Geodesic
A boundary uniqueness theorem in $\mathbf C^n$
Sbornik. Mathematics, Tome 30 (1976) no. 4, pp. 501-514 
A. S. Sadullaev. A boundary uniqueness theorem in $\mathbf C^n$. Sbornik. Mathematics, Tome 30 (1976) no. 4, pp. 501-514. http://geodesic.mathdoc.fr/item/SM_1976_30_4_a4/
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     title = {A~boundary uniqueness theorem in~$\mathbf C^n$},
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Voir la notice de l'article provenant de la source Math-Net.Ru

The classical boundary theorem of F. and M. Riesz asserts that, if the radial limits of a bounded holomorphic function $f(z)$ in the disk $|z|<1$ lie in a set of capacity zero for a set of positive measure on the circle $|z|=1$, then $f(z)\equiv\mathrm{constant}$. The main result of this paper is the proof of an analogous theorem for maps $F\colon D\to\mathbf C^n$, where $D$ is a domain in $\mathbf C^n$. We take as uniqueness set on the boundary any set of positive Lebesgue measure on a generating submanifold. Bibliography: 12 titles.

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[2] S. I. Pinchuk, “Granichnaya teorema edinstvennosti dlya golomorfnykh funktsii neskolkikh kompleksnykh peremennykh”, Matem. zametki, 15:2 (1974), 205–212 | Zbl

[3] A. Sadullaev, “Analog teoremy F. i M. Rissov”, DAN UzSSR, 1974, no. 7, 8–10 | MR | Zbl

[4] M. Brelo, Osnovy klassicheskoi teorii potentsiala, IL, Moskva, 1964 | MR

[5] L. I. Ronkin, Vvedenie v teoriyu tselykh funktsii mnogikh peremennykh, izd-vo «Nauka», Moskva, 1971 | MR

[6] P. Lelong, Fonctions plurisousharmoniques et formes differentielles positives, Dunod, 1968 | MR | Zbl

[7] P. Lelong, “Fonctions entieres ($n$ variables) et fonctions plurisousharmoniques de type exponentiel. Applications a l'analyse fonctionnelle”, Contemporary Problems in Theory Anal. Functions (Internat. Conf., Erevan, 1965), 1966 | MR

[8] E. Stein, “Boundary values of holomorphic fonctions”, Bull. Amer. Math. Soc., 76:6 (1970), 1292–1296 | DOI | MR | Zbl

[9] E. M. Chirka, “Teorema Lindelefa i Fatu v $\mathbf{C}^n$”, Matem. sb., 92 (134) (1973), 622–644 | Zbl

[10] E. Kollingvud, A. Lovater, Teoriya predelnykh mnozhestv, izd-vo «Mir», Moskva, 1971 | MR

[11] L. I. Ronkin, “O tselykh funktsiyakh konechnoi stepeni i o funktsiyakh vpolne regulyarnogo rosta ot neskolkikh peremennykh”, DAN SSSR, 119:2 (1958), 211–214 | MR | Zbl

[12] A. Sadullaev, “Kriterii algebraichnosti analiticheskikh mnozhestv”, Funkts. analiz, 6:1 (1972), 85–86 | MR | Zbl