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?