In page 250 of Real Analysis by Folland, it is stated that if $f$ belongs to the Schwartz space, then $\partial^{\alpha}f \in L^p$ for all $\alpha$ and all $p\in[1,\infty]$.
I think it is true for $p<1$ and here is my argument.
$$|\partial^{\alpha}f(x)|\leq C_N(1+|x|)^{-N}$$ for all $N$.
We need to show that $(1+|x|)^{-Np}\in L^1$ for some $N$.
If $B=\{x:|x|<2\}$, then $$(1+|x|)^{-Np}\leq 2^{-Np}|x|^{-Np}.$$ on $B^c.$ So for $N>n/p$, $(1+|x|)^{-Np} \in L^1(B^c)$.
On the other hand, it's clear that $(1+|x|)^{-Np}\in L^1(B).$
Therefore, $f\in L^p$ also for $p<1$.
So why does not the book state it for $p<1$?
I think that to say $(1+|x|)^{-Np} \leq 2^{-Np}|x|^{-Np}$ is to say that $2|x| \leq 1+|x|$, i.e. $|x| \leq 1$, is that what you meant to say?