Extension of Valdivia-Vogt isomorphism from $\mathscr{D}(K)$ to $\mathscr{E}'(K)$

Question

Let $M$ be a $d$-dimensional (say, Hausdorff, paracompact, connected and oriented) smooth manifold, and $K\subset M$ compact with $\mathring{K}\neq\varnothing$. M. Valdivia has shown (based on previous results by himself and D. Vogt, see e.g. M. Valdivia, A representation of the space $\mathscr{D}(K)$, J. reine angew. Math. 320 (1980) 97-98) that the nuclear Fréchet space $\mathscr{D}(K)$ of smooth functions supported in $K$ is topologically isomorphic to the space $s$ of rapidly decreasing sequences: $$s=\{(a_n)_{n\in\mathbb{N}}\ |\ ((1+n)^k a_n)_{n\in\mathbb{N}}\text{ is bounded for all }k\in\mathbb{N}\}\ .$$ Let $\Phi:\mathscr{D}(K)\cong s$ denote a Valdivia-Vogt isomorphism (see Edit below). It is clear that the transpose ${}^t\Phi$ of $\Phi$ yields a topological isomorphism between the dual $s'$ of $s$ $$s'=\{(a_n)_{n\in\mathbb{N}}\ |\ ((1+n)^{-k}a_n)_{n\in\mathbb{N}}\text{ is bounded for some }k\in\mathbb{N}\}$$ and the dual $\mathscr{D}(K)'$ of $\mathscr{D}(K)$, which may be identified as a vector space with $\mathscr{D}'(\wedge^d T^*M\rightarrow M)/\mathscr{D}(K)^\perp$, where $$\mathscr{D}(K)^\perp=\{u\in\mathscr{D}'(\wedge^d T^*M\rightarrow M)\ |\ u(\varphi)=0\text{ for all }\varphi\in\mathscr{D}(K)\}$$ is the annihilator of $\mathscr{D}(K)$. It is clear that $\mathscr{D}(K)'$ contains $$\mathscr{E}'(K)=\{u\in\mathscr{E}'(\wedge^d T^*M\rightarrow M)\ |\ \text{supp }u\subset K\}$$ as a (closed) subspace (I apologize for the slightly unconventional notation). Since the sequences $e_j=(e_{j,n})_{n\in\mathbb{N}}$ given by $$e_{j,n}=\begin{cases} 0 & (n\neq j) \\ 1 & (n=j) \end{cases}$$ form a Schauder basis of both $s$ and $s'$, it is clear that $s$ is dense in $s'$.

Question: Is there a choice of $\Phi$ (see Edit below) such that it extends to a topological isomorphism between $\mathscr{E}'(K)$ and $s'$? Likewise, does the restriction of ${}^t\Phi$ to $s$ for such a $\Phi$ yield another topological isomorphism between $s$ and $\mathscr{D}(K)$?

My question is inspired by the known characterization of $\mathscr{D}([0,1])$ and $\mathscr{E}'([0,1])$ through the decay / growth of their Fourier coefficients in $[0,1]$.

Edit: As suggested by Jochen Wengenroth in the comments below, the recent works of Bargetz

• C. Bargetz, Commutativity of the Valdivia-Vogt table of representations of function spaces. Math. Nachr. 287 (2014) 10-22
• C. Bargetz, Explicit representations of spaces of smooth functions and distributions, J. Math. Anal. Appl. 424 (2015) 1491-1505
• C. Bargetz, Completing the Valdivia-Vogt tables of sequence-space representations of spaces of smooth functions and distributions. Monatsh. Math. 177 (2015) 1-14

indicate that the answer to the question may be negative depending on the choice of Valdivia-Vogt isomorphism $\Phi$. More precisely, one needs first for ${}^t\Phi$ to map $\{e_j\ |\ j\in\mathbb{N}\}$ into $\mathscr{D}(K)$, which may not be the case.

Answer 1

Not a full answer but too long for am comment. For certain compact sets the answer might be positive when the isomorphism given by the Wilson basis constructed in the following paper is used:

C. Bargetz, A. Debrouwere, E. A. Nigsch: Sequence space representations for spaces of smooth functions and distributions via Wilson bases. arXiv: 2107.00245

In particular the isomorphisms there have the property that all of them arise from a single isomorphism between $\mathscr{D}'(\mathbb{R}^{n})$ and $s'\widehat{\otimes}\mathbb{C}^{\mathbb{N}}$.

Answer 2

This is just an addendum to the above answers so really a comment but too long--I hope that it will be of interest. At first, there is an abstract construction which associates to each self adjoint, unbounded operator on Hilbert space in a functorial way a densely embedded Fréchet space, whose dual space os identifiable as a superspace. The point is that if we take some of the classical such operators (notably, Laplace-Beltrami and Schrödinger operators), one obtains a unified approach to many (most?) of the standard spaces of distributions. Before giving example, let me note that the relevance to your question lies in the fact that if the operator has a discrete spectrum with eigenvalues that are asymptotically like a positive power of $n$, then we have precisely your situation--the ONB of eigenvectors is also a basis for the two derived spaces a d this establishes there isomorphism with $s$ and $s'$.

The canonical example is that of the Laplace-Beltrami operator on a closed compact Riemannian manifold where one gets the spaces of smooth functions and distributions thereon. The most important cases are the hypersphere and the higher dimensional tori. One can also consider the Laplacian on sufficiently regular open subsets of eucliidean space, with suitable boundary conditions (Dirichlet, Neumann, mixed). In the case of the distributions on the closed unit interval, one uses the Legendre differential operator with the Legendre function as basis. Many concrete compact subsets of euclidean space have been studied in detail (more recently, the case of fractals) but as mentioned above there doesn't seem to be a theory which covers your question in full generality.