Analytic discs attached to manifolds with boundary

Let X be a complex manifold of dimension n, M+ a closed half-space with boundary M, A an analytic disc of X "attached" to M+, tangent to M at some point z0 of dA n M, and intersecting M+ in any neighbourhood of z0. Then holomorphic functions extend from M+ to a full neighborhood of z0. This theorem refines the results of [1] where the boundary dA (instead of the whole A) was supposed to intersect M+ . The argument of the proof consists in constructing a (closed) manifold with boundary W, contained in the envelop of holomorphy of M+ and such that A c W but A <£ d W. In this situation it is easy to find a new small disc A i c A with 8A l <£ dW. We are therefore in a situation similar to [1], and get the conclusion by exhibiting a disc transversal to d W at z0. Extension of holomorphic functions 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; the new part of our theorem is that no assumptions on "defect" are made. This paper is tightly inspired to the results and the techniques by A. Tumanov [7]. We also owe to A. Tumanov a great help during private communications. §2.