Recall Theorem 34.5.1 which is listed here for convenience.
Theorem 37.1.1Suppose u,u_{,i}∈ L_{loc}^{p}
(ℝn )
for i = 1,
⋅⋅⋅
,n and p > n. Then u has arepresentative, still denoted by u, such that for allx,y ∈ℝ^{n},
( )1 ∕p
∫ p (1−n∕p)
|u(x) − u (y)| ≤ C B(x,2|y−x|)|∇u |dz |x − y| . (37.1.13)
(37.1.13)
This amazing result shows that every u ∈ W^{m,p}
(ℝn)
has a representative which is
continuous provided p > n.
Using the above inequality, one can give an important embedding theorem.
Definition 37.1.2Let X,Y be two Banach spaces and let f : X → Y be afunction. Then f is a compact map if whenever S is a bounded set in X, it followsthat f
(S )
is precompact in Y .
Theorem 37.1.3Let U be a bounded open set and for u a function defined onℝ^{n}, let r_{U}u
(x)
≡ u
(x )
for x ∈U. Then if p > n, r_{U} : W^{1,p}
n
(ℝ )
→ C
(-)
U
iscontinuous and compact.
Proof:First suppose u_{k}→ 0 in W^{1,p}
(ℝn)
. Then if r_{U}u_{k} does not converge to 0, it
follows there exists a sequence, still denoted by k and ε > 0 such that u_{k}→ 0 in
W^{1,p}
(ℝn )
but
||rUuk||
_{∞}≥ ε. Selecting a further subsequence which is still denoted by k,
you can also assume u_{k}
(x)
→ 0 a.e. Pick such an x_{0}∈ U where this convergence takes
place. Then from 37.1.13, for all x ∈U,
|uk (x)| ≤ |uk(x0)|+ C ||uk|| n diam (U)
1,p,ℝ
showing that u_{k} converges uniformly to 0 on U contrary to
||rUuk||
_{∞}≥ ε. Therefore, r_{U}
is continuous as claimed.
Next let S be a bounded subset of W^{1,p}
n
(ℝ )
with
||u||
_{1,p}< M for all u ∈ S. Then for
u ∈ S
∫ p
rpmn ([|u| > r]∩ U) ≤ |u|dmn ≤ M p
[|u|>r]∩U
and so
m ([|u| > r]∩ U ) ≤ M-p.
n rp
Now choosing r large enough, M^{p}∕r^{p}< m_{n}
(U)
and so, for such r, there exists x_{u}∈ U
such that
1−n∕p
|u (x)| ≤ |u(xu)|+CM diam (U)
≤ r+ CM diam (U )1− n∕p
showing that
{r u : u ∈ S}
U
is uniformly bounded. But also, for x,y∈U,37.1.13
implies
1− n
|u(x) − u (y)| ≤ CM |x − y | p
showing that
{rUu : u ∈ S}
is equicontinuous. By the Ascoli Arzela theorem, it follows
r_{U}
(S )
is precompact and so r_{U} is compact.
Definition 37.1.4Let α ∈ (0,1] and K a compact subset of ℝ^{n}
C α(K ) ≡ {f ∈ C (K ) : ρα(f)+ ||f || ≡ ||f||α < ∞ }
where
||f|| ≡ ||f||∞ ≡ sup{|f (x)| : x ∈ K}
and
{ }
ρ (f) ≡ sup |f (x)-− f-(y)|: x,y ∈ K, x ⁄= y .
α |x − y|α
Then
(C α(K ),||⋅||α)
is a complete normed linear space called a Holder space.
The verification that this is a complete normed linear space is routine and is left for
you. More generally, one considers the following class of Holder spaces.
Definition 37.1.5Let K be a compact subset of ℝ^{n}and let λ ∈ (0,1]. C^{m,λ}
(K)
denotesthe set of functions, u which are restrictions of functions defined on ℝ^{n}to Ksuch that for
which is obviously true. Suppose then that the inequality is valid for some n. Using
Fubini’s theorem, Holder’s inequality, and the induction hypothesis,