Surjectivity of the ∂-operator between weighted spaces of smooth vector-valued functions

Abstract

We derive sufficient conditions for the surjectivity of the Cauchy-Riemann operator ∂ between spaces of weighted smooth Fréchet-valued functions. This is done by establishing an analog of Hörmander's theorem on the solvability of the inhomogeneous Cauchy-Riemann equation in a space of smooth ℂ-valued functions whose topologyis given by a whole family of weights. Our proof relies on a weakened variant of weak reducibility of the corresponding subspace of holomorphic functions in combination with the Mittag-Leffler procedure. Using tensor products, we deduce the corresponding result on the solvability of the inhomogeneous Cauchy-Riemann equation for Fréchet-valued functions.