3.2 Global elliptic regularity
Let be a compact manifold, complex vector bundles, and an elliptic differential operator from to of order . By patching local results together, we conclude that
There is a constant such that for all and , we have
If , then . In particular, if , then .□
This implies a lot of very nice properties about elliptic operators. First observe the following result:
Let be Hilbert spaces and bounded, compact. If there is an such that
then is finite-dimensional and is closed.
We show that the unit ball of is compact. If is a sequence in the unit ball of , then
Since is compact, there is a subsequence such that is Cauchy. So is Cauchy. So we are done.
To show is closed, by restricting to the complement of the kernel, we may assume is injective. We will show that there is a such that
If not, pick a sequence with but . By compactness, we may assume that is Cauchy. Then we see that must also be Cauchy, and the limit must satisfy and , a contradiction.□
has finite-dimensional kernel and closed image.□
We want to show that in fact is Fredholm, i.e. both the kernel and the cokernel are compact. Here we simply use duality. We need to show that
is finite-dimensional. But this is exactly the kernel of . So we deduce
is Fredholm, and in fact
The first factor is independent of (since all elements are ), and taking the limit , we get