The inhomogeneous Cauchy-Riemann equation for weighted smooth vector-valued functions on strips with holes

Abstract

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator on spaces $\mathcal{E}\mathcal{V}(\Omega ,E)$ of ${\mathcal{C}}^{\infty }$-smooth vector-valued functions whose growth on strips along the real axis with holes $K$ is induced by a family of continuous weights $\mathcal{V}$. Vector-valued means that these functions have values in a locally convex Hausdorff space $E$ over $\mathbb{C}$. We characterise the weights $\mathcal{V}$ which give a counterpart of the Grothendieck-Köthe-Silva duality $\mathcal{O}(\mathbb{C}\setminus K)/\mathcal{O}(\mathbb{C})\cong \mathcal{A}(K)$ with non-empty compact $K\subset \mathbb{R}$ for weighted holomorphic functions. We use this duality to prove that the kernel $\ker \overline{\partial }$ of the Cauchy-Riemann operator $\overline{\partial }$ in $\mathcal{E}\mathcal{V}(\Omega ):=\mathcal{E}\mathcal{V}(\Omega ,\mathbb{C})$ has the property $(\Omega )$ of Vogt. Then an application of the splitting theory of Vogt for Fréchet spaces and of Bonet and Domański for (PLS)-spaces in combination with some previous results on the surjectivity of the Cauchy-Riemann operator $\overline{\partial }:\mathcal{E}\mathcal{V}(\Omega )\to \mathcal{E}\mathcal{V}(\Omega )$ yields the surjectivity of the Cauchy-Riemann operator on $\mathcal{E}\mathcal{V}(\Omega ,E)$ if $E:=F_b^{\prime }$ with some Fréchet space $F$ satisfying the condition $(DN)$ or if $E$ is an ultrabornological (PLS)-space having the property $(PA)$. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on $\mathcal{E}\mathcal{V}(\Omega )$.

References in corpus (4)

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

Cited by in corpus (4)

• Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions
• Asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions
• The abstract Cauchy problem in locally convex spaces
• Vector-valued Fourier hyperfunctions and boundary values