Mathematical Principles of Theoretical Physics

126 CHAPTER 3. MATHEMATICAL FOUNDATIONS

and[v]αis the H ̈older modulus:

[v]α= sup

x,y∈M,x 6 =y

|v(x)−v(y)|

|x−y|α

, 0 <α< 1.

A H ̈older continuous functionu∈C^0 ,α(M⊗pEN)is as shown in (3.2.6)-(3.2.7) with 0<
α<1.
The norm ofCk,α(M⊗pEN)( 0 <α≤ 1 )is given by

||u||Ck,α=||u||Ck+ [Dku]α ( 0 <α≤ 1 ),

where|| · ||Ckis the norm ofCk(M⊗pEN):

||u||Ck=

k

∑

|β|= 0

sup

M

|∂βu|.

3.2.2 Sobolev embedding theorem

In (3.2.6)-(3.2.7) we see that forΩ⊂Rn, a functionu∈W^1 ,p(Ω)does’t imply thatu∈Lq(Ω)
for anyq>p. For example, for the function

u=|x|−α, 0 <α<

1

2

, x∈Ω⊂R^3 , Ωbounded,

it is known that
∇u=|x|−

α+ 24

x∈L^2 (Ω).

Obviously we have

u∈W^1 ,^2 (Ω), u6∈Lq(Ω) ∀q>

n

α

.

The following embedding problem of Sobolev spaces providesa solution for this problem.

Theorem 3.7(Sobolev Embedding Theorem).LetMbe an n-dimensional compact mani-
fold. Then we have the embeddings:

(3.2.11) W^1 ,p(M⊗pEN)֒→











Lq(M⊗pEN) for 1 ≤q≤

np

n−p

if n>p,

Lq(M⊗pEN) for 1 ≤q<∞ if n=p,

C^0 ,α(M⊗pEN) forα= 1 −n/p if n<p.

Here C^0 ,α(M⊗pEN)are the Holder spaces. Moreover we have the following inequalities ̈
for the norms:

(3.2.12)

||u||Lq≤C||u||W 1 ,p for 1 ≤q≤

np

n−p

if n>p,

||u||Lq≤C||u||W 1 ,p for 1 ≤q<∞ if n=p,

||u||C 0 ,α≤C||u||W 1 ,p forα= 1 −n/p if n<p,

where C> 0 are constants depending on n,p and M.