Riemann-Lebesgue lemma and regularity for partial Fourier transform

Question

Consider a function $u $ on $\mathbb{R}^d \times \Omega$, where $\Omega \subseteq \mathbb{R}^k$ is an open subset of $\mathbb{R}^k$. We write $u = u(x, t)$ with "spatial" coordinates $x \in \mathbb{R}^d$ and "time" coordinates $t \in \Omega$ in mind.

For a fixed time $t \in \Omega$, the partial Fourier transform on spatial coordinates is defined formally as: $$ \mathcal{F}_x u(\cdot, t) : \mathbb{R}^d \to \mathbb{C} \ , \quad \mathcal{F}_x u(\xi, t) = \int_{\mathbb{R}^d} u(x, t) e^{-2\pi i x \cdot \xi} dx \quad \forall \xi \in \mathbb{R}^d $$ We want to view $\mathcal{F}_x u$ as a function on $\mathbb{R}^d \times \Omega$, but $\mathcal{F}_x u$ may not be well-defined a priori, because we didn't specify what function space $u$ lives in. My goal in this post is to clarify this issue.

Definition

Let $\Omega \subseteq \mathbb{R}^k$ be an open subset of $\mathbb{R}^k$. Let $\mathcal{S}(\mathbb{R}^d)$ be the Schwartz space on $\mathbb{R}^d$.

1. The space $C_0(\mathbb{R}^d, L^1(\Omega))$ is the collection of all functions $v$ on $\mathbb{R}^d \times \Omega$ such that $v(\xi, \cdot) \in L^1(\Omega)$ for every $\xi \in \mathbb{R}^d$ (this only requires $v(\xi, t)$ defined for almost every $t \in \Omega$) and the map $\xi \mapsto v(\xi, \cdot)$ is continuous and vanishes at infinity as a map $\mathbb{R}^d \to L^1(\Omega)$ between banach spaces.
2. The space $C^\infty(\Omega, \mathcal{S}(\mathbb{R}^d))$ is the collection of all functions $u$ on $\mathbb{R}^d \times \Omega$ such that $u(\cdot, t) \in \mathcal{S}(\mathbb{R}^d)$ for every $t \in \Omega$ and the map $t \mapsto u(\cdot, t)$ is "infinitely differentiable" as a map $\Omega \to \mathcal{S}(\mathbb{R}^d)$ from an open subset of $\mathbb{R}^k$ to a Frechet space, for some appropriate sense of differentiability (I haven't figured this out).

My claim is the following:

Claim

The partial Fourier transform can be viewed as:

1. a continuous linear map $ \mathcal{F}_x: L^1(\mathbb{R}^d \times \Omega) \to C_0(\mathbb{R}^d, L^1(\Omega)) $, and
2. a continuous linear map $ \mathcal{F}_x: C^\infty(\Omega, \mathcal{S}(\mathbb{R}^d)) \to C^\infty(\Omega, \mathcal{S}(\mathbb{R}^d)) $, given some appropriate sense of differentiability of any map $\Omega \to \mathcal{S}(\mathbb{R}^d)$.

Questions:

1. What should be the "appropriate sense of differentiability" to make the claim work?
2. Could you sketch a proof of the claim?

Answer 1

He means the sense of weak derivative (Distributional derivative). You can use the spaces $\mathcal(D)$ (or $\mathcal D'$) and work with Fourier transform as a tempered distribution on $\mathcal S '$.