A problem regarding a generalization of Schwartz space

Question

Consider a space of functions $f:\mathbb{R}^n\to \mathbb{C}$ satisfying $$\sup_{x\in \mathbb{R}^n}|x^\alpha f(x)|<\infty, $$ for any multi index $\alpha$.

1. This space contains Schwartz space. Does this has any name?
2. Does this space contained in $L^p(\mathbb{R}^n), 1\leq p\leq \infty?$ (I proved that this is true but not sure)
3. Does this space dense in $L^p(\mathbb{R}^n), 1\leq p\leq \infty?$ (I guess this will not be true)
4. If (3) not true then will it become true if we add that the functions in the space are continuous?

Answer 1

1. These are the functions with "superpolynomial decay". I have not seen the whole space studied much as a whole, because it does not have particularly good regularity properties (at least, no better than Schwartz or $L^p$ spaces), but I have seen various linear subspaces with greater regularity studied more often. You can equivalently write your defining condition by saying that for every $f$ in your space and every $k \geq 1$, there exists a constant $C_{f,k} > 0$ that, for all $x \in \mathbb{R}^n$, $$ |f(x)| \leq C_{f,k} \exp(-k \log |x|) $$ It's a lot easier to get work done if you have some uniform regularity, e.g. requiring that you have some uniform $C_k > 0$ such that $C_{f,k} \leq C_k$ for all $f$. Another option: there exist some $b > 0$ and some $t > 1$, such that for any $f$, there exists $C_f > 0$ with $$ |f(x)| \leq C_f \exp(-b \log^t|x|) $$ This kind of bound comes up a lot when estimating decay of correlations in statistical physics and dynamical systems. Edit in response to a comment: For example, this paper by Mark Holland (ETDS 25 (1), 2005, pp. 131-159). I found that paper by following a reference in a paper by Benoît R. Kloeckner (ETDS 40 (3), 2020, pp. 714-750), which I found by Googling "sub-polynomial decay".

2. Yes, this space is contained in $L^p(\mathbb{R}^n)$. Same argument as for Schwartz space. Edit: not quite right, see the comments: you need measurability.

3. and 4. Since it contains Schwartz space (which is dense in $L^p(\mathbb{R}^n)$ for every $n$ and $p\in[1,\infty)$, this space is certainly dense in $L^p(\mathbb{R}^n)$. It is not dense in $L^\infty$ (again, see comments; adding continuity would make it worse).