(Ω)

≤ ckωk

Lr(Ω)

.

Proof. By the raising steps Theorem 2.1 we have a u ∈ L

t0

(Ω) such that Du = ω + ˜ ω with ω ˜ ∈ L

s

(Ω)

and ω ˜ ⊥ K; by hypothesis (iv) we can solve D˜ v = ˜ ω with v ˜ ∈ L

s

(Ω) so it remains to set v := u − v ˜

to get v ∈ L

t

(Ω) and Dv = ω.

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If moreover (i’) is true then we have u ∈ W

α,r

(Ω). If (iv’) is true then we can solve D˜ v = ˜ ω with

˜

v ∈ W

α,s

(Ω). Hence, because r ≤ s, with v := u − v ˜ we get v ∈ W

α,r

(Ω) and Dv = ω. The proof is complete.

3 Application to Poisson equation on a compact riemannian manifold.

3.1 Local existence with increasing regularity.

In order to have the local result, we choose a chart (V, ϕ := (x

1

, ..., x

n

)) so that g

ij

(y) = δ

ij

and ϕ(V ) = B where B = B

e

is an Euclidean ball centered at ϕ(y) = 0 and g

ij

are the components of the metric tensor w.r.t. ϕ.

Because I was unable to find an easy proof of the following theorem in the literature, I reprove it for the reader’s convenience.

Theorem 3.1. For any y ∈ M, there are open sets W ⊂ W ¯ ⊂ V ⊂ M, y ∈ W, such that we have:

∀ω ∈ L

rp

(W ), ∃u ∈ W

p2,r

(W ) :: ∆u = ω, kuk

W2,r(W)

≤ Ckωk

Lr(W)

.

Proof. Of course the operator d on p-forms is local and so is d

as a first order differential operator.

We start with a chart (V, ϕ) of M such that ϕ(y) = 0 ∈ R

n

and the metric tensor read in this chart at y is the identity.

Then the Hodge laplacian ∆

ϕ

read by ϕ in a ball B := B(0, R) ⊂ R

n

is not so different from that of R

n

in B when acting on p-forms in B. We set ∆

ϕ

ω

ϕ

= ∆

R

ω

ϕ

+ Aω

ϕ

, where ω

ϕ

is the p-form ω read in the chart (V, ϕ) and A is a matrix valued second order operator with C

smooth coefficients such that A : W

2,r

(B) → L

r

(B) with, for a R small enough kAvk

Lr(B)

≤ ckvk

W2,r(B)

.

This is true because at the point y ∈ V we are in the flat case and if R is small enough, the difference A := ∆

ϕ

− ∆

R

in operator norm W

2,r

(B) → L

r

(B) goes to 0 when R goes to 0, because ϕ ∈ C

2

and the metric tensor g is also C

2

.

We know that ∆

R

operates component-wise on the p-form γ, so we have

∀γ ∈ L

rp

(B), ∃v

0

∈ W

p2,r

(B) :: ∆

R

v

0

= γ, kv

0

k

W2,r(B)

≤ Ckγ k

Lr(B)

,

simply setting the component of v

0

to be the Newtonian potential of the corresponding component of γ in U. This way v

0

is linear with respect to γ. These non trivial estimates are coming from Gilbarg and Trudinger [6, Thorem 9.9, p. 230] and the constant C = C(n, r) depends only on n and r.

So we get ∆

R

v

0

+ Av

0

= γ + γ

1

, with

γ

1

= Av

0

⇒ kγ

1

k

Lr(B)

≤ ckv

0

k

W2,r(B)

≤ cCkγk

Lr(B)

. We solve again

∃v

1

∈ W

p2,r

(B) :: ∆

R

v

1

= γ

1

, kv

1

k

W2,r(B)

≤ Ckγ

1

k

Lr(B)

= C

2

ckγk

Lr(B)

, and we set

γ

2

:= Av

1

⇒ kγ

2

k

Lr(B)

≤ ckv

1

k

W2,r(B)

≤ Ckγ

1

k

Lr(B)

≤ C

2

c

2

kγk

Lr(B)

. And by induction:

∀k ∈ N , γ

k

:= Av

k−1

⇒ kγ

k

k

Lr(B)

≤ ckv

k−1

k

W2,r(B)

≤ Ckγ

k−1

k

Lr(B)

≤ C

k

c

k

kγk

Lr(B)

and

∃v

k

∈ W

p2,r

(B) :: ∆

R

v

k

= γ

k

, kv

k

k

W2,r(B)

≤ Ckγ

k

k

Lr(B)

≤ C

k+1

c

k

kγk

Lr(B)

.

