Abstract
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.
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References
Bruning J. and Lesh M., “Hilbert complexes,” J. Funct. Anal., 108, 88–132 (1992).
Kuz?minov V. I. and Shvedov I. A., “Homological aspects of the theory of Banach complexes,” Sibirsk. Mat. Zh., 40, No. 4, 893–904 (1999).
Dodziuk J., “Sobolev spaces of differential forms and de Rham-Hodge isomorphism,” J. Differential Geometry, 16, 63–73 (1981).
Kuz?minov V. I. and Shvedov I. A., “Separation of variables in the problems of normal and compact solvability of the operator of exterior derivation,” Sibirsk. Mat. Zh., 41, No. 2, 385–396 (2000).
Gol?dshte?n V. M., Kuz?minov V. I., and Shvedov I. A., “On normal and compact solvability of linear operators,” Sibirsk. Mat. Zh., 30, No. 5, 49–59 (1989).
Riesz F. and Szökefalvi-Nagy B., Lectures on Functional Analysis [Russian translation], Mir, Moscow (1979).
Gol?dshte?n V. M., Kuz?minov V. I., and Shvedov I. A., “On normal and compact solvability of the operator of exterior derivation under homogeneous boundary conditions,” Sibirsk. Mat. Zh., 28, No. 4, 82–96 (1987).
Kuz?minov V. I. and Shvedov I. A., “On orthogonal direct sum decomposition of de Rham complexes for warped products of Riemannian manifolds,” Sibirsk. Mat. Zh., 39, No. 2, 354–368 (1998).
Kato T., Perturbation Theory for Linear Operators [Russian translation], Mir, Moscow (1972).
Reed M. and Simon B., Methods of Modern Mathematical Physics. Vol. 1: Functional Analysis [Russian translation], Mir, Moscow (1977).
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Glotko, N.V. On the Complex of Sobolev Spaces Associated with an Abstract Hilbert Complex. Siberian Mathematical Journal 44, 774–792 (2003). https://doi.org/10.1023/A:1025924417135
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DOI: https://doi.org/10.1023/A:1025924417135