Structural theorems for quasiasymptotics of ultradistributions

We provide complete structural theorems for the so-called quasiasymptotic behavior of non-quasianalytic ultradistributions. As an application of these results, we obtain descriptions of quasiasymptotic properties of regularizations at the origin of ultradistributions and discuss connections with Gelfand-Shilov type spaces

Convolutors of translation-modulation invariant Banach spaces of ultradistributions

We study the space of tempered ultradistributions whose convolutions with test functions are all contained in a given translation-modulation invariant Banach space of ultradistributions. Our main result will be the first structural theorem for the aforementioned space. As an application we consider several extensions of convolution.Comment: 37 page

Kernel theorems for Beurling-Bj\"orck type spaces

We prove new kernel theorems for a general class of Beurling-Bj\"orck type spaces. In particular, our results cover the classical Beurling-Bj\"orck spaces $\S^{(\omega)}_{(\eta)}$ and $\S^{\{\omega\}}_{\{\eta\}}$ defined via weight functions $\omega$ and $\eta$.Comment: 17 page

Weighted (PLB)(PLB)-spaces of ultradifferentiable functions and multiplier spaces

We study weighted $(PLB)$-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight system. This generalizes Grothendieck's classical result that the space $\mathcal{O}_M$ of slowly increasing smooth functions is ultrabornological to the context of ultradifferentiable functions. Furthermore, we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the above result, characterize when such spaces are ultrabornological. In particular, we show that the multiplier space of the space of Fourier ultrahyperfunctions is ultrabornological, whereas the one of the space of Fourier hyperfunctions is not

An extension result for (LB)(LB)-spaces and the surjectivity of tensorized mappings

We study an extension problem for continuous linear maps in the setting of $(LB)$-spaces. More precisely, we characterize the pairs $(E,Z)$, where $E$ is a locally complete space with a fundamental sequence of bounded sets and $Z$ an $(LB)$-space, such that for every exact sequence of $(LB)$-spaces $$0 \rightarrow X \xrightarrow{\iota} Y \rightarrow Z \rightarrow 0$$ the mapping $$L(Y,E) \to L(X, E), ~ T \mapsto T \circ \iota$$ is surjective, meaning that each continuous linear map $X \to E$ can be extended to a continuous linear map $Y \to E$ via $\iota$, under some mild conditions on $E$ or $Z$ (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized mappings between Fr\'echet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [23].Comment: 28 page

Asymptotic boundedness and moment asymptotic expansion in ultradistribution spaces

We obtain structural theorems for the so-called S-asymptotic and quasiasymptotic boundedness of ultradistributions. Using these results, we then analyze the moment asymptotic expansion (MAE), providing a full characterization of those ultradistributions satisfying this asymptotic formula in the one-dimensional case. We also introduce and study a uniform variant of the MAE