Mathematical Principles of Theoretical Physics

3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 125

The spacesWk,p(M⊗pEN)are called Sobolev spaces, and the norms are defined by

(3.2.9) ||u||Wk,p=

k

∑

β= 0

[∫

M

|Dβu|p

√

−gdx

] 1 /p

.

It is clear that

u∈Wk,p(M⊗pEN)⇒uisk−th weakly differentiable.

In addition, we introduce the spacesW 0 k,p(M⊗pEN)as

W 0 k,p(M⊗pEN) =Closure ofC∞ 0 (M⊗pEN)underWk,pnorm( 3. 2. 9 ).

HereC 0 ∞(M⊗pEN)is as defined in (3.2.4).

If∂M=/0, thenW 0 k,p(M⊗pEN) =Wk,p(M⊗pEN), and if∂M 6 =/0 then foru∈

W 0 k,p(M⊗pEN)we have

u|∂M= 0 ,···,∂βu|∂M= 0 ∀|β| ≤k− 1.

4.Hkspaces. AsMis a Riemannian manifold,M⊗pENand its dual bundleM⊗p

(EN)∗are isomorphic. In this case, the spacesWk,^2 (M⊗pEN)are Hilbert spaces, denoted

by

(3.2.10)

Hk(M⊗pEN) =Wk,^2 (M⊗pEN),

H 0 k(M⊗pEN) =W 0 k,^2 (M⊗pEN).

The inner products of (3.2.10) are defined by

〈u,v〉Hk=

∫

M

k

∑

|β|= 0

Dβu·Dβv∗

√

−gdx

wherev∗∈Hk(M⊗p(EN)∗)is the dual field ofv∈Hk(M⊗pEN).

5.Lipschitz spaces.Letk≥0 be integrals. The Lipschitz spaceCk,^1 (M⊗pEN)consists

of allk-th order continuously differentiable functionsuwithDkubeing Lipschitz continuous:

Ck,^1 (M⊗pEN) ={u∈Ck(M⊗pEN)|[∂ku]Lip<∞},

where[∂ku]Lipis the Lipschitz modulus, defined by

[v]Lip= sup

x,y∈M,x 6 =y

|v(x)−v(y)|

|x−y|

.

A Lipschitz continuous functionu∈C^0 ,^1 (M⊗pEN)is as shown in (3.2.6)-(3.2.7) withα=1.

6.Holder spaces. ̈ The H ̈older spaceCk,α(M⊗pEN) ( 0 <α< 1 )consists of allk-th

order continuously differentiable functionsuwithDkubeing H ̈older continuous:

Ck,α(M⊗pEN) ={u∈Ck(M⊗pEN)|[Dku]α<∞},