CR extension from hypersurfaces of higher type

Abstract

We prove extension of CR functions from a hypersurface M of C N in presence of the so-called sector property. If M has finite type in the Bloom–Graham sense, then our result is already contained in [C. Rea, Prolongement holomorphe des fonctions CR, conditions suffisantes, C. R. Acad. Sci. Paris 297 (1983) 163–166] by Rea. We think however, that the argument of our proof carries an expressive geometric meaning and deserves interest on its own right. Also, our method applies in some case to hypersurfaces of infinite type; note that for these, the classical methods fail. CR extension is treated by many authors mainly in two frames: extension in directions of iterated of commutators of CR vector fields (cf., for instance, [A. Boggess, J. Pitts, CR extension near a point of higher type, Duke Math. J. 52 (1) (1985) 67–102; A. Boggess, J.C. Polking, Holomorphic extension of CR functions, Duke Math. J. 49 (1982) 757–784. [4]; M.S. Baouendi, L. Rothschild, Normal forms for generic manifolds and holomorphic extension of CR functions, J. Differential Geom. 25 (1987) 431–467. [1]]); extension through minimality towards unprecised directions [A.E. Tumanov, Extension of CR-functions into a wedge, Mat. Sb. 181 (7) (1990) 951–964. [6]; A.E. Tumanov, Analytic discs and the extendibility of CR functions, in: Integral Geometry, Radon Transforms and Complex Analysis, Venice, 1996, in: Lecture Notes in Math., vol. 1684, Springer, Berlin, 1998, pp. 123–141].