Separate holomorphic extension of CR functions

Abstract

We consider a real analytic foliation of \({\mathbb{C}^n}\) by complex analytic manifolds of dimension m issued transversally from a CR generic submanifold \({M\subset\mathbb{C}^n}\) of codimension m. We prove that a continuous CR function f on M which has separate holomorphic extension along each leaf, is holomorphic. When the leaves are cartesian straight planes, separate holomorphic extension along suitable selections of these planes suffices and f turns out to be holomorphic in a neighbourhood of their union. If M is a hypersurface we can also specify the side of the extension, regardless the leaves are straight or not.