Title: Are nuclear spaces used in creating variant theories of distributions?
Tags: schwartz-distributions;nuclear-spaces
Question: Laurent Schwartz proved his Kernel Theorem in 1952  to justify extending his theory of distributions to several variables. Then he and Jean Dieudonne gave Alexander Grothendieck the assignment to explain what was really going on in the Kernel Theorem.  Grothendieck did that to their very great satisfaction by creating the concept of nuclear spaces and generalizing the Kernel Theorem to them.
I suppose that one motivation for better understanding,and then generalizing, the Kernel Theorem was to help in finding other variant theories of distributions, based on other  spaces of test functions than Schwartz's,  that might have other good features than Schwartz distributions.  Certainly, very many people have used other spaces of test functions for various purposes.
My question is, does the theory of nuclear spaces get used in developing these variant theories of generalized functions?  I do not find it prominently used in the sources I find.
Comment: This is just a suggestion which might interest you.  If $T$ is an unbounded, self adjoint operator on Hilbert space, then there is a canonical (and functorial) way to embed $H$ into a locally convex space on which $T$ operates continuously and which is nuclear if and only if $T$ has a discrete spectrum with eigenvalues such that $|\lambda|^\alpha$ is summable for some negative $\alpha$.  By choosing many classical differential operators (Sturm-Liouville, Laplace-Beltrami, Schrödinger ...), one obtains a whole palette of nuclear distribution-type spaces on various manifolds.


Here is another answer: Since this question has not been answered, I have decided to expand my comment.  A well documented construction allows one to associate a generalised space of distributions to each unbounded self-adjoint operator $T$ on a Hilbert space $H$.  It is motivated by the elementary and axiomatic approach of the portuguese mathematician J. Sebastião e Silva who showed that one can correct the basic “fault” of the differential operator on spaces of continuous functions by embedding them (in a unique manner) into larger superspaces on which it is everywhere defined.  These are the Schwartzian distributions.  In the general situation described above, a simple construction shows we can embed $H$ in a (in a suitable sense) unique manner into a vector space $H^{-\infty}$ on which $T$ is everywhere defined.  If we choose for $T$ any of the classical self-adjoint differential operators, we obtain a unified approach to many known, but also many new, spaces of distribution, including those introduced by Schwartz.  We can also vary the construction in simple ways to obtain other variants, including some which involve differential operators of infinite order.
The space $H^{-\infty}$ has a natural lc structure of a well-studied type (as the inductive limit of a sequence of Banach spaces with weakly compact linking mappings—Komatsu).  If $T$ has a discrete spectrum and its sequence $(\lambda_n)$ of eigenvalues converges to infinity (i.e., it is genuinely unbounded), then it is a Silva space (defined as above but with compact links).  Finally, if $(|\lambda_n|^\alpha)$ is summable for some negative $\alpha$, then it is nuclear, with all resulting benefits.  This latter condition is fulfilled if the eigenvalues are asymptotically like a positive power of $n$ as is the case for most of the classical differential operators which are relevant here.
References as requested:
Komatsu spaces: Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan 19 (1967), 366-383.
Silva spaces (not under that name of course): J. Sebastião e Silva, Su certe classi di spazi localmente convessi importanti per le applicazioni, Rend. Mat. e Appl. 14 (1955), 388-410. (Now in the secondary literature as Silva spaces, notably in the first volume of Köthe’s monumental “Topological Vector Spaces”).
His works on distribution theory can be found at the site “jss100.campus.ciencias.ulisboa.pt”  For an elementary overview (in english), go to “publicações”, then “Textos Didáticos”.
The abstract construction is in “Normal operators and spaces of distributions”, Collectanea Math. (1975), 257-284.  This contains the criterion for nuclearity.
You can then construct ready made distribution spaces by consulting the long tradition of computing the spectral properties of the classical self-adjoint differential operators, a tradition which goes back to Fourier, Thomson and Tait.  For the Sturm-Liouville operators, you could use the good old Courant and Hilbert. A more modern treatment with emphasis on connections with functional analysis is in Triebel’s “Höhere Analysis”.
For the spectral properties required for distributions on manifolds, see Berger et al, “Le Spectre d’une Variété Riemannienne”.  Corresponding results for Schrödinger operators have been investigated by Barry Simon amongst others.

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