# 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:

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?

• Convolution is only needed for the "only if" part, isn't it? math.chalmers.se/~hasse/distributioner_eng.pdf p. 63+ – md2perpe Jun 25 '17 at 20:59
• Ow, you're right. Yet I need the theorem 13.2 (from such file) which depends on the Prop. 13.1, and it uses both parts (if and only if). Also, to my purposes, I really need the entire result 13.2; but as it was sugessted, if I restrict to the case $H_0 = \mathbb{C}$, it works very well. – Alex Pereira Jun 27 '17 at 18:13