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On the complex of Sobolev spaces associated with an abstract Hilbert complex. (Russian, English) Zbl 1037.46029
Sib. Mat. Zh. 44, No. 5, 992-1014 (2003); translation in Sib. Math. J. 44, No. 5, 774-792 (2003).
The author associates a cochain complex of Sobolev spaces to a given cochain complex of Hilbert spaces and studies the cohomology groups of these complexes. He shows that the differentials of the original complex are normally solvable simultaneously with the differentials of the associated Sobolev complex and that the reduced cohomologies of these complexes coincide. He also constructs the Hodge decomposition in the Hilbert complexes and indicates some applications of his results to the theory of differential forms on Riemannian manifolds, especially on warped products.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
55U05 Abstract complexes in algebraic topology
58A10 Differential forms in global analysis
58A14 Hodge theory in global analysis
58J10 Differential complexes
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