A result on extension of C.R. functions

Derridj, Makhlouf; Fornaess, John Erik

Annales de l'Institut Fourier, Tome 33 (1983) no. 3, pp. 113-120.

Soit $\Omega$ un ouvert de ${\mathbf{C}}^n$, de classe $C^4$ près de $z_0\in \partial \Omega$, $\lambda$ 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 : $\overline{\partial }u=\lambda f\left(f\right(0,1)$ forme donnée dans $U\left(z_0\right)$, $\overline{\partial }$ fermée) dans $U\left(z_0\right)$ avec supp$\left(u\right)\subset \overline{\Omega }\cap U(z_0)$, on déduit un résultat d’extension de fonctions $C.R.$ sur $\partial \Omega \cap U\left(z_0\right)$, en fonctions holomorphes dans $\Omega \cap V\left(z_0\right)$.

Let $\Omega$ an open set in ${\mathbf{C}}^4$ near $z_0\in \partial \Omega$, $\lambda$ 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)]) : $\overline{\partial }u=\lambda f$, ($f$ is a $(0,1)$ form, $\overline{\partial }$ closed in $U\left(z_0\right)$ in $U\left(z_0\right)$ with supp$\left(u\right)\subset \overline{\Omega }\cap U(z_0)$, then we deduce an extension result for $C.R.$ functions on $\partial \Omega \cap U\left(z_0\right)$, as holomorphic fonctions in $\Omega \cap V\left(z_0\right)$.

[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

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

[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 | Zbl

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

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

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

[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

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

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

[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 | Zbl

[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 | Zbl

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