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 weighted spaces of 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 -valued functions whose topology is 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.