(8)

Now we set v := P

j∈N

(−1)

j

v

j

. This series converges in norm W

2,r

(B ), provided that we choose the radius of the ball B small enough to have cC

2

< 1, and we get:

ϕ

v = ∆

R

v + Av = X

j∈N

(−1)

j

(∆

R

v

j

+ Av

j

) = γ, the last series converging in L

r

(B ).

In fact every step is linear and we get that v is linear in γ.

Going back to the manifold M with γ := ω

ϕ

and setting u

ϕ

:= v, W := ϕ

−1

(B ), we get the right estimates:

∃u ∈ W

2,r

(W ) :: ∆u = ω in W, kuk

W2,r(W)

≤ Ckωk

Lr(W)

,

because the Sobolev spaces for B go to the analogous Sobolev spaces for W in M.

Now the next corollary is precisely the result we are searching for.

Corollary 3.2. For any y ∈ M, there are open sets V, W ⋐ V, y ∈ W, such that we have for r ≤ 2:

∀ω ∈ L

rp

(V ), ∃u ∈ W

2,r

(W ) ∩ L

tp

(W ) :: ∆u = ω, kuk

W2,r(W)

≤ Ckωk

Lr(V)

, kuk

Lt

p(W)

≤ Ckωk

Lr p(V)

with 1 t = 1

r − 2

n , and ∇u ∈ W

p1,r

(W ).

Proof. The Theorem 3.1 gives u ∈ W

2,r

(W ) such that ∆u = ω, kuk

W2,r(W)

≤ Ckωk

Lr(V)

, hence we get that ∇u ∈ W

p1,r

(W ) with the same control: k∇uk

W1,r(W)

≤ Ckωk

Lr(V)

.

For the first statement it remains to apply the Sobolev embedding theorems which are true here.

So we are in a special case of the previous section with D := ∆ and, because ∆ is essentially self adjoint, we have here K := H

p

(M ) where H

p

(M ) is the space of C

harmonic p-forms in M.

We shall need the following lemma.

Lemma 3.3. Let ∆

ϕ

be a second order elliptic matrix operator with C

coefficients operating on p- forms v defined in U ⊂ R

n

. Let B := B(0, R) a ball in R

n

, B

:= B(0, R/2) and suppose that B ⋐ U.

Then we have an interior estimate: there are constants c

1

, c

2

depending only on n = dim

R

M, r and the C

1

norm of the coefficients of ∆

ϕ

in B ¯ such that

∀v ∈ W

p2,r

(B), kvk

W2,r(B)

≤ c

1

kv k

Lr(B)

+ c

2

k∆

ϕ

vk

Lr(B)

. (3.3) Proof. For a 0-form this lemma is exactly [6, Theorem 9.11].

For p-forms we cannot avoid the use of deep results on elliptic systems of equations.

Let v be a p-form in B ⊂ R

n

. We use the interior estimates in [10, §6.2, Thm 6.2.6]. In our context, second-order elliptic system, and with our notations, with r > 1, we get:

∃C > 0, ∀v ∈ W

p2,r

(B), kv k

W2,r(B)

≤ c

1

R

−2

kvk

Lr(B)

+ c

2

k∆

ϕ

vk

Lr(B)

, already including the dependence in R.

The constants c

1

, c

2

depend only on r, n := dimM and the bounds and moduli of continuity of all the coefficients of the matrix ∆

ϕ

. (In [10], p. 213: the constant depends only on E and on E

.)

In particular, if ∆

ϕ

has its coefficients near those of ∆

R

in the C

1

norm, then the constants c

1

, c

2

are near the ones obtained for ∆

R

.

(9)

Now we deduce from it local interior regularity for the laplacian on a smooth compact manifold without boundary.

Lemma 3.4. Let (M, g) be a riemannian manifold. For x ∈ M, R > 0, we take a geodesic ball B(x, R) such that, read in a chart (V, ϕ), B(x, R) ⋐ V, the metric tensor at the center is the identity.

We have a local Calderon Zygmund inequality on the manifold M. For any r > 1, there are constants c

1

, c

2

depending only on n = dim

R

M, r and R, such that:

∀u ∈ W

2,r

(B (x, R)), kuk

W2,r(B(x,R/2))

≤ c

1

kuk

Lr(B(x,R))

+ c

2

k∆uk

Lr(B(x,R))

.

