$E\subset
\mathbb{R}^{n}$ will be denoted by $|E|$.

The *Hardy–Littlewood maximal operator*, $M$, is defined for every locally integrable function $f$ on $\mathbb{R}^{n}$ and every $x\in \mathbb{R}^{n}$ by $$\begin{aligned}
Mf(x)=\sup _{Q\owns x}\frac{1}{|Q|}\int _{Q}|f(y)|\,{\mathrm{d}}y,\end{aligned}$$ where the supremum is extended over all cubes $Q\subset \mathbb{R}
^{n}$, whose edges are parallel to the coordinate axes of $\mathbb{R}
^{n}$, that contain $x$.

The operator $M$ is merely sublinear, rather than linear, and it is clearly a contraction on $L^{\infty }$. On the other hand, $Mf$ is never integrable unless $f\equiv 0$. For every locally-integrable function $f$ on $\mathbb{R}^{n}$, one has $|f|\leq Mf$ almost everywhere. The most important information (for our purpose) concerning the operator $M$, now classical, states that there exist positive constants $c,c'$, depending only on $n$, such that $$\begin{aligned}
\label{E:herz}
c(Mf)^{*}(t)\leq f^{**}(t)\leq c'(Mf)^{*}(t), \quad t\in (0,\infty ),\end{aligned}$$ for every locally integrable function $f$ on $\mathbb{R}^{n}$. The first inequality in  was established during the 1930s in works of R.M. Gabriel [@G], F. Riesz [@R] and N. Wiener [@W], while the second was added later through the efforts of C. Herz [@H] (for one dimension) and C. Bennett and R. Sharpley [@BS-paper] (for higher dimensions). The result is summarized and proved in [@BS Chapter 3, 