# Interpolation of Sobolev spaces

I know quite a bit about the abstract theory of Interpolation of Banach spaces. Today I had a colleague from Environmental sciences (who used to be in our Applied Maths department) come and ask me about (complex) interpolation of Sobolev spaces. I was, in the end, able to explain enough to give him a "black box" which seemed to do enough (recover norm estimates for odd cases from formula established for even cases by partial integration techniques).

Now, the book which he'd been pointed to (by another book) was "Interpolation spaces" by Bergh and Lofstrom. I've read this, of course, but it takes a (almost caricatured) pure maths approach: you have to read it cover to cover to catch all the definition etc. So my question is:

Does anyone know of a "friendly" (applied maths style) approach to complex interpolation of Sobolev (and related function) spaces?

I'm guessing that this must exist, as it's only a small step from the classical Riesz-Thorin interpolation, which must be used by lots of people who don't particularly care about abstract Banach spaces.

Edit: Perhaps I'm being dismissive or confusing or something about "applied maths style". I don't wish to be! The book my colleague showed me said something like: "The odd case follows by interpolation. (This is not an easy argument, and we do not give it. See, for example, the book of Bergh+Lofstrom.)" I'm sort of taking this as a baseline. Many thanks for the suggestions so far-- I'll leave this open a bit longer, and then accept an answer.

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All the applied mathematicians I know use the modern approach to complex interpolation; i.e., construct the complex interpolation spaces of a compatible couple of Banach spaces and then, when the spaces have a functional representation, try to find functional representations for the interpolation spaces. This uses just a bit of complex analysis and the definition of Banach spaces. It is no more difficult than classical proofs of Riesz-Thorin and isolates the difficulty where it should be--identifying the interpolations spaces. That is easy for $L_p$ spaces but can be very tricky for others. –  Bill Johnson Feb 8 '11 at 14:42
@Bill: So you're saying that, basically, I should encourage my colleague just to bite the bullet and learn (a little of) the abstract theory? –  Matthew Daws Feb 8 '11 at 15:09
Yes, Matt. It is no more difficult that Riesz-Thorin itself. –  Bill Johnson Feb 8 '11 at 15:59

## 2 Answers

What exactly does your colleague need interpolation for? I guess he just needs to extend some inequalities to intermediate values of the parameters. Then he can use the black box approach and his problem is reduced to computing interpolation spaces between given couples of Banach spaces. Then, two possibilities:

1) The interpolation spaces have already been computed. There is a vast literature on this, and he would not need to really study it but just check the statements. Besides the books already mentioned I would add Bennett and Sharpley, Interpolation of Operators, and a few books by H.Triebel with a similar name (Interpolation is the keyword).

2) In the unlucky case his spaces have not been considered, then he has to delve into the theory a little, and try to compute the spaces himself. Bergh and Lofstrom is a strange book, full of results, but with several imprecisions which can cause the beginner a few nightmares. Better start with Bennett and Sharpley which is crystal clear and reliable, keeping BL on the side for a comparison. Lunardi's book is also quite good but less comprehensive (at least from the early version I have).

3) If all else fails, try real interpolation, Peetre style. The theory is much easier to grasp, and closer to approximation and convexity methods he might be familiar with.

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If it's just interpolating Sobolev norms, it's not clear to me that complex interpolation is needed at all. –  Deane Yang Feb 8 '11 at 18:45
If he needs a lot (it does not seem so) he should look at S. G. Krein, Ju.I. Petunin, E.M. Semenov, Interpolation of linear operators. –  Bill Johnson Feb 9 '11 at 12:09
Oh, I forgot to mention the excellent 'An Introduction to Sobolev Spaces and Interpolation Spaces' by Luc Tartar. He is quite a character, and it shows from his witty commentaries, but the book is very good. –  Piero D'Ancona Feb 9 '11 at 20:17

The book Interpolation Theory by Alessandra Lunardi seems quite readable, though perhaps not really “applied maths style”, whatever that might mean. See MR2523200 for a review. ISBN: 978-88-7642-342-0 or 88-7642-342-0.

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