Proof. We transcribe the problem in R

n

by use of a coordinates path (V, ϕ) exactly the same way we did to prove Theorem 3.1. The Hodge laplacian is the second order elliptic matrix operator

ϕ

with C

coefficients operating in ϕ(V ) ⊂ R

n

. By the choice of a R small enough we can apply Lemma 3.3, to the euclidean balls B

:= B

e

(0, R

e

) ⊂ ϕ(B(x, R/2)), B := B

e

(0, R

e

) ⊂ ϕ(B (x, R)) and we get, with u

ϕ

the p-form u read in the chart (V, ϕ),

ku

ϕ

k

W2,r(B)

≤ c

1

R

−2

ku

ϕ

k

Lr(B)

+ c

2

k∆

ϕ

u

ϕ

k

Lr(B)

.

The Lebesgue measure on U and the canonical measure dv

g

on B(x, R) are equivalent; so the Lebesgue estimates and the Sobolev estimates up to order 2 on U are valid in B (x, R) up to a constant.

So passing back to M, we get, with A := ϕ

−1

(B), A

:= ϕ

−1

(B

) kuk

W2,r(A)

≤ c

1

kuk

Lr(A)

+ c

2

k∆uk

Lr(A)

).

So taking a smaller ball centered at x we end the proof of the lemma.

Theorem 3.5. Let M be a compact C

Riemannian manifold without boundary. We have:

∀ω ∈ L

rp

(M) ∩ H

p

(M )

, r ∈ (1, n

2 ); ∃u ∈ L

sp

(M ) :: ∆u = ω, with 1

s = 1 r − 2

n . Moreover u ∈ W

2,r

(M), kuk

W2,r(M)

≤ ckωk

Lr(M)

.

Proof. First by duality we get the range r > 2. For this we shall proceed as we did in [1], using an avatar of the Serre duality [13]. We take t as in corollary 2.2.

Let g ∈ L

tp

(M) ∩ H

p

(M )

, we want to solve ∆v = g, with t

> 2 and t

conjugate to t.

We know by the previous part that, with r ≤ 2,

∀ω ∈ L

rp

(M ) ∩ H

p

(M )

, ∃u ∈ L

tp

(M), ∆u = ω, kuk

Lt(M)

≤ ckωk

Lr(M)

. (3.4) Consider the linear form

∀ω ∈ L

rp

(M ), L(ω) := hu, gi,

where u is a solution of (3.4); in order for L(ω) to be well defined, we need that if u

is another solution of ∆u

= ω, then hu − u

, gi = 0; hence we need that g must be "orthogonal" to p-forms ϕ such that ∆ϕ = 0 which is precisely our assumption.

Hence we have that L(f) is well defined and linear; moreover

|L(f )| ≤ kuk

Lt(M)

kgk

Lt(M)

≤ ckωk

Lr(M)

kgk

Lt(M)

.

So this linear form is continuous on ω ∈ L

rp

(M) ∩ H

p

(M )

. By the Hahn Banach Theorem there is

a form v ∈ L

rp

(M ) such that:

(10)

∀ω ∈ L

rp

(M ) ∩ H

p

(M )

, L(ω) = hω, vi = hu, gi.

But ω = ∆u, so we have, because ∆ is essentially self adjoint and M is compact, hω, vi = h∆u, vi = hu, ∆vi = hu, gi, for any u ∈ L

tp

(M) :: ∆u ∈ L

rp

(M ). In particular for u ∈ C

p

(M ). Now the hypothesis (iii) gives that ∆v = g in L

tp

(M ), with v ∈ L

rp

(M ). So we get:

∀g ∈ L

tp

(M ) ∩ H

p

(M )

, ∃v ∈ L

rp

(M), ∆v = g.

It remains to prove the moreover. The condition (i’) is true by the local existence Theorem 3.1.

The condition (iv’) is true in the case of the Hodge Laplacian on a compact boundary-less manifold by Morrey’s results for L

2

(M), so we are done for r ≤ 2.

For r > 2, Lemma 3.4 gives us by compactness that there is a smaller R > 0 and bigger constants c

1

, c

2

such that:

∀x ∈ M, kuk

W2,r(B(x,R/2))

≤ c

1

kuk

Lr(B(x,R))

+ c

2

k∆uk

Lr(B(x,R))

. (3.5) We take for u our global solution in L

rp

(M). We have that ∆u = ω ∈ L

rp

(M ) hence we can apply the estimate (3.5) to u:

∀x ∈ M, kuk

W2,r(B(x,R/2))

≤ c

1

kuk

Lr(B(x,R))

+ c

2

k∆uk

Lr(B(x,R))

≤ Ckωk

Lr(M)

.

