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

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator on spaces $\mathcal{EV}(\Omega,E)$ 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-K\"othe-Silva duality $\mathcal{O}(\mathbb{C}\setminus K)/\mathcal{O}(\mathbb{C})\cong\mathscr{A}(K)$ with non-empty compact $K\subset\mathbb{R}$ for weighted holomorphic functions. We use this duality to prove that the kernel $\operatorname{ker}\overline{\partial}$ of the Cauchy-Riemann operator $\overline{\partial}$ in $\mathcal{EV}(\Omega):=\mathcal{EV}(\Omega,\mathbb{C})$ has the property $(\Omega)$ of Vogt. Then an application of the splitting theory of Vogt for Fr\'{e}chet spaces and of Bonet and Doma\'nski for (PLS)-spaces in combination with some previous results on the surjectivity of the Cauchy-Riemann operator $\overline{\partial}\colon\mathcal{EV}(\Omega)\to\mathcal{EV}(\Omega)$ yields the surjectivity of the Cauchy-Riemann operator on $\mathcal{EV}(\Omega,E)$ if $E:=F_{b}'$ with some Fr\'{e}chet 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 $\mathcal{EV}(\Omega)$

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

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

Extension of vector-valued functions and sequence space representation

We give a unified approach to handle the problem of extending functions with values in a locally convex Hausdorff space $E$ over a field $\mathbb{K}$, which have weak extensions in a space $\mathcal{F}(\Omega,\mathbb{K})$ of scalar-valued functions on a set $\Omega$, to functions in a vector-valued counterpart $\mathcal{F}(\Omega,E)$ of $\mathcal{F}(\Omega,\mathbb{K})$. The results obtained base upon a representation of vector-valued functions as linear continuous operators and extend results of Bonet, Frerick, Gramsch and Jord\'{a}. In particular, we apply them to obtain a sequence space representation of $\mathcal{F}(\Omega,E)$ from a known representation of $\mathcal{F}(\Omega,\mathbb{K})$.Comment: The former version arXiv:1808.05182v2 of this paper is split into two parts. This is the first par

The abstract Cauchy problem in locally convex spaces

We derive sufficient criteria for the uniqueness and existence of solutions of the abstract Cauchy problem in locally convex Hausdorff spaces. Our approach is based on a suitable notion of an asymptotic Laplace transform and extends results of Langenbruch beyond the class of Fr\'echet spaces

Weighted composition semigroups on spaces of continuous functions and their subspaces

This paper is dedicated to weighted composition semigroups on spaces of continuous functions and their subspaces. We consider semigroups induced by semiflows and semicocycles on Banach spaces $\mathcal{F}(\Omega)$ of continuous functions on a Hausdorff space $\Omega$ such that the norm-topology is coarser than the compact-open topology like the Hardy spaces, the weighted Bergman spaces, the Dirichlet space, the Bloch type spaces and weighted spaces of continuous or holomorphic functions. It was shown by Gallardo-Guti\'errez, Siskakis and Yakubovich that there are no non-trivial norm-strongly continuous weighted composition semigroups on Banach spaces $\mathcal{F}(\mathbb{D})$ of holomorphic functions on the open unit disc $\mathbb{D}$ such that $H^{\infty}\subset\mathcal{F}(\mathbb{D})\subset\mathcal{B}_{1}$ where $H^{\infty}$ is the Hardy space of bounded holomorphic functions on $\mathbb{D}$ and $\mathcal{B}_{1}$ the Bloch space. However, we show that there are non-trivial weighted composition semigroups on such spaces which are strongly continuous w.r.t. the mixed topology between the norm-topology and the compact-open topology. We study such weighted composition semigroups in the general setting of Banach spaces of continuous functions and derive necessary and sufficient conditions on the spaces involved, the semiflows and semicocycles for strong continuity w.r.t. the mixed topology and as a byproduct for norm-strong continuity as well. Moreover, we give several characterisations of their generator and their space of norm-strong continuity

On linearisation, existence and uniqueness of preduals

We study the problem of existence and uniqueness of preduals of locally convex Hausdorff spaces. We derive necessary and sufficient conditions for the existence of a predual with certain properties of a bornological locally convex Hausdorff space $X$. Then we turn to the case that $X=\mathcal{F}(\Omega)$ is a space of scalar-valued functions on a non-empty set $\Omega$ and characterise those among them which admit a special predual, namely a strong linearisation, i.e. there is a locally convex Hausdorff space $Y$, a map $\delta\colon\Omega\to Y$ and a topological isomorphism $T\colon\mathcal{F}(\Omega)\to Y_{b}'$ such that $T(f)\circ \delta= f$ for all $f\in\mathcal{F}(\Omega)$. We also lift such a linearisation to the case of vector-valued functions, covering many previous results on linearisations, and use it to characterise the bornological spaces $\mathcal{F}(\Omega)$ with (strongly) unique predual in certain classes of locally convex Hausdorff spaces