In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and associated theories of the Laplace operator on noncompact Lie groups. My expectation is that the theory should be reduced to coincide with that in $\mathbb{R}^n$ as a trivial case of noncompact Lie groups. But I fail to find useful references on this subject. Can someone kindly help me with this difficulty? Thanks in advance.

In this post, a question was raised to discuss the generalization of Sobolev spaces on locally compact Lie groups. Now my question is whether there exists a generalization of Sobolev spaces and associated theories of the Laplace operator on noncompact Lie groups. My expectation is that the theory should be reduced to coincide with that in $\mathbb{R}^n$ as a trivial case of noncompact Lie groups. But I fail to find useful references on this subject. Can someone kindly help me with this difficulty? Thanks in advance.