Schauder Theory on Manifolds
Let M be a compact Riemannian manifold of dimension n with possibly non-empty boundary. We can define Hölder norms and spaces on M as follows. First, recall the definition of the
space C k (M ) — as the set of k-times differentiable functions on M , equipped with the C k norm
corresponding to the metric g of M . This norm is
|u|C k (M ) = |u|0,M +
k
X
sup ∇l u(x)g ,
l=1 x∈M
where | · |0,M is the supremum norm on M and k∇l ukg refers to the norm of l tensor ∇l u induced
by the metric g.
Next, we define the Hölder coefficient. For x, y ∈ M , let d(x, y) denote the geodesic distance
between x and y. Then define
|u(x) − u(y)|
[u]α,M ≡ sup
α
x,y∈M (d(x, y)) )
for any function u ∈ C 0 (M ). Defining a Hölder semi-norm for tensors is a bit more complicated since
we can not write T (x)−T (y) for a tensor T . But we can proceed in this way: let τ(x,y) : Tx M → Ty M
denote the isomorphism of tangent spaces given by parallel transport along a unique geodesic from
x to y. (We’ll take our manifolds to be complete so that geodesics connecting any two points of M
∗
always exist.) Then τ(x,y) induces a map τ(x,y)
: Ty∗ M → Tx∗ M and thus also a map between the
fibers of any bundle obtained from tensor powers of T ∗ M . We can now define the Hölder coefficient
of a tensor T ∈ Γ(T k,0 M ) as
[T ]α,M ≡ sup
∗
kT (x) − τ(x,y)
(T (y))kg
(d(x, y))α
x,y∈M
,
define the k th order Hölder coefficient as
[u]k,α,M ≡ [∇k u]α,M
and the C k,α Hölder norm of a function u ∈ C k (M ) as
|u|C k,α (M ) ≡ |u|C k (M ) + [u]k,α,M .
Finally, the space C k,α (M ) is define as the space of C k functions on M whose C k,α norm is finite.
To complete the picture of Hölder spaces on manifolds, we’d like to be able to define a C k,α
section T of a tensor bundle B over M . We already know how to define the supremum norm
of T and its covariant derivatives. Define the Hölder coefficient of T using parallel transport as
above. Then the space C k,α Γ(B) consists of C k sections of the bundle B with finite (k, α) Hölder
coefficient.
Note that if Omega ⊆ M is a domain, then we can consider the Hölder norms on Ω just by
taking all suprema in the above definitions to be over Ω and not M . Recall that in a sufficiently
small domain Ω, we can define Riemannian normal coordinates in which the metric and its first
derivatives can be well-controlled. In particular, we find that gij (x) = δij and gij,k (x) = 0 at a
point x in the centre of the normal coordinate chart, and so by continuity, we can say that there
1
exists a number r(x) such that if Ω is any domain centred on x of diameter less than r(x) then
1
in Ω. Moreover, by compactness of M , r(x) is bounded below, so that
|gij − δij |0 + |gij,k |0 ≤ 10
there is a minimal diamter rg with the property that the metric bounds given above hold in any
domain of radius less than rg . Call rg the radius of uniformity of the metric g. We now have this
result.
Lemma. Let Ω be any domain of diameter less than rg . Then there exists a constant C depending
only on n so that
1 δ
|u|
≤ |u|C k,α (Ω) ≤ C|u|δk,α,Ω
C k,α,Ω
for any function u ∈ C k,α (M ). Here, | · |δk,α,Ω refers to the C k,α norm with respect to the Euclidean
metric δ possessed by Ω when viewed as a subset of Rn using the local Riemannian normal coordinate
system.
Proof. First switch to Riemannian normal coordinates. Now the estimates on the supremum norm
terms is straightforward: express covariant derivatives in terms of ordinary derivatives and the
Christoffel symbols, and then apply the metric bounds. The Hölder coefficient bounds are somewhat
more complicated since they involve parallel transport. But recall that parallel transport is defined
in terms of solving an ODE with initial conditions. Thus according to the existence/uniequeness
theorem for first order ODEs we have bounds on the parallel transport in terms of the initial
conditions. Thus we can use these to get estimates on the Hölder coefficient.
The reason for proving this lemma is to be able to apply the Schauder theory for domains in
Rn that we developed earlier to the manifold setting. We’ll always start with quantities globally
defined on M , but then analyze their local properties using normal coordinates. We’ll then need
the lemma to pass our bounds back and forth between the local coordinate patch and the manifold.
Consider now the differential operator L = aij ∇i ∇j + bi ∇i + c where a is a C 0,α section of the
bundle of symmetric, positive definite tensors over M , b is a C 0,α vector field on M and c ∈ C 0,α (M ).
Furthermore, suppose L satisfies the structure conditions
(E-S1 )
kakC 0,α (M ) + kbkC 0,α (M ) + kckC 0,α (M ) ≤ Λ
(E-S2 )
λkξk2 ≤ aij (x)ξi ξj ≤ Λkξk2
for all x ∈ M and ξ ∈ Rn .
We will study the Dirichlet problem
Lu = f
in M
u=g
on ∂M
where f ∈ C 0,α (M ) and g ∈ C 2,α (M ).
We would like to prove the following elliptic estimate.
Theorem. Let L be as above. Then there exists a constant C = C(M, λ, Λ) so that the estimate
|u|C 2,α (M ) ≤ C |Lu|C 0,α (M ) + |u|0,M
holds for any function u ∈ C 2,α (M ).
2
Proof. Let (Ui , φi ) be a covering of M by a finite number (by compactness) of Riemannian normal
coordinate charts of diameter less than rg . Now,
X
|u|C 2 (M ) ≤
|u|C 2 (Ui )
i
≤C
X
δ
|u ◦ φ−1
i |C 2 (φi (Ui ))
(by the metric bounds)
i
≤C
X
≤C
X
δ
|u ◦ φ−1
i |k,α,φi (Ui )
i
−1
δ
|φ∗i L(u ◦ φ−1
i )|0,α,φi (Ui ) + |u ◦ φi |0,φi (Ui )
i
where φ∗i L denotes the operator L written in local coordinates in φi (Ui ). By the metric bounds
once again,
X
|u|C 2 (M ) ≤ C
|Lu|C 0,α (Ui ) + |u|0,Ui
i
≤ C |Lu|C 0,α (M ) + |u|0,M .
Next, we must estimate the Hölder coefficient. As usual, we consider two cases: if x and y
are two points in M with d(x, y) < rg and d(x, y) ≥ rg . In the first case, let (Ω, φ) be a normal
coordinate neighbourhood containing both x and y. Then if T denotes ∇2 u,
∗
kT (x) − τ(x,y)
T (y)kg
(d(x, y))α
≤ [T ]gα,Ω
≤ C[φ∗ (T )]δα,φ(Ω)
(by the metric bounds)
≤ C |φ∗ L(u ◦ φ)|δ0,α,φ(Ω) + |u ◦ φ|0,φ(Ω)
≤ C |Lu|C 0,α (Ω) + |u|0,Ω
by the metric bounds)
≤ C |Lu|C 0,α (M ) + |u|0,M .
In the second case,
∗
kT (x) − τ(x,y)
T (y)kg
(d(x, y))α
≤ C|T |0,M rgα
≤ C|u|C k (M )
≤ C |Lu|C 0,α (M ) + |u|0,M
as before. To conclude, we take the supremum of the difference quotient above, and combine this
Hölder coefficient estimate with the ordinary derivative estimates.
3