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I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square integrable. This can be generalized to function spaces with $k$ not an integer by appealing to Fourier transforms, by using the Fourier identity for distributional derivatives; we get the norm $ \|f\|_{H^s} = \| (1 + |t|^2)^{s/2} \hat{f}(t)\|_2$. We also know we can generalize to functions on Riemannian manifolds.

Is there any way to generalize $H^s$ to function spaces on locally compact groups $G$? We already know we can define an extension of the Fourier transform to $L^2(G)$. However, we need to define some expression that is like $(1 + |t|^2)^{s/2} \hat{f}(t)$. How can we arrange for $G$ to have something analogous? Is there any literature on this? Are there any interesting/useful PDEs that may arise in such contexts, defined on such function spaces?

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I believe that if you google "Sobolev spaces compact Lie groups", you'll find lots of hits and if you look at them as well as references contained in them, you'll find what you want. – Deane Yang Nov 18 '11 at 20:02
It seems like those developments are based on the fact that such groups can be made into Riemannian manifolds, in some way reducing it to a known case. Am I wrong in this assessment? – Christopher A. Wong Nov 18 '11 at 20:48
I heard that the analogy of tempered distributions in this setting is called Harish-Chandra functions. To define them, one must introduce some kind of decay conditions, so it might give something useful if you Google this term. – timur Jun 18 '14 at 12:28
There is some literature on Sobolev spaces in arbitrary metric measure spaces:… . For a locally compact group, one can use Haar measure for the measure, and if the group is second countable one can use Birkhoff-Kakutani to get a metric (but the choice of metric is not unique, and this can lead to different Sobolev spaces, e.g. a Riemannian metric gives different results to a Carnot-Caratheodory metric). – Terry Tao Aug 17 '14 at 15:52
For further references about Sobolev spaces in an arbitrary metric space, I would look for this book: Furthermore, the theory has been generalized even more in here:…. – Juhana Siljander Jan 14 '15 at 14:47

Yes, look at the papers of Gorka and Reyes.

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Aren't they limited to the LCA case? – Yemon Choi Aug 17 '14 at 11:58

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