Invariant subspaces in functional spaces of polynomial growth on $\mathbb{R}^{N}$

S. S. Platonov

Problemy analiza, no. 4 (1997), pp. 105-124.

Résumé

Let $G$ be a transitive group of transformations of a set $M, \mathcal{F}$ be some locally convex space consisting of complex-valued functions on $M, \pi(g): f(x)\to f(g^{-1}x), f(x)\in \mathcal{F}$ be the quasiregular representation of $G$. A linear subspace $H\subseteq \mathcal{F}$ we call an invariant subspace if $H$ is closed and invariant with respect to the representation $\pi$. In the paper we consider the case when $M$ is $n$-dimensional Euclidean space $R^{n}, G$ is the group of all orientation-preserving isometries. The function spaces are spaces of polynomial growth, for example $\mathcal{F}=S'$ is the space of tempered distributions on $R^{n}$. The main result of the paper is the complele description of invariant subspaces of this function spaces. In particular we obtain the description of irreductible and indecomposable subspaces.