Surjectivity of the $\bar{\mathrm{\partial }}$-operator between weighted spaces of smooth vector-valued functions

We derive sufficient conditions for the surjectivity of the Cauchy-Riemann operator $\bar{\mathrm{\partial }}$ between spaces of weighted smooth Fr'echet-valued functions. This is done by establishing an analog of H"ormander’s theorem on the solvability of the inhomogeneous Cauchy-Riemann equation in a space of smooth $\mathbb{C}$-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 Fr'echet-valued functions.