Global Analysis: Global Sobolev spaces

2.2 Global Sobolev spaces

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

Lemma

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

The proof is an exercise in the chain rule (for

s \geq 0

) and duality (for

s < 0

). From now on, we always assume

s

is an integer.

This means it makes sense to define

Definition

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

\| u\| _s^2 = \sum _i \| \rho _i u \circ \varphi _i^{-1}\| _s^2,

and define $H^s(M; E)$ to be the completion of $\Gamma (M, E)$ with respect to $\| \cdot \| _k^s$ .

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

One should check for themselves that

Lemma

The norm $\| \cdot \| _s$ is well-defined up to equivalence, i.e. does not depend on the choice of the $U_i, \phi _i, \rho _i$ .

One sees that our local theorems generalize easily to

Lemma

There are bounded inclusions $H^s \to H^t$ for $s > t$ which are compact.
There are bounded inclusions $H^s \to C^k$ for $s > \frac{n}{2} + k$ .
There is a natural duality pairing $H^s(M; E) \times H^{-s}(M; E) \to \mathbb {C}$ .
Any differential operator $L: \Gamma (M, E_0) \to \Gamma (M, E_1)$ of order $k$ induces a continuous map $H^{s + k}(E_0) \to H^s(M; E_1)$ .