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I'm looking some book from which I could learn more about Sobolev spaces. I'm interested rather in abstract theory: some topics which I would like understand in detail include:

  • general construction of Sobolev spaces of sections of vector bundles over manifold

  • the equivalence of norms: one defined in terms of Fourier transform and the other being the sum of $L^2$ norms of derivatives

  • embedding theorems (condition for an embedding to be compact)

  • theorems about regularity of Sobolev spaces (which Sobolev spaces consist of $C^k$ functions)

I would like to avoid any detailed discussions about motivations for Sobolev spaces (coming from differential equations). My goal is to understand some topics about Sobolev spaces and then move to pseudodifferential operators (on general vector bundles) and finally prepare to study index theorem.

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Necas' book "Direct Methods in the Theory of Elliptic Equations" is a wonderful guide for the topics that you mentioned above, although the proofs are very abstract and most steps are omitted. Evans' book is more understandable and also includes the topics above, but not in the most general settings of the theorems. If you just start studying the Sobolev space theory, Evans' book would be better.

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  • $\begingroup$ I second Evan's book. It is an extremely readable introduction. The Sobolev spaces are introduced in Chapter 5 of the book. $\endgroup$ – Joel Jun 26 '15 at 22:32
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Answer with four years delay: Besides Evans book, the book I recommend everyone for the detail introduction on Sobolev spaces is:

"Robert Adams, John Fournier, Sobolev spaces, 2003".

It is very good for the compact embeddings and the regularity part of your question. But I am not sure about the first two topics you mentioned. I assume that good sources could be found in the references of that book.

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