I am learning about the Lorentz spaces $L^{p,\infty}(\mathbb{R}^n)$.
If I can bound a function $f(x)$ by $C|x|^{-\frac{n}{p}}$ with $C>0$, i.e. $$f(x)\leq C|x|^{-\frac{n}{p}}, \quad \text{for all}\ \ x\in \mathbb{R}^n,$$
Can I conclude that $f\in L^{p,\infty}(\mathbb{R}^n) ?$
How can I prove it ?