My question here relates to an argument that I have seen used many a time, for example to prove that $\mathbb{L}^p$ spaces define tempered distributions in the following sense : $\varphi \mapsto \int \varphi f$ is a tempered distribution for $f \in \mathbb{L}^p$
The argument is, if $f \in \mathbb{L}^p$ then for any function $\varphi \in \mathbb{L}^q$ (hence for any Schwartz function) $$ |\varphi (x)|=(1+|x|)^{N+1} \cdot |\varphi (x)| \frac{1}{(1+|x|)^{N+1}} $$ where $N$ is the dimension of the euclidean space, and $|\cdot |$ denotes the euclidean norm. Then, $\frac{1}{(1+|x|)^{N+1}}$ is in $\mathbb{L}^q$ and one can write $$ ||\varphi (x)||_q \leq \sup|(1+|x|)^{N+1} \varphi (x)| \cdot ||\frac{1}{(1+|x|)^{N+1}}||_q $$
What I don't get is how this $\sup$ expression can be upper-bounded by any of the Schwartz norms since $(1+|x|)^{N+1}$ may well not be a polynomial of $x$ with whole powers because of the square root. Especially confused as I've just worked out that $|x|^{|\alpha|} \geq |x^\alpha|$ ?