Extension of Paley-Wiener-Schwartz theorem to vector-valued distributions

Let $H_{j} := (H_{j}, \| \cdot \|_{H_{j}} ), j=0,1$ be a Hilbert space, and set \begin{equation*} {\mathscr S}'(\mathbb{R}^{n}, H_0; H_1) := {\mathscr L}( {\mathscr S}(\mathbb{R}^{n}, H_0), H_1) \end{equation*} where ${\mathscr S}(\mathbb{R}^{n}, H_0)$ denotes the space of Schwartz functions $\phi \colon \mathbb{R}^{n} \to H_0$. (Bergh & Lofstrom's book: Interpolation Spaces, page 134)

Then clearly ${\mathscr S}(\mathbb{R}^{n}, {\mathscr L}(H_0,H_1)) \subset L^{p}(\mathbb{R}^{n}, {\mathscr L}(H_0,H_1)) \subset {\mathscr S}'(\mathbb{R}^{n}, H_0; H_1)$.

I wonder whether the Paley-Wiener-Schwartz theorem can be extended to this space of vector-valued tempered distributions. I imagine a statement like this one:

Theorem: An entire analytic fucntion $U \colon \mathbb{C}^{n} \to {\mathscr L}(H_0,H_1)$ is the Fourier-Laplace transform of a distribution $u \in {\mathscr E}'(\mathbb{R}^{n}, H_0; H_1)$ with support in the ball $B[0,R]$ if and only if for some positive constants $c$ and $N$ we have $\| U(\zeta) \|_{{\mathscr L}(H_0,H_1)} \leqslant c (1+|\zeta|)^{N}\exp(R |Im \, \zeta|)$, for every $\zeta \in \mathbb{C}^{n}$.

Here, ${\mathscr E}'(\mathbb{R}^{n}, H_0; H_1):={\mathscr L}( C^{\infty}(\mathbb{R}^{n}, H_0), H_1)$, unsurprisingly.

I could not do it because I do not know how convolutions can be defined in this context and be used to obtain density results that appear in the proof of Paley-Wiener-Schwartz theorem.

Is there some good reference for vector-valued distributions as defined above? Bergh & Lofstrom is too succinct and does not provide a reference for this subject, particularly.

• What exactly is the problem you are having with convolutions? The obvious one I would think is about integrating vector-valued functions, but doesn't that problem also come up when you even try to define Fourier transform? – Willie Wong Apr 5 '17 at 18:12
• We define ${\mathscr F} \colon {\mathscr S}'(\mathbb{R}^{n}, H_0; H_1) \to {\mathscr S}'(\mathbb{R}^{n}, H_0; H_1)$ by $({\mathscr F} u ) \phi = \langle {\mathscr F} u, \phi \rangle := \langle u, {\mathscr F} \phi \rangle \in H_1$. Since ${\mathscr S}(\mathbb{R}^n, H_0; H_1) \subset {\mathscr S}'(\mathbb{R}^n, H_0; H_1)$, it is natural to wonder if it is dense on it. Given $\phi \in {\mathscr S}$ and $u \in {\mathscr S}'$, we wish to define $u \ast \phi$. To do so, we need to define a integral where these two objects interact correctly. May you suggest how I do it? – Alex Pereira Apr 13 '17 at 18:58
• If you want to use $\phi$ to mollify/smooth $u$, so as then to represent $u$ as a limit of smooth functions, then it is sufficient for $\phi$ to be scalar valued, not $H_0$-valued. Then convolution between a scalar valued Schwartz function and an $H_0$-valued distribution is defined in the obvious way. – Igor Khavkine May 1 '17 at 9:08
• It seems good to me. I'll check it out. Thanks – Alex Pereira May 9 '17 at 20:50