Suppose $f: \mathbb{R} \rightarrow\ \mathbb{R}$ is Lebesgue-measurable. Define $f^{**}$ as
$$f^{**}(x) = \sup_{t>0}\left\{ \frac{1}{t} \int_{I_t}|f| : I_t \subseteq \mathbb{R}, |I_t| = t , x \in I_t \right\},
$$
where $I_t$ means an interval of length $t$.
Is there a (useful) bound on $\big|\left\{ x \in \mathbb{R}:f^{**}(x)>c\right\}\big|$ similar to the bound for the Hardy-Littlewood maximal function $\big(\;\big|\{ x \in \mathbb{R}:f^{*}(x)>c \}\big| \le \frac{3}{c}\int_{\mathbb{R}} |f|\;$ where $f^{*}(x) =\sup_{t>0}\left\{ \frac{1}{2t} \int^{x+t}_{x-t}|f| : t>0 \right\}\big)$ ?
Best Answer
Whenever $t > 0$ you have $$\int_{I_t} |f| \le \int_{x-t}^{x+t} |f|$$ for any interval $I_t$ with $x \in I_t$ and $|I_t| = t$, so that $$\frac 1t \int_{I_t} |f| \le \frac{2}{2t} \int_{x-t}^{x+t} |f| \le 2 f^*(x).$$ Now take the sup over all $t > 0$ and corresponding intervals $I_t$ to find that $f^{**}(x) \le 2 f^*(x)$. This means that $$\{x : f^{**}(x) > c\} \subset \{x : f^*(x) > c/2\}$$ and thus $$|\{x : f^{**}(x) > c\}| \le |\{x : f^*(x) > c/2\}| \le \frac 6c \int |f|.$$