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There is a well known geometric characterization of tempered distributions on $\mathbb{R}^n$.

A distribution $T\in \mathcal{D}'(\mathbb{R}^n)$ is an element of $\mathcal{S}'(\mathbb{R}^n)$ if and only if $T$ is the restriction of a distribution $\tilde{T}$ defined on the sphere $S^n$. (See "Theorie des distributions" from L.Schwartz p. 238)

In a paper of 1958, Silva defined a new notion of distribution using infra-exponential growth conditions. These are more general then tempered distributions and are used in ultrahyperfonction fourier theory. Here are the definitions :

For a convex compact set $K$, I will note $h_K$ the support function of $K$.

Let $K$ be a convex compact set of $\mathbb{R}^n.$ The set $H_b(\mathbb{R}^n,K)$ is the set of functions $f\in C_{\infty}(\mathbb{R}^n)$ such that $$||f||_{K,n} :=\sup_{x\in \mathbb{R}^n ; \alpha \leq n} \{e^{h_K(x)}|D^{\alpha}f(x)|\} < \infty,$$ for all $n\in \mathbb{N}.$ Equipped with these semi-norms, the space is a Frechet space. For $O$ a convex open set of $\mathbb{R}^n$, we set $$H(\mathbb{R}^n, O) = \varprojlim_{K \subset O}H_b(\mathbb{R}^n,K).$$ The Silva distributions with $O$ conditions are the element of the dual space $$H'(\mathbb{R}^n,O) \subset \mathcal{D}'(\mathbb{R}^n).$$

Is there any geometric characterization of the Silva distributions as the one for tempered distributions ? (For example, using the sphere) Any help or references will be much appreciated.

ADDENTUM The following caracterisation is already known. An element $T \in \mathcal{D}'(\mathbb{R}^n)$ is an element of $H'(\mathbb{R}^n,O)$ if and only if there is a multi-index $\alpha$, a convex compact set $K \subset O$ and a continuous fonction $F$ such that $$T = D^{\alpha}(\exp(h_K(x))F(x)).$$ However, for me it is not a geometric caracterisation as the one given by L. Schwartz in his book.

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    $\begingroup$ I hear about it for the first time, but could they be restrictions of distributions on compactifications of $\mathbb R^n$? $\endgroup$ – მამუკა ჯიბლაძე Oct 17 '17 at 9:36
  • $\begingroup$ I doubt about it since it is already the criterion for tempered distributions and Silva distributions are different from tempered one. $\endgroup$ – C. Dubussy Oct 17 '17 at 13:00
  • $\begingroup$ Oh perhaps you mean more exotic compactifications. Since it explicitely uses the convexity, I thought about convex compactification but I don't really now theses theories. $\endgroup$ – C. Dubussy Oct 17 '17 at 13:07
  • $\begingroup$ Isn't it enough to take $K$ as a ball of radius $r$ so that $h_K(x)=r\|x\|$? $\endgroup$ – Jochen Wengenroth Oct 17 '17 at 13:47
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    $\begingroup$ @მამუკაჯიბლაძე How would one define distributions on a space like $\beta \mathbb{R}^n$ that is only compact+Haussdorff but does not have any notion of differentiation? $\endgroup$ – Johannes Hahn Oct 17 '17 at 22:11

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