Are Schwartz functions in $L^{p}$ for $0 < p < 1$?

Question

Let $S(\mathbb{R}^{d})$ denote the Schwartz functions in $\mathbb{R}^{d}$. I know that $S(\mathbb{R}^{d}) \subset L^{p}(\mathbb{R}^{d})$ for $1 \leq p < \infty$. Is $S(\mathbb{R}^{d}) \subset L^{p}(\mathbb{R}^{d})$ for $0 < p < 1$? If not, what is an example of a Schwartz function which is not in some $L^{p}, 0 < p < 1$?

Answer 1

To elaborate on the comment, let $N > 1/p$ be a natural number and $f$ a Schwartz function.

Then $|f(x)| \leq C (1+|x|)^{-nN}$ for some $C>0$ and all $x$. Hence,

$$ \int |f(x)|^p \, dx \leq C \int (1+|x|)^{-n \cdot Np} <\infty, $$

where the last integral is finite because of $-n \cdot Np < -n$.