About the holomorphic extension of CR functions on real hypersurfaces in complex space (Q1058022)

Without use of the method of analytic discs, the authors derive significant and general results on local holomorphic extensions of CR functions on a \(C^{\infty}\) hypersurface, \(\Sigma\), in \({\mathbb{C}}^{n+1}\). Suppose \(\Sigma\) is of finite type m (tangent to \({\mathbb{R}}^{2n+1}\) to order m) at 0. The authors prove the following results: (1) If m is odd then every CR function has an ambient holomorphic extension near 0. - (2) If m is even then every CR function has a holomorphic extension to at least one side of \(\Sigma\). These results follow from a deep analysis which first shows a sector property of Cauchy characteristics is sufficient for such extension results. Then the type conditions (even or odd) are shown to yield the appropriate sector properties.