On the complex of Sobolev spaces associated with an abstract Hilbert complex

We consider the complexes of Hilbert spaces whose differentials are closed densely-defined operators. A peculiarity of these complexes is that from their differentials we can construct Laplace operators in every dimension. The Laplace operator together with a sufficiently “nice” measurable function enables us to define a “generalized Sobolev space”. There exist pairs of measurable functions allowing us to construct some “canonical” mappings of the corresponding Sobolev spaces. We find necessary and sufficient conditions for those mappings to be compact. In some cases for a given Hilbert complex we can construct an associated Sobolev complex. We show that the differentials of the original complex are normally solvable simultaneously with the differentials of the associated complex and that the reduced cohomologies of these complexes coincide.