Definition. Let $f$ be a measurable function defined on $\mathbb{R}^n$ such that $\int_{\mathbb{R}}\frac{|f(x)|}{(1+|x|)^N}dx<\infty$ for some positive integer $N$. Then we call $f$ a tempered function.
Why any $f\in L^p(\mathbb{R}^n)$ is a tempered function?
I imagine that it should start like this,
$\int_{\mathbb{R}^n}\frac{|f(x)|}{(1+|x|)^N}dx=\int_{\mathbb{R}^n}\frac{|f(x)|^p|f(x)|^{1-p}}{(1+|x|)^N}dx$
but I do not know how to continue
For $p=1$ the claim is obvious (choose for example $N=1$). So let $p>1$ and $q=\frac{p}{p-1}$.
Note that $q>1$. With the Hölder inequality we get \begin{align} \int_{\mathbb{R}^n}\frac{|f(x)|}{(1+|x|)^N}\mathrm dx &=\left\|\frac{f(x)}{(1+|x|)^N}\right\|_{L^1} \\ &\leq\|f\|_{L^p}\left\|\frac{1}{(1+|x|)^N}\right\|_{L^q} \end{align} Moreover we have \begin{align} \left\|\frac{1}{(1+|x|)^N}\right\|_{L^q}^q &=\int_{\mathbb R^n}\left(\frac{1}{1+|x|}\right)^{Nq}\mathrm dx \\ &=\int_0^\infty\int_{\partial B_r(0)}\frac{1}{(1+r)^{Nq}}\mathrm dr \\ &=\int_0^\infty\frac{\omega_{n-1} r^{n-1}}{(1+r)^{Nq}}\mathrm dr \\ &\leq\int_0^\infty\frac{\omega_{n-1} r^{n-1}}{(1+r)^{2n}}\mathrm dr \\ &\leq\int_0^\infty\frac{\omega_{n-1}}{(1+r)^2}\mathrm dr \\ &=\omega_{n-1}<\infty \end{align} where $\omega_{n-1}$ is the surface area of the unit $n-1$-sphere viewed as a subset of $\mathbb R^n$ The latter of course only holds if we choose $N$ large enough, but we are free to do so to show that $f$ is a tempered function (note that $N=2n$ would be sufficient).
Hence $\int_{\mathbb{R}^n}\frac{|f(x)|}{(1+|x|)^N}\mathrm dx<\infty$ for some $N\in\mathbb N$, so $f$ is a tempered function.