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

Abstract

We derive sufficient conditions for the surjectivity of the Cauchy-Riemann operator $\overline{\partial }$ 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 $\mathbb{C}$-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 Fréchet-valued functions.

References in corpus (4)

• Weighted vector-valued functions and the $\epsilon$-product
• The approximation property for weighted spaces of differentiable functions
• Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions
• On the nuclearity of weighted spaces of smooth functions

Cited by in corpus (4)

• The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes
• Asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions
• On the surjectivity of the Cauchy-Riemann and Laplace operators on weighted spaces of smooth functions
• Vector-valued Fourier hyperfunctions and boundary values