[Math] Hardy-Littlewood maximal function not integrable in B(0,1)

Question

It is well known that if $f\in L^1(\mathbb{R}^n)$, then the Hardy-Littlewood maximal function:
$$
Mf=\sup_{r>0}\frac{1}{|B(x,r)|}\int_{B(x,r)}|f(y)|dy
$$
is not in $L^1(\mathbb{R}^n)$. Does there exist a $f$ such that $Mf$ is not integrable in $B(0,1)$?

Answer 1

The answer to your question is no by theorem of Stein given on the screen shot for the full proof see here