First I should apologise if this is a bit of a vague question, but I could not find any references for the explicit construction.
I've seen it stated in several places that a differential operator on sections of a vector bundle $E$ over a compact manifold $M$ can be extended to a Fredholm operator between appropriate Sobolev spaces. However, I am unsure how exactly this is done.
1) How does one define the appropriate Sobolev spaces? I understand that ordinarily (for functions on open subsets of $\mathbb{R}^n$) this is done via weak derivatives, but what would be the analogous thing to do on a manifold? How similar are these two constructions in terms of their respective properties (eg is completeness proved in different ways)?
2) Once the $H^i$ are defined, how do we construct a Fredholm operator based on our differential operator $\Gamma(E)\rightarrow\Gamma(E)$? Is it true that the index (which, I think, is only defined for Fredholm operators) of this Fredholm operator is equal to the difference in dimension of kernel and cokernel of our differential operator?
3) In the case that $d: \Gamma(\Lambda^*(T^*M))\rightarrow \Gamma(\Lambda^*(T^*M))$, I'm assuming $d$ determines an operator $H^1\rightarrow H^0$. What are $H^0$ and $H^1$ in this case?
I'd appreciate any comments, explicit constructions, or suggestions for references on this topic. Thanks!
1) Partitions of unity and local trivializations. Do the standard Sobolev norms and add them. This will depend on the choice of trivializations; up to equivariance it will not.
2) With he above norm, you can define Sobolev completions of the appropriate spaces of sections. Because locally your operator is just a differential operator in the standard sense, you can continuously extend it to the completion.
3) I don't really understand the question. They're the spaces of $L^2$ and $L^2_1$ sections of the appropriate bundles, where a function being $L^2$ or whatever is defined with respect to a local trivialization.