A result on extension of C.R. functions
Derridj, Makhlouf ; Fornaess, John Erik
Annales de l'Institut Fourier, Tome 33 (1983), p. 113-120 / Harvested from Numdam
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Soit Ω un ouvert de C n , de classe C 4 près de z 0 ∈∂Ω, λ une fonction holomorphe convenable près de z 0 . Sachant que l’on sait résoudre (voir [M. Derridj, Annali.Sci. Norm. Pisa, Série IV, vol. IX (1981)]) le problème : ∂ ¯u=λf(f(0,1) forme donnée dans U(z 0 ), ∂ ¯ fermée) dans U(z 0 ) avec supp(u)⊂Ω ¯∩U(z 0 ), on déduit un résultat d’extension de fonctions C.R. sur ∂Ω∩U(z 0 ), en fonctions holomorphes dans Ω∩V(z 0 ).

Let Ω an open set in C 4 near z 0 ∈∂Ω, λ a suitable holomorphic function near z 0 . If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : ∂ ¯u=λf, (f is a (0,1) form, ∂ ¯ closed in U(z 0 ) in U(z 0 ) with supp(u)⊂Ω ¯∩U(z 0 ), then we deduce an extension result for C.R. functions on ∂Ω∩U(z 0 ), as holomorphic fonctions in Ω∩V(z 0 ).

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     author = {Derridj, Makhlouf and Fornaess, John Erik},
     title = {A result on extension of C.R. functions},
     journal = {Annales de l'Institut Fourier},
     volume = {33},
     year = {1983},
     pages = {113-120},
     doi = {10.5802/aif.933},
     mrnumber = {85f:32031},
     zbl = {0518.32010},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1983__33_3_113_0}
}

Derridj, Makhlouf; Fornaess, John Erik. A result on extension of C.R. functions. Annales de l'Institut Fourier, Tome 33 (1983) pp. 113-120. doi : 10.5802/aif.933. http://gdmltest.u-ga.fr/item/AIF_1983__33_3_113_0/

[1] A. Andreotti and C.D. Hill, E.E. convexity and the H. Lewy problem. Part I : Reduction to vanishing theorems, Ann. Scuola Normale Sup. di Pisa, 26 (1972). | Numdam | Zbl 0256.32007

[2] A. Andreotti and E. Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Publ. IHES, vol. 24-25. | Numdam | Zbl 0138.06604

[3] M.S. Baouendi and F. Treves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. Math., 113 (1981). | MR 82f:35057 | Zbl 0491.35036

[4] E. Bedford and J.E. Fornaess, Local extension of C.R. function from weakly pseudoconvex boundaries, Michigan Math. J., 25. | MR 80c:32014 | Zbl 0401.32007

[5] A. Bogges, C.R. extendability near a point where the first leviform vanishes, Duke Math. J., 43 (3). | Zbl 0509.32006

[6] M. Derridj, Inégalités de Carleman et extension locale des fonctions holomorphes, Annali. Sci. Norm. Pisa, Serie IV vol. IX (1981). | Numdam | Zbl 0548.32013

[7] C.D. Hill and Taïani, Families of analytic discs in Cn with boundaries on a prescribed C.R. submanifold, Ann. Scuola Norm. Pisa, 4-5 (1978). | Numdam | Zbl 0399.32008

[8] L. Hörmander, Introduction to complex analysis in several variables, van Nostrand. | Zbl 0138.06203

[9] L. Hörmander, L²-estimates and existence theorems for the ∂-operator, Acta Math., 113 (1965). | MR 31 #3691 | Zbl 0158.11002

[10] J.J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. Math., 81 (1965). | MR 31 #1399 | Zbl 0166.33802

[11] H. Lewy, On the local character... Ann. Math., 64 (1956).

[12] R. Nirenberg, On the Lewy extension phenomenon, Trans. Amer. Math. Soc., 168 (1972). | MR 46 #392 | Zbl 0241.32006

[13] R.O. Wells, On the local holomorphic hull... Comm. P.A.M., vol. XIX (1966). | Zbl 0142.33901