arXiv subjects
Publications and source records attributed to Dietmar Vogt.
Dronov and Kaplitzki showed that every complemented subspace of a nuclear Köthe space E with a regular basis of type ($d_1$) has a basis so, in particular, solving the long standing problem whether any complemented subspace of the space (s) of rapidly decreasing sequences has a basis. We present a slightly modified version of their proof which shows that the range of every closed-range operator in $E$ has a basis
We study continuous linear operators on $\mathscr{D}'(\mathbb{R}^d)$ which admit all monomials as eigenvectors, that is, operators of Hadamard type. Such operators on $C^\infty(\mathbb{R}^d)$ and on the space $A(\mathbb{R}^d)$ of real analytic functions on $\mathbb{R}^d$ have been investigated by Domanski, Langenbruch and the author. The situation in the present case, however, is quite different and also the characterization. An operator $L$ on $\mathscr{D}'(\mathbb{R}^d)$ is of Hadamard type if there is a distribution T, the support of which has positive distance to all coordinate hyperplanes and which has a certain behaviour at infinity, such that $L(S) = S \star T$ for all $S \in \mathscr{D}'(\mathbb{R}^d)$. Here $(S \star T)φ= S_y(T_x φ(xy))$ for all $φ\in \mathscr{D}(\mathbb{R}^d)$. To describe the behaviour at infinity we introduce a class $\mathscr{O}_H'(\mathbb{R}^d)$ of distributions defined by the same conditions like in the description of class $\mathscr{O}_C'(\mathbb{R}^d)$ of Laurent Schwartz, but derivatives replaced with Euler derivatives.
We study Hadamard operators on $S'(R^d)$ and give a complete characterization. They have the form $L(S)=S*T$ where * here means the multiplicative convolution and T is in the space of distributions which are $θ$-rapidly decreasing in infinity and at the coordinate hyperplanes. To show this we study and characterize convolution operators on the space $Y(R^d)$ of exponentially decreasing $C^\infty$-functions on $R^d$. We use this and the exponential transformation to characterize the Hadamard operators on $S'(Q)$, $Q$ the positive quadrant, and this result we use as a building block for our main result.
We study the algebra $\mathscr{E}'(\mathbb{R}^d)$ equipped with the multiplication $(T\star S)(f)=T_x(S_y(f(xy))$ where $xy=(x_1y_1,\dots,x_dy_d)$. This allows us a very elegant access to the theory of Hadamard type operators on $C^\infty(Ω)$, $Ω$ open in $\mathbb{R}^d$, that is, of operators which admit all monomials as eigenvectors. We obtain a representation of the algebra of such operators as an algebra of holomorphic functions with classical Hadamard multiplication. Finally we study global solvability for such operators on open subsets of $\mathbb{R}_+^d$.
Euler operators are partial differential operators of the form $P(θ)$ where $P$ is a polynomial and $θ_j = x_j \partial/\partial x_j$. We show that every non-trivial Euler operator is surjective on the space of temperate distributions on $R^d$. This is in sharp contrast to the behaviour of such operators when acting on spaces of differentiable or analytic functions.
Euler operators are partial differential operators of the form $P(θ)$ where $P$ is a polynomial and $θ_j = x_j \partial/\partial x_j$. They are surjective on the space of temperate distributions on $R^d$. We show that this is, in general, not true for the space of Schwartz distributions on $R^d$, $d\ge 3$, for $d=1$, however, it is true. It is also true for the space of distributions of finite order on $R^d$ and on certain open sets $Ω\subset R^d$, like the euclidian unit ball.
For open sets $Ω\subset \mathbb{R}^d$ we study Hadamard operators on $\mathscr{D}'(Ω)$, that is, continuous linear operators which admit all monomials as eigenvectors. We characterize them as operators of the form $L(S)=S\star T$ where $T$ is a distribution and $\star$ the multiplicative convolution. This extends previous results for the case of $Ω=\mathbb{R}^d$ but requires essentially different methods.
Let $K\subset\R^d$ be compact and $A(K)$ the space of germs of real analytic functions on $K$ with its natural (LF)-topology. This topology can be given by $A(K)=\limind_{k\to+\infty} A_k$ where $A_k=\{(f_α)_{α\in\N_0^d}\in C(K)^{\N_0^d}\,:\, \|f\|_k:=\sup_{x\in K} \frac{|f^{(α)}(x)|}{α!} k^{-|α|}< +\infty\}.$ Based on this description we give in the present note an explicit fundamental system of seminorms for $A(K)$.
It is shown that under certain stability conditions a complemented subspace of the space $s$ of rapidly decreasing sequences is isomorphic to $s$ and this condition characterizes $s$. This result is used to show that for the classical Cantor set $X$ the space $C_\infty(X)$ of restrictions to $X$ of $C^\infty$-functions on $\R$ is isomorphic to $s$, so completing the theory developed in "Restriction spaces of $A^\infty$", to appear in Rev. Mat. Iberoamericana 29.4 (2013)