2.2 Global Sobolev spaces
To define Sobolev spaces on manifolds, first observe the following lemma:
Let be open and a diffeomorphism. If is precompact and , then the induced map is a bounded isomorphism for all .□
The proof is an exercise in the chain rule (for
) and duality (for
). From now on, we always assume
is an integer.
This means it makes sense to define
Let be a compact manifold and a Hermitian vector bundle. Fix an open cover and a partition of unity subordinate to . For any , we define the Sobolev norm by
and define to be the completion of with respect to .
There is an obvious inner product that gives rise to the Sobolev norm.
One should check for themselves that
The norm is well-defined up to equivalence, i.e. does not depend on the choice of the .□
One sees that our local theorems generalize easily to
There are bounded inclusions for which are compact.
There are bounded inclusions for .
There is a natural duality pairing .
Any differential operator of order induces a continuous map .□