# The Size of Intervals in the Hardy-Littlewood Maximal Function

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Posted online: 2024-03-20 15:17:20Z by Stefan Steinerberger17

Cite as: P-240320.2

• Classical Analysis and ODEs

### Problem's Description

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is a continuous function (more smoothness can be assumed if deemed necessary). A classical object is the Hardy-Littlewood maximal function $\mathcal{M}f: \mathbb{R} \rightarrow \mathbb{R}$ defined via $$(\mathcal{M} f)(x) = \sup_{r >0} \frac{1}{2r} \int_{x-r}^{x+r} |f(z)| dz.$$

There is a naturally associated object that is less frequently studied: the size of the shortest interval on which the maximal value is attained. Let us define $r_f(x)$ as the length of the shortest interval on which the maximal value is being attained (note the absence of absolute values)

$$r_f(x) = \inf_{r>0} \left\{ \frac{1}{2r} \int_{x-r}^{x+r} f(z) dz = \sup_{s>0} \frac{1}{2s} \int_{x-s}^{x+s} f(z) dz \right\}$$

One might be naturally inclined to believe that $r_f(x)$ assumes many different values: however, this is not necessarily the case. Consider the function

$$f(x) = 1 + \sin{(x)}$$

Then $r_f(x)$ is either $0$ (when $\sin{(x)} > 0$) or it is 4.49341... (the smallest positive solution of $\tan{x} = x$).

Question. Does $r_f(x)$ `typically' assume many different values (with $f(x) = 1 + \sin{(x)}$ being a rare counterexample) or are there many such counterexamples?

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