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

TL;DR

This paper establishes conditions under which the Cauchy-Riemann operator is surjective between weighted spaces of smooth vector-valued functions, extending classical results to more general function spaces.

Contribution

It generalizes H"ormander's theorem to weighted Fréchet-valued function spaces using tensor products and a weakened reducibility condition.

Findings

Derived sufficient conditions for surjectivity of ar in weighted spaces

Extended solvability results to vector-valued functions via tensor products

Provided a new approach using weak reducibility and Mittag-Leffler procedures

Abstract

We derive sufficient conditions for the surjectivity of the Cauchy-Riemann operator $\overline{\partial}$ 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.