Functional Inequalities and Relative Capacities

V. S. Klimov ; E. S. Panasenko

Matematičeskie zametki, Tome 72 (2002) no. 2, pp. 216-226.

Résumé

In this paper, we study functional inequalities of the form $$ \|f;Q\| \le C\varphi (\|\nabla f;P\|,\|f;R\|), $$ where $P$, $Q$, and $R$ are Banach ideal spaces of functions on a domain $\Omega \subset \mathbb R^n$, the constant $C$ is the same for all compactly supported functions $f$ satisfying the Lipschitz condition, $\nabla f$ is the gradient of $f$, and $\varphi $ is a continuous degree one homogeneous function. We give compatibility conditions for norms on the spaces $P$, $Q$, and $R$ that ensure the equivalence of the inequality in question to an isoperimetric inequality between the norms of indicators and relative capacities of compact subsets of the domain $\Omega $.

Bibliographie

[1] Mazya V. G., “O nekotorykh integralnykh neravenstvakh dlya funktsii mnogikh peremennykh”, Problemy matem. analiza, no. 3, Izd-vo LGU, L., 1972, 33–68

[2] Mazya V. G., Prostranstva S. L. Soboleva, Izd-vo LGU, Leningrad, 1985 | Zbl

[3] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1977 | Zbl

[4] Krein S. G., Petunin Yu. I., Semenov E. M., Interpolyatsiya lineinykh operatorov, Nauka, M., 1978

[5] Brudnyi Yu. A., Petunin Yu. I., Semenov E. M., “Interpolyatsiya lineinykh operatorov”, Itogi nauki i tekhniki. Matem. analiz, 24, VINITI, M., 1986, 3–163 | MR

[6] Berezhnoi E. I., “Tochnye otsenki operatorov na konusakh v idealnykh prostranstvakh”, Tr. MIAN, 204, Nauka, M., 1993, 3–34 | MR | Zbl

[7] Zabreiko P. P., “Idealnye prostranstva vektor-funktsii”, Dokl. AN BSSR, 31:4 (1987), 298–301 | MR | Zbl

[8] Krasnoselskii M. A., Rutitskii Ya. B., Vypuklye funktsii i prostranstva Orlicha, Fizmatgiz, M., 1958

[9] Levin V. L., Vypuklyi analiz v prostranstvakh izmerimykh funktsii i ego primeneniya v matematike i ekonomike, Nauka, M., 1985

[10] Rokafellar R. G., Vypuklyi analiz, Mir, M., 1973

[11] Klimov V. S., “Emkosti mnozhestv i teoremy vlozheniya dlya idealnykh prostranstv”, Dokl. RAN, 341:5 (1995), 588–589 | MR | Zbl

[12] Klimov V. S., “Funktsionalnye neravenstva i obobschennye emkosti”, Matem. sb., 187:1 (1996), 41–54 | MR | Zbl

[13] Kalderon A. P., “Promezhutochnye prostranstva i interpolyatsiya. Kompleksnyi metod”, Matematika. Sb. perevodov, 9:3 (1965), 56–129

[14] Dmitriev V. I., Krein S. G., Ovchinnikov V. I., “Osnovy teorii interpolyatsii lineinykh operatorov”, Geometriya lineinykh prostranstv i teoriya operatorov, Izd-vo Yaroslavskogo un-ta, Yaroslavl, 1977, 31–74 | MR

[15] Lozanovskii G. Ya., “Preobrazovanie banakhovykh idealnykh prostranstv s pomoschyu vypuklykh funktsii”, Kachestvennye i priblizhennye metody issledovaniya operatornykh uravnenii, no. 3, Izd-vo Yaroslavskogo un-ta, Yaroslavl, 1978, 122–148 | MR

[16] Polia G., Segë G., Izoperimetricheskie neravenstva v matematicheskoi fizike, Mir, M., 1962

[17] Goldshtein V. M., Reshetnyak Yu. G., Vvedenie v teoriyu funktsii s obobschennymi proizvodnymi i kvazikonformnye otobrazheniya, Nauka, M., 1983

[18] Brelo M., Osnovy klassicheskoi teorii potentsiala, Mir, M., 1964 | Zbl

[19] Klimov V. S., “Teoremy vlozheniya i geometricheskie neravenstva”, Izv. AN SSSR. Ser. matem., 40:3 (1976), 645–671 | MR | Zbl

[20] Klimov V. S., “O perestanovkakh differentsiruemykh funktsii”, Matem. zametki, 9:6 (1971), 629–638 | MR | Zbl