I am working on the following problem from Stein and Shakarchi:

Let $f$ be an integral function on $\mathbb{R}^d$ such that $\|f\|_{L^1} = 1$ and let $f^*$ by the Hardy-Littlewood maximal function corresponding of $f$.

- Prove that if $f$ is integrable on $\mathbb{R}^d$, and $f$ is not identically zero, then $f^*(x) \geq c/|x|^d $ for some $c > 0$ and all $|x| > 1$.
- Conclude that $f^*$ is not integrable on $\mathbb{R}^d$.
- Then, show that the weak type estimate $m(\{x:f^*(x) > \alpha \}) \geq c/\alpha$ for all $\alpha > 0$ whenever $\int|f| = 1$, is best possible in the following sense: if $f$ is supported in the unit ball with $\int|f| = 1$, then $m(\{x:f^*(x) > \alpha \}) \geq c'/\alpha$ for some $c' > 0$ and all sufficiently small $\alpha$.

[Hint: For the first part, use the fact that $\int_B|f| > 0$ for some ball $B$.]

I am not really sure where to begin with this. Any help would be greatly appreciated.