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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
    
Christopher, for a compact Lie group $G$, you can define a Sobolev space on $G$ via charts, reducing it to $\mathbb R^n$, or you can define it analogously to $\mathbb R^n$ by using the Plancherel theorem for $G$ (replacing the Laplacian on $\mathbb R^n$ with the Casimir operator on $G$ and using a sum over the dual space of $G$, rather than an integral over $\mathbb R^n$). When $G$ is non-abelian, I don't think these are equivalent, though they might be. –  B R Nov 18 '11 at 21:51
    
In another direction, you can ask about $p$-adic Sobolev spaces. –  B R Nov 18 '11 at 21:59
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