Global Analysis — Global Sobolev spaces

2.2 Global Sobolev spaces

To define Sobolev spaces on manifolds, first observe the following lemma:

Lemma

Let U,VRnU', V' \subseteq \mathbb {R}^ n be open and ϕ:UV\phi : U' \to V' a diffeomorphism. If UUU \subseteq U' is precompact and V=ϕ(U)VV = \phi (U) \subseteq V', then the induced map Hs(V)Hs(U)H^ s(V) \to H^ s(U) is a bounded isomorphism for all sZs \in \mathbb {Z}.

The proof is an exercise in the chain rule (for s0s \geq 0) and duality (for s<0s < 0). From now on, we always assume ss is an integer.

This means it makes sense to define

Definition

Let MM be a compact manifold and EME \to M a Hermitian vector bundle. Fix an open cover {φi:RnUiM}\{ \varphi _ i: \mathbb {R}^ n \overset {\sim }{\to } U_ i \subseteq M\} and a partition of unity {ρi}\{ \rho _ i\} subordinate to φi\varphi _ i. For any uΓ(M,E)u \in \Gamma (M, E), we define the Sobolev norm by

us2=iρiuφi1s2, \| u\| _ s^2 = \sum _ i \| \rho _ i u \circ \varphi _ i^{-1}\| _ s^2,

and define Hs(M;E)H^ s(M; E) to be the completion of Γ(M,E)\Gamma (M, E) with respect to ks\| \cdot \| _ k^ s.

There is an obvious inner product that gives rise to the Sobolev norm.

One should check for themselves that

Lemma

The norm s\| \cdot \| _ s is well-defined up to equivalence, i.e. does not depend on the choice of the Ui,ϕi,ρiU_ i, \phi _ i, \rho _ i.

One sees that our local theorems generalize easily to

Lemma
  1. There are bounded inclusions HsHtH^ s \to H^ t for s>ts > t which are compact.

  2. There are bounded inclusions HsCkH^ s \to C^ k for s>n2+ks > \frac{n}{2} + k.

  3. There is a natural duality pairing Hs(M;E)×Hs(M;E)CH^ s(M; E) \times H^{-s}(M; E) \to \mathbb {C}.

  4. Any differential operator L:Γ(M,E0)Γ(M,E1)L: \Gamma (M, E_0) \to \Gamma (M, E_1) of order kk induces a continuous map Hs+k(E0)Hs(M;E1)H^{s + k}(E_0) \to H^ s(M; E_1).