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There is extensive literature on Sobolev spaces on complete Riemannian manifolds but are there any standard references regarding the definition and properties of Sobolev spaces on domains (possessing certain regularity, at least) in complete Riemannian manifolds?

In particular, I'm interested in a definition of such spaces that has the following two properties:

  1. The Sobolev embedding theorems hold (assuming the usual geometric restrictions on the ambient manifold).
  2. Passing to a smaller domain induces a restriction map on the corresponding Sobolev spaces.
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    $\begingroup$ If I understand correctly, you're asking essentially for Sobolev spaces on a Riemannian manifold with boundary? $\endgroup$
    – Deane Yang
    Feb 7, 2022 at 23:50
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    $\begingroup$ Following up on @DeaneYang's comment-question: are you wanting a "trace theorem" for behavior at the boundary? $\endgroup$
    – paul garrett
    Feb 8, 2022 at 0:32
  • $\begingroup$ @DeaneYang, having the Sobolev spaces defined on manifolds with boundary may be helpful but it's not quite what I'm asking. I'd like to have the exact analogue of the situation in R^n: One has the Sobolev spaces defined on the whole of R^n but also on open subsets of R^n. $\endgroup$
    – S.Z.
    Feb 8, 2022 at 4:39
  • $\begingroup$ @paulgarrett, No, I'm not interested in what happens at the boundary or in restricting to the boundary, I simply want a definition which assigns to an inclusion of open subsets restriction between the corresponding Sobolev spaces. I believe this holds in the case of subsets of R^n. $\endgroup$
    – S.Z.
    Feb 8, 2022 at 4:47
  • $\begingroup$ Could you you say more precisely what you think holds in $\mathbb{R}^n$ and would like to extend to a manifold? And what do you mean by the usual geometric restrictions? $\endgroup$
    – Deane Yang
    Feb 8, 2022 at 13:52

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