We characterize possible bounds for representing functions of arbitrary hyperfunctions. Specifically, there are always representing functions decreasing rapidly outside each strip near $\mathbb{R}$. Also, exponential decrease of any type on any strip $\mathbb{R}\times \pm i[c,C], 0<c<C<\infty,$ can be achieved. This will be used in [10] to define an asymptotic Fourier and Laplace transformation on the space of hyperfunctions.

A Fréchet space of distributions which is stable under differential operators is continuously included in ${\mathcal C}^\infty$. We give extensions of this result to the ultradifferentiable setting and show their connection with the problem of iterates of differential operators

Given $p \in \mathbb{N}$, a non empty open subset $\Omega$ of $\mathbb{R}^k$ and a semi-regular matrix $\mathfrak{M}$, we characterize the elements of the duals of the Beurling classes $\mathcal{D}{(\mathfrak{M})}{\Omega}$ and $\\mathcal{D}_{L_p}{(\mathfrak{M})}{\Omega}$ of ultradifferentiable functions. We provide a global representation of these ultradistributions with and without compact support by means of series involving measures in the first case and elements of $\L_{loc}^{q}{\Omega}$ in the second.

We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation $P(D)u=f$, i.e., the question if for every family $(f_\lambda)\subseteq \mathscr_{\{\omega\}}(\Omega)$ of ultradifferentiable functions of Roumieu type (in particular, of real analytic functions or of functions from Gevrey classes) depending in a real analytic way on $\lambda\in U$, $U$ a real analytic manifold, there is a family of solutions $(u_\lambda)\subseteq \mathscr_{\{\omega\}}(\Omega)$ also depending analytically on $\lambda$ such that $$ P(D)u_\lambda=f_\lambda \text{for every $\lambda\in U$}, $$ where $\Om\subseteq \mathbb{R}^d$ an open set. We solve the problem for many types of differential operators following a similar method as in the earlier paper of the same author for operators acting on spaces of distributions. We show for an operator $P(D)$ on the space of real analytic functions $\mathscr{A}(\Omega)$, $\Omega \subseteq \mathbb{R}^d$ open convex, that it has real analytic parameter dependence if and only if its principal part $P_p(D)$ has a continuous linear right inverse on the space $C^\infty(\Omega)$ (or, equivalently, on $\mathscr{D}'(\Omega)$). In particular, the property does not depend on the set of parameters $U$. Surprisingly, in all solved non-quasianalytic cases, it follows that the solution is positive if and only if $P(D)$ has a linear continuous right inverse.

Let $P \in \mathbb{C}[\tau, \zeta_1, \ldots, \zeta_n]$ be a quadratic polynomial for which the $\tau$-variable is non-characteristic. We characterize when the zero-variety $V(P)$ of $P$ satisfies the Phragmén-Lindelöf condition $PL(\omega)$ or equivalently when the pair $(\mathbb{R}_x^n, \mathbb{R}_\tau \times \mathbb{R}_x^n)$ is of evolution in the class ${\mathcal E}_\omega$ for the partial differential operator $P(D)$ with symbol $P$.

We present a new formula for the Fourier transform of a Lorentz invariant temperate distribution. The formula is applied so as to yield the temperate fundamental solution of the Klein-Gordon operator.

Let $C[0,1]$ be the Banach algebra of real valued continuous functions on $[0,1]$, provided with the supremum norm. For $f,g\in C[0,1]$ and balls $B_{f}$, $B_{g}$ with center $f$ and $g$, respectively, it is not necessarily true that $f\cdot g$ is in the interior of $B_{f}\cdot B_{g}$. In the present paper we characterize those pairs $f,g$ where this is the case. The problem is illustrated by using a suitable translation. One studies walks in a landscape with hills and valleys where an accompanying dog can move in a certain prescribed way.