Hilbert-valued Schwartz functions

Question

Let $H$ be a separable complex Hilbert space. We can define Schwartz functions $f\colon\mathbb R^n\to H$ to be the smooth functions for which $$ \sup_{x\in\mathbb R^n}\|(1+|x|^2)^mD^\alpha f(x)\|_H<\infty $$ for all $m\in\mathbb N$ and all multi-indices $\alpha$. I would like to use the space $S(\mathbb R^n,H)$ of $H$-valued Scwhartz functions and its dual $S'(\mathbb R^n,H)$, but I don't want to build the theory from scratch.

Is there a good book or other reference material for this kind of thing? I was unable to find anything. Here are some examples of what I would like the material to cover:

• Let $f\in S(\mathbb R^n,H)$ and let $\{e_k;k\in\mathbb N\}$ be an orthonormal basis for $H$. Then is each $f_k(x)=\langle f(x),e_k\rangle_H$ a Schwartz function in the usual sense and does the series $\sum_kf_k(x)e_k$ converge to $f$ in the space? Do similar results work for $f\in S'$?
• Do all the familiar results for Fourier transforms in $S$ and $S'$ hold the way one expects?
• How do I define things by duality on $S'$ once I have defined them on $S$?
• How do continuous linear functions between Hilbert spaces work together with the Schwartz structure?

Please do not answer these specific questions here; these are just examples of what I would like the material to cover. I will ask separate questions about more specific things if I find a material but there are holes.

About the sources I have found so far: The book Interpolation Spaces (section 6.1) by Bergh and Löfström discusses such spaces briefly, but the kind of basic theory I am after is not developed there. I also found this treatise on vector-valued distributions, but it does not seem to develop any Fourier theory or answer my questions. This MathOverflow question is related, but it asks for something different. One of the answers mentions "Vector-Valued Distributions And Fourier Multipliers" by Herbert Amann, and it is the most promising source so far. If you think I'm unlikely to find a better source, that would make a decent answer. Please correct me if I have misread my sources.

Answer 1

Unfortunately, the theory of vector-valued distributions is extremely underdeveloped. So I'm afraid that if Amann is not enough for your purposes, your only options are:

1. Develop the theory you need from scratch, which you're trying to avoid
2. Learn French and read the two long AIF articles by Schwartz
3. Do a toy model first

Option 3 which might be practically useful to you is as follows.

Let $\mathfrak{s}$ be the space rapidly decaying sequences $(x_n)_{n\ge 0}$ with defining seminorms $$ ||x||_k=\sup_n\ (1+n)^k|x_n|\ . $$ Try to develop a theory of "${\rm Hom}(\mathfrak{s},l^2)$" / "$\mathfrak{s}\otimes l^2$" as well as "${\rm Hom}(\mathfrak{s}',l^2)$" / "$\mathfrak{s}'\otimes l^2$".

This amounts to studying doubly infinite matrices $(a_{m,n})_{m,n}$, for example, those which satisfy $$ \sup_m\ \left[ (1+m)^k \sum_{n} |a_{m,n}|^2\right]\ <\infty\ . $$

You should be able to develop this toy theory from scratch and quickly.

Finally, use the isomorphism of $S(\mathbb{R}^d)$ with $\mathfrak{s}$, via the basis of Hermite functions, to transmute the theorems of the toy theory into theorems of the theory you want. It might even work for the Fourier transform, because Hermite functions are eigenvectors of this transformation.