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

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 $\mathcal{EV}(\Omega)$

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 surjectivity of the Borel mapping in the mixed setting for ultradifferentiable ramification spaces

We consider r-ramification ultradifferentiable classes, introduced by J. Schmets and M. Valdivia in order to study the surjectivity of the Borel map, and later on also exploited by the authors in the ultraholomorphic context. We characterize quasianalyticity in such classes, extend the results of Schmets and Valdivia about the image of the Borel map in a mixed ultradifferentiable setting, and obtain a version of the Whitney extension theorem in this framework.Comment: 31 pages; this version has been accepted for publication in Monatsh. Mat

On the Life and Work of S. Helgason

This article is a contribution to a Festschrift for S. Helgason. After a biographical sketch, we survey some of his research on several topics in geometric and harmonic analysis during his long and influential career. While not an exhaustive presentation of all facets of his research, for those topics covered we include reference to the current status of these areas.Comment: Final versio

Notions of Stein spaces in non-archimedean geometry

Let $k$ be a non-archimedean complete valued field and $X$ be a $k$-analytic space in the sense of Berkovich. In this note, we prove the equivalence between three properties: 1) for every complete valued extension $k'$ of $k$, every coherent sheaf on $X \times_{k} k'$ is acyclic; 2) $X$ is Stein in the sense of complex geometry (holomorphically separated, holomorphically convex) and higher cohomology groups of the structure sheaf vanish (this latter hypothesis is crucial if, for instance, $X$ is compact); 3) $X$ admits a suitable exhaustion by compact analytic domains considered by Liu in his counter-example to the cohomological criterion for affinoidicity. When $X$ has no boundary the characterization is simpler: in~2) the vanishing of higher cohomology groups of the structure sheaf is no longer needed, so that we recover the usual notion of Stein space in complex geometry; in 3) the domains considered by Liu can be replaced by affinoid domains, which leads us back to Kiehl's definition of Stein space. v2: major revision to handle also the case of spaces with boundaryComment: 31 page