Abstract
We study the problem of completely describing the domains that enjoy the generalized multiplicative inequalities of the embedding theorem type. We transfer the assertions for the Sobolev spaces L p 1(Ω) to the function classes that result from the replacement of L p (Ω) with an ideal space of vector-functions. We prove equivalence of the functional and geometric inequalities between the norms of indicators and the capacities of closed subsets of Ω. The most comprehensible results relate to the case of the rearrangement invariant ideal spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Sobolev S. L., Some Applications of Functional Analysis in Mathematical Physics [in Russian], Nauka, Moscow (1988).
Besov O. V., Il'in V. P., and Nikol'skii S. M., Integral Representations of Functions and Embedding Theorems [in Russian], Nauka, Moscow (1975).
Gol'dshtein V. M. and Reshetnyak Yu. G., An Introduction to the Theory of Functions with Generalized Derivatives and the Quasiconformal Mappings [in Russian], Nauka, Moscow (1983).
Maz'ya V. G., Sobolev Spaces [in Russian], Leningrad Univ., Leningrad (1985).
Brudnyi Yu. A., Krein S. G., and Semenov E. M., “Interpolation of linear operators,” in: Mathematical Analysis. Vol. 24 [in Russian], VINITI, Moscow, 1986, pp. 3-64 (Itogi Nauki i Tekhniki).
Zabreiko P. P., “Ideal spaces of vector-functions,” Dokl. Akad. Nauk SSSR, 31,No. 4, 298-301 (1987).
Kantorovich L. V. and Akilov G. P., Functional Analysis [Russian], Nauka, Moscow (1977).
Krein S. G., Petunin Yu. I., and Semenov E. M., Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).
Berezhnoi E. I., “Exact estimates for operators on cones in ideal spaces,” Trudy Mat. Inst. Steklov, 204, 3-36 (1993).
Klimov V. S., “Functional inequalities and generalized capacities,” Mat. Sb., 187,No. 1, 41-54 (1996).
Klimov V. S. and Panasenko E. S., “Geometric properties of ideal spaces and the capacities of sets,” Sibirsk. Mat. Zh., 40,No. 3, 573-586 (1999).
Krasnosel'skii M. A. and Rutitskii Ya. B., Convex Functions and Orlicz Spaces [in Russian], Fizmatgiz, Moscow (1958).
Rockafellar R. T., Convex Analysis [Russian translation], Mir, Moscow (1973).
Levin V. L., Convex Analysis in Spaces of Measurable Functions and Its Application in Mathematics and Economics [in Russian], Nauka, Moscow (1985).
Lozanovskii G. Ya., “Transformation of Banach ideal spaces by concave functions,” in: Qualitative and Approximate Methods for Studying Operator Equations [in Russian], Yaroslav. Univ., Yaroslavl', 1979, Vol. 3, pp. 122-148.
Calderón A. P., “Intermediate spaces and interpolation, the complex method,” Matematika, 9,No. 3, 56-129 (1965).
Maz'ya V. G., “Weak solutions of the Dirichlet and Neumann problems,” Trudy Moskov. Mat. Obchsh., 20, 137-172 (1969).
Klimov V. S., “Embedding theorems for Orlicz spaces and some of their applications to boundary value problems,” Sibirsk. Mat. Zh., 13,No. 2, 334-348 (1972).
Kolyada V. I., “Permutations of functions and embedding theorems,” Uspekhi Mat. Nauk, 44,No. 5, 61-95 (1989).
Klimov V. S., “To embedding theorems of anisotropic function classes,” Mat. Sb., 127,No. 2, 198-208 (1985).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Klimov, V.S. Generalized Multiplicative Inequalities for Ideal Spaces. Siberian Mathematical Journal 45, 112–124 (2004). https://doi.org/10.1023/B:SIMJ.0000013016.80981.08
Issue Date:
DOI: https://doi.org/10.1023/B:SIMJ.0000013016.80981.08