Now it remains to cover M with a finite set of balls of the type B(x, R/2) to end the proof.

3.2 The L

r

Hodge decomposition.

In order to deduce the Hodge decomposition from the existence of a good solution to the Poisson equation, we shall need a little bit more material.

3.2.1 Basic facts.

Let H

p2

be the set of harmonic p-forms in L

2

(M ), i.e. p-form ω such that ∆ω = 0, which is equivalent here to dω = d

ω = 0.

The classical L

2

theory of Morrey [10] gives, on a compact manifold M without boundary:

H

p

:= H

2p

⊂ C

(M ) [ [10], (vi) p. 296]

dim

R

H

p

< ∞ [ [10], Theorem7.3.1].

This gives the existence of a linear projection from L

rp

(M ) → H

p

:

∀v ∈ L

rp

(M ), H (v) :=

N

X

j=1

hv, e

j

ie

j

where {e

j

}

j=1,...,N

is an orthonormal basis for H

p

. This is meaningful because v ∈ L

rp

(M) can be integrated against e

j

∈ H

p

⊂ C

(M ). Moreover we have v − H(v) ∈ L

rp

(M ) ∩ H

p

in the sense that

∀h ∈ H

p

, hv − H(v ), hi = 0; it suffices to test on h := e

k

. We get hv − H(v), e

k

i = hv, e

k

i −

*

N

X

j=1

hv, e

j

ie

j

, e

k

+

= hv, e

k

i − hv, e

k

i = 0.

Let v ∈ L

rp

(M). Set h := H(v) ∈ H

p

, and ω := v − h. We have that ∀k ∈ H

p

, hω, ki = hv − H(v), ki = 0. Hence we can solve ∆u = ω with u ∈ W

p2,r

(M ) ∩ L

sp

(M ). So we get v = h + ∆u which means:

L

rp

(M ) = H

pr

⊕ Im∆(W

p2,r

(M )).

We have a direct decomposition because if ω ∈ H

p

∩ Im∆(W

p2,r

(M )), then ω ∈ C

(M ) and

ω = ∆u ⇒ ∀k ∈ H

p

, hω, ki = 0

(11)

so choosing k = ω ∈ H

p

we get hω, ωi = 0 hence ω = 0.

Now we are in position to prove:

Theorem 3.6. If (M, g) is a compact riemannian manifold without boundary; we have the strong L

r

Hodge decomposition:

∀r, 1 ≤ r < ∞, L

rp

(M ) = H

rp

⊕ Imd(W

p1,r

(M )) ⊕ Imd

(W

p1,r

(M )).

Proof. We already have

L

rp

(M ) = H

pr

⊕ Im∆(W

p2,r

(M ))

where ⊕ means uniqueness of the decomposition.

So: ∀ω ∈ L

rp

(M ) ∩ H

r⊥p

, ∃u ∈ W

p2,r

(M) :: ∆u = ω.

From ∆ = dd

+ d

d we get ω = d(d

u) + d

(du) with du ∈ W

p1,r

(M ) and d

u ∈ W

p1,r

(M), so

∀ω ∈ L

rp

(M ) ∩ H

r⊥p

, ∃α ∈ W

p−11,r

(M), ∃β ∈ W

p+11,r

(M) :: ω = α + β,

simply setting α = d(d

u), β = d

(du), which gives α ∈ d(W

p−11,r

(M )) and β ∈ d

(W

p+11,r

(M )) and this proves the existence.

The uniqueness is given by Lemma 6.3 in [12] and I copy this simple (but nice) proof here for the reader’s convenience.

Suppose that α ∈ W

p−11,r

(M ), β ∈ W

p+11,r

(M ), h ∈ H

p

satisfy dα + d

β + h = 0.

Let ϕ ∈ C

p

(M ), because of the classical C

-Hodge decomposition, there are η ∈ C

p−1

(M ), ω ∈ C

p+1

(M ) and τ ∈ H

p

satisfying ϕ = dη + d

ω + τ.

