## Analytic Discs Attached to Half Spaces of $\Bbb C^n$ and Extension of Holomorphic Functions

J. Math. Sci. Univ. Tokyo
Vol. 8 (2001), No. 2, Page 317--327.

Baracco, Luca ; Zampieri, Giuseppe
Analytic Discs Attached to Half Spaces of $\Bbb C^n$ and Extension of Holomorphic Functions
Let $M$ be a real hypersurface of $\C^n$, $M^+$ a closed half space with boundary $M$, $z_o$ a point of $M$. We prove that the existence of a disc $A$ tangent to $M$ at $z_o$, attached to $M^+$ but not to $M$ (i.e.$\partial A \subset M^+$ but $\partial A \not\subset M$), entails extension of holomorphic functions from the interior of ${M^+}$ to a full neighborhood of $z_o$. This result covers a result in \cite{9}, where the disc $A$ is assumed to lie on one side $M^+$ of $M$. The basic idea, which underlies to the whole paper, is due to A. Tumanov [8] and consists in attaching discs to manifolds with boundary. Further, holomorphic extendability by the aid of tangent discs attached to $M$ and of "defect 0" is a particular case of a general theorem of "wedge extendibility" of CR--functions by A. Tumanov.