Abstract. We prove local approximation and extension theorems for C submanifolds M of C" (CR submanifolds). Under some conditions on M, any smooth solution of the induced Cauchy-Riemann equations can be extended holomorphically to bigger (and sometimes open) sets.
Introduction.A characteristic feature of the theory of functions of several complex variables is the holomorphic extendibility of functions from a set to a bigger set. This phenomenon is intimately related to the vanishing of certain spaces of cohomology with compact support. The first example of this is the well-known theorem of Hartogs: any function defined and holomorphic on a neighborhood of the boundary of a ball in Cn (n> 1) can be extended holomorphically to the interior of the ball. A stronger form of this result was given by Bochner (see [ From then on, much work has been done on the local extendibility properties of real submanifolds M of C with real dimension less than 2n -1, but always in the setup of Hartogs' result, that is, one looks at functions which are holomorphic on some open neighborhood of M in Cn, and tries to extend them to a fixed open set of Cn. In other words, one extends germs of holomorphic functions on M. The pioneer work in this direction was done by Bishop [3]. Further results are found in
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