3Function spaces
III Analysis of Partial Differential Equations
3.1 The H¨older spaces
The most straightforward class of functions paces is the
C
k
spaces. These are
spaces based on classical continuity and differentiability.
Definition
(
C
k
spaces)
.
Let
U ⊆ R
n
be an open set. We define
C
k
(
U
) to be
vector space of all
u
:
U → R
such that
u
is
k
-times differentiable and the partial
derivatives D
α
u : U → R are continuous for |α| ≤ k.
We want to turn this into a Banach space by putting the supremum norm
on the derivatives. However, even
sup |u|
is not guaranteed to exist, as
u
may
be unbounded. So this doesn’t give a genuine norm. This suggests the following
definition.
Definition
(
C
k
(
¯
U
) spaces)
.
We define
C
k
(
¯
U
)
⊆ C
k
(
U
) to be the subspace of
all
u
such that D
α
u
are all bounded and uniformly continuous. We define a
norm on C
k
(
¯
U) by
kuk
C
k
(
¯
U)
=
X
|α|≤k
sup
x∈U
kD
α
u(x)k.
This makes C
k
(
¯
U) a Banach space.
In some cases, we might want a “fractional” amount of differentiability. This
gives rise to the notion of H¨older spaces.
Definition
(H¨older continuity)
.
We say a function
u
:
U → R
is H¨older
continuous with index γ if there exists C ≥ 0 such that
|u(x) − u(y)| ≤ C|x −y|
γ
for all x, y ∈ U.
We write
C
0,γ
(
¯
U
)
⊆ C
0
(
¯
U
) for the subspace of all H¨older continuous functions
with index γ.
We define the γ-H¨older semi-norm by
[u]
C
0,γ
(
¯
U)
= sup
x6=y∈U
|u(x) − u(y)|
|x − y|
γ
.
We can then define a norm on C
0,γ
(
¯
U) by
kuk
C
(0,γ
(
¯
U)
= kuk
C
0
(
¯
U)
+ [u]
C
0,γ
(
¯
U)
.
We say
u ∈ C
k,γ
(
¯
U
) if
u ∈ C
k
(
¯
U
) and D
α
u ∈ C
0,γ
(
¯
U
) for all
|α|
=
k
, and we
define
kuk
C
k,γ
(
¯
U)
= kuk
C
k
(
¯
U)
+
X
|α|=k
[D
α
u]
C
0,γ
(
¯
U)
.
This makes C
k,γ
(
¯
U) into a Banach space as well.
Note that C
0,1
(
¯
U) is the set of (uniformly) Lipschitz functions on U.