Dual/Fourier Transform of Smooth Functions of Moderate Growth

Question

If we consider $\mathcal{D}(\mathbb{R}^d)$ to be the space of compactly supported smooth functions (with the LF topology) and $\mathcal{E}(\mathbb{R}^d)$ to be the space of smooth functions with the Fréchet topology, then it is well known that if $ \cdot$ and $*$ represent multiplication and convolution, and $\mathcal{D}'$ is the space of distributions, $\mathcal{E'}$ is the space of compactly supported distributions, then $$ u\in \mathcal{D}',\phi\in \mathcal{D}\implies u* \phi\in \mathcal{E}, $$ $$ u\in \mathcal{D}',\phi\in \mathcal{D}\implies u\cdot \phi\in \mathcal{E}'. $$ The spaces $\mathcal{E}$ and $\mathcal{E}'$ are also related to $\mathcal{D}$, $\mathcal{D}'$ via $$ u\in \mathcal{D}', v\in \mathcal{E}'\implies u* v\in \mathcal{D}'. $$ $$ \phi\in \mathcal{D}, \psi\in \mathcal{E}\implies \phi\cdot \psi\in \mathcal{D}. $$ If we replace $\mathcal{D}$ by $\mathcal{S}$, the space of Schwartz functions, I am curious about the existence of natural spaces $\mathcal{V}$, $\mathcal{W}$ (corresponding to $\mathcal{E},\mathcal{E}'$) large enough such that $$ u\in \mathcal{S}',\phi\in \mathcal{S}\implies u* \phi\in \mathcal{V}, $$ $$ u\in \mathcal{S}',\phi\in \mathcal{S}\implies u\cdot \phi\in \mathcal{W}. $$ and small enough such that $$ u\in\mathcal{S}', v\in\mathcal{W}\implies u*v\in\mathcal{S}' $$ $$ \phi\in \mathcal{S}, \psi\in\mathcal{V}\implies u\cdot v\in\mathcal{S} $$ I think that $\mathcal{V}$ should be "smooth functions of moderate growth" (this is proved in Stein and Shakarchi's Functional Analysis) and $\mathcal{W}$ should be "linear combinations of higher derivatives of rapidly decaying continuous functions" (based on the structure theorem for tempered distributions presented in Strichartz's A Guide to Distribution Theory and Fourier Transforms along with the fact that the distribution should be rapidly decaying). My questions are

1. Is this guess for characterizations of $\mathcal{V}$ and $\mathcal{W}$ are correct?
2. What is the natural topology for these spaces?
3. Is the dual of $\mathcal{V}$ identifiable with $\mathcal{W}$?
4. Is the Fourier transform a continuous isomorphism between these spaces? (or even continuous in one direction?)
5. What would be a good (online accessible preferred) reference for these sorts of questions.

Edit: Thanks for the recommendation to Horvath's "Topological Vector Spaces" it seems that to justify the relations (1), (3) and (4) one should use the spaces $\mathcal{V}=\mathcal{O}_C$, $\mathcal{W}=\mathcal{O}_C'$ and $\mathcal{V}=\mathcal{O}_M$. It seems that the image of Schwartz space multiplied by tempered distributions should be the space $\mathcal{U}$ of tempered distributions whose Fourier Transform lies in $\mathcal{O}_C$, and the space should have a final topology. An updated version of this question would be

1. Is there a nice characterization of $\mathcal{F}(\mathcal{O}_C)\subset \mathcal{S}'$, with $\mathcal{O}_C$ defined as in Horvath's Topological Vector Spaces? Does this space of distributions have a final topology as described in Ch 2.12?
2. Is the dual of this unknown space identifiable with $\mathcal{O}_M$?