Dual representation of superlinear functionals and its applications in function theory.~II

B. N. Khabibullin

Izvestiya. Mathematics , Tome 65 (2001) no. 5, pp. 1017-1039.

Résumé

The results of the first part of this work (see [1]) are used only in § 7 of this paper, from which subsequent results follow. We pose new dual problems for weight spaces of holomorphic functions of one and several variables defined on a domain in $\mathbb C^n$, namely, the problem of non-triviality of a given space, description of zero sets, description of sets of (non-)uniqueness, existence of holomorphic functions of certain classes that “suppress” the growth of a given holomorphic function, and representation of meromorphic functions as quotients of holomorphic functions contained in a given space.

Bibliographie

[1] Khabibullin B. N., “Dvoistvennoe predstavlenie superlineinykh funktsionalov i ego primeneniya v teorii funktsii, I”, Izv. RAN. Ser. matem., 65:4 (2001), 205–224 | MR | Zbl

[2] Khabibullin B. N., “Naimenshaya plyurisupergarmonicheskaya mazhoranta i multiplikatory tselykh funktsii, I”, Sib. matem. zhurn., 33:1 (1992), 173–178 | MR

[3] Lelon P., Gruman L., Tselye funktsii mnogikh kompleksnykh peremennykh, Mir, M., 1989 | MR | Zbl

[4] Burbaki N., Mery, integrirovanie mer, Nauka, M., 1967 | MR

[5] Landkof N. S., Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR | Zbl

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

[7] Khermander L., Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, T. I, Mir, M., 1986

[8] Azarin V. S., “Ob asimptoticheskom povedenii subgarmonicheskikh funktsii konechnogo poryadka”, Matem. sb., 108:2 (1979), 147–167 | MR | Zbl

[9] Shabat B. V., Vvedenie v kompleksnyi analiz, Ch. II, Nauka, M., 1985 | MR

[10] Khabibullin B. N., “O nulevykh mnozhestvakh dlya tselykh funktsii i predstavlenii meromorfnykh funktsii”, Matem. zametki, 59:4 (1996), 611–617 | MR | Zbl

[11] Khabibullin B. N., “Mnozhestva edinstvennosti v prostranstvakh tselykh funktsii odnoi peremennoi”, Izv. AN SSSR. Seriya matem., 55:5 (1991), 1101–1123

[12] Markushevich A. I., Teoriya analiticheskikh funktsii, Nauka, M., 1968 | Zbl

[13] Khabibullin B. N., “Naimenshaya plyurisupergarmonicheskaya mazhoranta i multiplikatory tselykh funktsii. II: Algebry funktsii konechnogo $\lambda$-tipa”, Sib. matem. zhurn., 33:3 (1992), 186–191 | MR

[14] Khabibullin B. N., “O tipe tselykh i meromorfnykh funktsii”, Matem. sb., 183:11 (1992), 35–44 | Zbl

[15] Khabibullin B. N., “Otsenki ob'ema nulevykh mnozhestv golomorfnykh funktsii”, Izv. vuzov. Matematika, 1992, no. 3(358), 58–63 | MR | Zbl

[16] Khabibullin B. N., “Teorema o naimenshei mazhorante i ee primeneniya. II: Tselye i meromorfnye funktsii konechnogo poryadka”, Izv. RAN. Ser. matem., 57:3 (1993), 70–91 | MR | Zbl

[17] Khabibullin B. N., “Nekonstruktivnye dokazatelstva teoremy Berlinga–Malyavena o radiuse polnoty i teoremy needinstvennosti dlya tselykh funktsii”, Izv. RAN. Ser. matem., 58:4 (1994), 125–148 | MR | Zbl

[18] Khermander L., Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968 | MR

[19] Khabibullin B. N., “Dvoistvennoe predstavlenie superlineinykh funktsionalov”, Kompleksnyi analiz, differentsialnye uravneniya, chislennye metody i prilozheniya. I. Kompleksnyi analiz, RAN i Institut matematiki s VTs, Ufa, 1996, 122–131

[20] Cole B. J., Ransford T. J., “Subharmonicity without upper semicontinuity”, J. Funct. Anal., 147 (1997), 420–442 | DOI | MR | Zbl

[21] Larusson F., Sigurdsson R., “Plurisubharmonic functions and analytic discs on manifolds”, J. reine angew. Math., 501 (1998), 1–39 | MR | Zbl

[22] Poletsky E. A., “Disk envelopes of functions, I”, Complex Analysis in Contemporary Mathematics, ed. E. M. Chirka, Fasis, M., 1997

[23] Poletsky E. A., “Disk envelopes of functions, II”, J. Funct. Anal., 163 (1999), 111–132 | DOI | MR | Zbl