# A question about Hardy-Littlewood maximal function and a characterization of measurable sets.

Let $M$ be the Hardy-Littlewood maximal operator: $$Mf(x) = \sup_{r>0} \frac{1}{|B(x,r)|}\int_{B(x,r)}|f(y)|dy,$$ where $f\in L^1_{loc}(\mathbb{R}^n)$.

I have the following question:

How can I characterize for which measurable sets $E\subset \mathbb{R}^n$ such that the following condition holds:there exist constants $0<C_1<C_2<\infty$ such that $$C_{1}\chi_{E}(x)\leq M(\chi_{E})(x)\leq C_{2}\chi_{E}(x),$$ for almost everywhere $x\in \mathbb{R}^n$?

Thank you.

Hint: If $m(E) > 0,$ then $M(\chi_E)(x) > 0$ for every $x.$
• Dear @zhw Thank you for your answer. I can´t follow why this implies my problem. I only know that, by the Lebesgue´s differentiation theorem, $\chi_{E}(x)\leq M(\chi_{E})(x)$, but I still have trouble with the other inequality. – Claudia Reyes Dec 9 '17 at 4:10
• @ClaudiaReyes Yes the other inequality is the problem. Example: On $\mathbb R,$ let $E=[0,1].$ Then for $x>1,$ $M(\chi_E)(x) \ge 1/x.$ So we have a counterexample – zhw. Dec 9 '17 at 16:46