Notice that hd

β, dηi = hβ, d

2

ηi = hβ, 0i = 0 and hh, dηi = hd

h, ηi = 0, by the duality between d and d

. Linearity then gives

hdα, dηi = hdα + d

β + h, dηi = h0, dηi = 0. (3.6) Finally we have

hdα, ϕi = hdα, dηi + hdα, d

ωi + hdα, τ i

= 0 +

α, d

∗2

ω

+ hα, dτ i by (3.6)

= hα, 0i + hα, 0i because d

∗2

= 0 and τ ∈ H

p

= 0.

Since C

p

(M ) is dense in L

rp

(M ), r

being the conjugate exponent of r, and ϕ is arbitrary, we see that dα = 0. Analogously, we see that d

β = 0 and it follows that h = 0.

4 Case of Ω a domain in M.

Let Ω be a domain in a C

smooth complete riemannian manifold M, compact or non compact;

we want to show how the results in case of a compact boundary-less manifold apply to this case.

A classical way to get rid of a "annoying boundary" of a manifold is to use its "double". For instance: Duff [4], Hörmander [9, p. 257]. Here we copy the following construction from [7, Appendix B].

Let N be a relatively compact domain of M such that ∂N is a smooth hypersurface and Ω ¯ ⊂ N.

The "Riemannian double" D := D(N) of N, obtained by gluing two copies of N along ∂N, is a

compact Riemannian manifold without boundary. Moreover, by its very construction, it is always

possible to assume that D contains an isometric copy of the original domain N, hence of the original

Ω. We shall also write Ω for its isometric copy to ease notations.

(12)

We shall need the following difficult result by N. Aronszajn, A. Krzywicki and J. Szarski [3], a strong continuation property, which says that if, for any compact set K ⊂ M, the p-form ω satisfies the following inequality

hdω, dωi + hd

ω, d

ωi ≤ C(K)hω, ωi (4.7) uniformly on K, then if ω is zero to infinite order at a point x

0

∈ M, we have that ω ≡ 0. The regularity conditions on ω are to be L

2

(M ) with strong L

2

derivatives. The p-form ω must vanish at x

0

with all derivatives in "L

1

mean", which is also much weaker than the usual notion.

We shall apply it to the compact manifold D and with a C

harmonic p-form h, hence which satisfies inequality (4.7), and which is zero on an open non void set, which also implies a zero of infinite order.

The main lemma of this section is:

Lemma 4.1. Let ω ∈ L

rp

(Ω), then we can extend it to ω

∈ L

rp

(D) such that: ∀h ∈ H

p

(D), hω

, hi

D

= 0.

Proof. Recall that H

p

(D) is the vector space of harmonic p-form in D, it is of finite dimension K

p

and H

p

(D) ⊂ C

(D).

Make an orthonormal basis {e

1

, ..., e

Kp

} of H

p

(D) with respect to L

2p

(D), by the Gram-Schmidt procedure.We get he

j

, e

k

i

D

:= R

D

e

j

e

k

dV = δ

jk

.

Set λ

j

:= hω 1

, e

j

i = hω, e

j

1

i, j = 1, ..., K

p

, which makes sense since e

j

∈ C

(D) ⇒ e

j

∈ L

(D), because D is compact.

We shall see that the system {e

k

1

D\Ω

}

k=1,...,Kp

is a free one. Suppose this is not the case, then it will exist γ

1

, ..., γ

Kp

, not all zero, such that P

Kp

k=1

γ

k

e

k

1

D\Ω

= 0 in D\Ω. But the function h :=

Kp

X

k=1

γ

k

e

k

is in H

p

(D) so if h is zero in D\Ω which is non void, then h ≡ 0 in D by the N. Aronszajn, A.

Krzywicki and J. Szarski [3] result. This is not possible because the e

k

make a basis for H

p

(D). So the system {e

k

1

D\Ω

}

k=1,...,Kp

is a free one.

We set γ

jk

:=

e

k

1

D\Ω

, e

j

1

D\Ω

hence we have that det{γ

jk

} 6= 0. So we can solve the linear system to get {µ

k

} such that

∀j = 1, ..., K

p

,

Kp

X

k=1

µ

k

e

k

1

D\Ω

, e

j

= λ

j

. (4.8)

We put ω

′′

:= P

Kp

j=1

µ

j

e

j

1

D\Ω

and ω

:= ω 1

− ω

′′

1

D\Ω

= ω − ω

′′

. From (4.8) we get

∀j = 1, ..., K

p

, hω

, e

j

i

D

= hω, e

j

i − hω

′′

, e

j

i = λ

j

Kp

X

k=1

µ

k

e

k

1

D\Ω

, e

j

= 0.

