On the Central Maximal Function in ${\mathbb{R}}^n$

We study the central Hardy–Littlewood maximal operator $M_c$ in ${\mathbb{R}}^n$ for all $n\ge 1$, with particular emphasis on the question of its injectivity. More precisely, we consider whether there exist two nonnegative functions $f,g\in L^1({\mathbb{R}}^n)$ such that $\mathrm{\parallel }f-g{\mathrm{\parallel }}_1>0$ and $M_{c}f=M_{c}g$. Along the way, we obtain an elementary explicit formula for $M_c({\chi }_U)$, where $U$ is the unit ball in ${\mathbb{R}}^3$. We also derive an explicit formula, in terms of the incomplete beta function, for the $n$-dimensional volume of the intersection of two balls with arbitrary radii and centers.