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.



Interpolation of Sobolev spaces
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.


