Baouendi–Trèves approximation theorem for CR functions with values in a complex Fréchet space

Abstract

We study CR functions with values in a complex Frechet space X. We prove a vector valued analog to a result by Baouendi and Treves (Ann Math 113:387–421, 1981), i.e. any X-valued CR function of Teodorescu class B1 may be locally approximated by X-valued holomorphic functions on \({{\mathbb C}^n}\). We show that any CR function \({u \in C^\omega (M, X)}\) on a real analytic CR hypersurface \({M \subset {\mathbb {C}}^n}\) admits a unique holomorphic extension \({f \in {\mathcal {O}}(\Omega, X)}\) to some open neighborhood \({\Omega \supset M}\).