Inclusion of Schwartz space on $L^p$

To make Davide Giraudo's comment more clear, here are the steps:

$$ \int |f(x)|^p = \int \left((1+x^2)|f(x)|\right)^p\frac{1}{(1+x^2)^p} $$

From here, since $(1+x^2)f(x)$ is bounded (because $f\in \mathcal{S}$), and $\frac{1}{(1+x^2)^p}\leq \frac{1}{1+x^2}\in L^1$ (for $p\geq 1$), one gets

$$ \int |f(x)|^p \leq \|(1+x^2)f(x)\|^p_\infty \int \frac{1}{1+x^2}\leq \pi (\|f\|_{0,0}+\|f\|_{2,0})^p $$

where

$$ \|f\|_{\alpha,\beta} = \underset{x\in\mathbb{R}^n}{\sup} |x^\alpha D^\beta f(x)| $$

Since $f\in \mathcal{S}$, $\|f\|_{\alpha,\beta}$ is finite for all $\alpha,\beta\geq 0$.