So the p-form ω

is orthogonal to H

p

. Moreover ω

|Ω

= ω and clearly ω

′′

∈ L

rp

(D) being a finite combination of e

j

1

D\Ω

, so ω

∈ L

rp

(D) because ω itself is in L

rp

(D). The proof is complete.

Now let ω ∈ L

r

(Ω) and see Ω as a subset of D; then extend ω as ω

to D by Lemma 4.1.

(13)

By the results on the compact manifold D, because ω

⊥ H

p

(D), we get that there exists u

∈ W

p2,r

(D), u

⊥ H

p

(D), such that ∆u

= ω

; hence if u is the restriction of u

to Ω we get u ∈ W

p2,r

(Ω), ∆u = ω in Ω.

Hence we proved

Theorem 4.2. Let Ω be a domain in the smooth complete riemannian manifold M and ω ∈ L

rp

(Ω), then there is a p-form u ∈ W

p2,r

(Ω), such that ∆u = ω and kuk

Wp2,r(Ω)

≤ c(Ω)kωk

Lr

p(Ω)

. As for domains in R

n

, there is no constrain for solving the Poisson equation in this case.

Corollary 4.3. (Of the proof ) Let M be a smooth compact riemannian manifold with smooth boundary ∂M. Let ω ∈ L

rp

(M ). There is a form u ∈ W

p2,r

(M), such that ∆u = ω and kuk

W2,r

p (M)

≤ ckωk

Lr

p(M)

.

Proof. We can build the "double manifold" D := D(M ) which is compact without boundary.

Copying the proof of Theorem 4.2 we extend the form ω defined on M, viewed as a subset in D, to a form ω

in L

rp

(D) orthogonal to H

p

(D) so there is a u

:: ∆u

= ω

in D. We just take u := u

|M

to finish the proof.

References

[1] E. Amar. The raising steps method. Application to the ∂ ¯ equation in Stein manifolds. J.

Geometric Analysis, 26(2):898–913, 2016. 1, 8

[2] E. Amar. On estimates for the ∂ ¯ equation in Stein manifolds. J. London Math. Soc., 2017. To appear. 1

[3] N. Aronszajn, A. Krzywicki, and J. Szarski. A unique continuation theorem for exterior differ- ential forms on riemannian manifolds. Ark. Mat., 4:417–453, 1962. 11

[4] G.F.D. Duff. Differential forms in manifolds with boundary. Ann. of Math., 56:115–127, 1952.

10

[5] M. P. Gaffney. Hilbert space methods in the theory of harmonic integrals. Amer. Math. Soc., 78:426–444, 1955. 2

[6] D. Gilbarg and N. Trudinger. Elliptic Partial Differential equations, volume 224 of Grundlheren der mathematischen Wissenschaften. Springer, 1998. 2, 6, 7

[7] B. Guneysu and S. Pigola. Calderon-Zygmund inequality and Sobolev spaces on noncompact riemannian manifolds. Advances in Mathematics, 281:353–393, 2015. 10

[8] E. Hebey. Sobolev spaces on Riemannian manifolds., volume 1635 of Lecture Notes in Mathe- matics. Springer-Verlag, Berlin, 1996. 2

[9] L. Hörmander. The Analysis of Linear Partial Differential Operators III, volume 274 of

Grundlehren der mathematischen Wissenschften. Springer, 1994. 10

(14)

[10] C. B. Morrey. Multiple Integrals in the Calculus of Variations, volume 130 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag Berlin Heidelberg New York, 1966. 2, 7, 9

[11] G. Schwarz. Hodge decomposition - A method for solving boundary values problems, volume 1607 of Lecture Notes in Mathematics. Springer - Verlag Berlin Heidelberg, 1995. 3

[12] C. Scott. L

p

theory of differential forms on manifolds. Transactions of the Americain Mathe- matical Society, 347(6):2075–2096, 1995. 2, 10

[13] J-P. Serre. Un théorème de dualité. Comment. Math. Helv., 29:9–26, 1955. 8

[14] C. Voisin. Théorie de Hodge et géométrie algébrique complexe., volume 10 of Cours spécialisé.

S.M.F., 2002. 3

[15] F. W. Warner. Foundations of Differentiable Manifolds and Lie Groups, volume 94 of Graduate

texts in mathematics. Springer-Verlag, 1983. 3

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