I am looking for references and materials that discuss the following aspects of weighted Sobolev spaces $ W^{k,2}_\rho(\mathbb{R}) $ defined on the entire real line $ \mathbb{R} $:
Hilbert Space Structure:
- Under what conditions on the weight $ \rho(x) > 0 $ (e.g., local integrability $ \rho(x) \in L^1_{\text{loc}}(\mathbb{R}) $), does the space $ W^{k,2}_\rho(\mathbb{R}) $ become a Hilbert space?
- How are completeness and reflexivity established in the unbounded setting of $ \mathbb{R} $?
Density of Smooth Functions:
- Are there sufficient conditions on $ \rho(x) $ to ensure that $ C^\infty_c(\mathbb{R}) $, the space of compactly supported smooth functions, is dense in $ W^{k,2}_\rho(\mathbb{R}) $?
- Are there works that provide explicit estimates for the approximation of functions in the weighted Sobolev norm by smooth compactly supported functions?
I am aware of some works treating weighted Sobolev spaces in bounded domains, but I am particularly focused on the entire real line $ \mathbb{R} $. For example:
- The monograph by Alois Kufner and Bohumír Opic, "How to Define Reasonably Weighted Sobolev Spaces", primarily considers bounded domains.
- V. V. Zhikov's "Weighted Sobolev Spaces" link mentions conditions in Section 1.1 but appears to treat this topic only briefly.
Since I am not an expert in the field, I would greatly appreciate references (books, papers, or lecture notes) that provide a more detailed or explicit treatment of these questions for the full space $ \mathbb{R} $. Any clarification of general techniques or methods to approach these problems would also be helpful.
Thank you!
