There exists a isomorphism given by the Fourier transform on $\mathbb{R}^{n}\times \mathbb{R}$ and I would like to find its equivalent on $M\times \mathbb{R}$ where $M$ is a smooth manifold.
Let me explain the set-up, first on $\mathbb{R}^{n}$:
Let $h \in (0,1)$. For any $s, k \in \mathbb{R}$, we define semiclassical Sobolev spaces $ \, h^k H^s_h(\mathbb{R}^{n}) \,$ as a vector space that we equip with the norm
$$ \vert\vert u \vert\vert_{ h^k H^s_h(\mathbb{R}^{n})} := h^{-k} \vert\vert \langle hD \rangle^s u \vert\vert_{L^2(\mathbb{R}^{n})}, $$
where $\langle hD \rangle^s := \mathcal{F}^{-1}\langle h \xi \rangle^s \mathcal{F} $ and $\mathcal{F}$ denotes the usual Fourier transform on $\mathbb{R}^{n}$ ($\mathcal{F}^{-1}$ its inverse). The notation $ \langle \cdot \rangle^s $ represents the Japanese bracket.
The isomorphism I am interested in is the following: for $x’ = (x,y) \in \mathbb{R}^{n}\times \mathbb{R}$, $s \in \mathbb{R}$, the Fourier transform in $y$,
$$ (\mathcal{F}u)(x,\sigma) := \int_{\mathbb{R}} e^{-iy\sigma}u(x,y)\, dy$$
extends from $\mathcal{S}(\mathbb{R}^{n+1}) \rightarrow \mathcal{S}(\mathbb{R}^{n+1})$ to an isomorphism $$ H^s(\mathbb{R}^{n}\times \mathbb{R} ) \longrightarrow L^2(\mathbb{R}_{\sigma}, \langle \sigma \rangle^{-s} H^s_h(\mathbb{R}^{n})) $$
where the space on the right is defined as the completion of $\mathcal{S}(\mathbb{R}^{n}\times\mathbb{R})$ with respect to the norm
$$ \vert\vert v \vert\vert := \int_{\mathbb{R}} \vert\vert v(\cdot \,, \sigma) \vert\vert^2_{ \langle \sigma \rangle ^{-s} H^s_{\langle \sigma \rangle ^{-1}}(\mathbb{R}^{n})} \, d\sigma,$$
and the proof is very direct.
My goal is to a way to define such a map for $M$ a smooth manifold. For a (possibly non-compact) smooth manifold, such results do not exist, but it has been proven that such isomorphisms are true on manifolds $M$ that have bounded geometry (with more complicated Sobolev spaces). But in my case I am only interested in a potential well-defined inverse Fourier transform of such a map on $M$, so a priori I do not want to add any extra bounded geometry structure on $M$.
More precisely, I would like to know for what functions $u_{\sigma}$ (I mean from what space, like $L^2(\mathbb{R}_{\sigma}, \langle \sigma \rangle^{-s} H^s_h(M))$ in the case $M = \mathbb{R}^{n}$ above) we can arrive in $H^s(M \times \mathbb{R})$, after having applied the inverse Fourier transform. Such a map may not exist on $M$, so my idea is to consider local Sobolev spaces. Let me say that I do not hope to get an isomorphism.
The map I would like to define would look like
$$ L^2(\mathbb{R}_{\sigma}, \langle \sigma \rangle^{-s} H^s_{h,\mathrm{loc}}(M)) \longrightarrow H^s_{\mathrm{loc}}(M \times \mathbb{R} ) .$$
The definitions of $H^s_{h,\mathrm{loc}}(M)$ and $H^s_{\mathrm{loc}}(M \times \mathbb{R})$ are based on the ones on $\mathbb{R}^{n}$ through a pull-back defined with coordinate charts and cutoff functions (defining seminorms), with a notion of uniform boundedness in the case of the semiclassical local Sobolev spaces. I can write it down if ever.
Is it possible to find such a well-defined map without having to use these annoying definitions with charts, knowing that I am working with local Sobolev spaces now?
On the top of that, I do not really care about Sobolev regularity in the $y$ variable in $ H^s_{\mathrm{loc}}(M \times \mathbb{R} ) $ (meaning that the resulting function could be $-N$-regular for any fixed $N \in \mathbb{R},$ possibly positive, with respect to $y \in \mathbb{R}$). In other words the choice of the $L^2$ space at the beginning could be simpler, such as $L^2(\mathbb{R}_{\sigma}, H^s_{\mathrm{loc}}(M))$ for instance. Then in my case I would be interested in the anisotropic Sobolev space $H^{s,-N}(M\times \mathbb{R})$, even though I would prefer to get a map that goes into $ H^s_{\mathrm{loc}}(M \times \mathbb{R} ) $.
Any ideas or results close to such considerations would be very appreciated, as well as references that might help, thank you!