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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.

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Hint: If $m(E) > 0,$ then $M(\chi_E)(x) > 0$ for every $x.$

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  • $\begingroup$ 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. $\endgroup$ – Claudia Reyes Dec 9 '17 at 4:10
  • $\begingroup$ @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 $\endgroup$ – zhw. Dec 9 '17 at 16:46

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