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Given $f: R^n\to R $ an integrable function. Then the centered maximal operator is defined by $$M^{*}f(x)= \sup_{r>0} \dfrac{1}{\mu(B(x,r))} \int_{B(x,r)}|f(y)|\,d\mu(y) .$$

Here the supremum is taken over all the balls in $\mathbb{R}^{n}$ centred at the point $x$.

I did prove that if the function $f$ is continuous the non-centered maximal function is continuous. But it is not true in the case of the centered maximal function. I spent the last week trying to find a counter example but I could not. Can some one give me a hint how to find one? I will list some of the functions that I tried.

$$ f(x)=\begin{cases} 2x, & x \in [0,1]\\ 2, & x \in [1,2]\\ 0, & \text{otherwise} \end{cases}.$$

$$f(x)=\begin{cases} x^2, & x \in [0,1]\\ \dfrac{1+x}{2}, & x \in [1,2]\\ 0, & \text{otherwise} \end{cases}.$$

Adrian Keister
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majduleen zeyadeh
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1 Answers1

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You can get an easy counterexample with $n=1$ and $f(x)=1_{[0,1]}$. Then $$ Mf(x)=\begin{cases} \frac1{2|x-1|},&\ x\leq0,\\ \ \\ 1,&\ x\in(0,1)\\ \ \\ \frac1{2|x|},&\ x\geq1 \end{cases} $$

Martin Argerami
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  • The function $f$ is not cont at $x=0 , 1$. I need an example of $f$ to be a continuous function, where $M*f$ is not continuous. – majduleen zeyadeh May 11 '18 at 14:54
  • Neither are the examples you wrote in your question, so why would you consider them? – Martin Argerami May 11 '18 at 15:06
  • both of the examples are continuous at $x=1$. But the centred maximal function is continuous is also continuous . I need an example for a continuous function at a point where the centred maximal function at that point is not continuous. – majduleen zeyadeh May 11 '18 at 15:10