Mean oscillation bounds on rearrangements

We use geometric arguments to prove explicit bounds on the mean oscillation for two important rearrangements on ${\mathbb{R}}^n$. For the decreasing rearrangement $f^{\ast }$ of a rearrangeable function $f$ of bounded mean oscillation (BMO) on cubes, we improve a classical inequality of Bennett–DeVore–Sharpley, $\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_{BMO({\mathbb{R}}_{+})}\le C_n\mathrm{\parallel }f{\mathrm{\parallel }}_{BMO({\mathbb{R}}^n)}$, by showing the growth of $C_n$ in the dimension $n$ is not exponential but at most of the order of $\sqrt{n}$. This is achieved by comparing cubes to a family of rectangles for which one can prove a dimension-free Calder'{o}n–Zygmund decomposition. By comparing cubes to a family of polar rectangles, we provide a first proof that an analogous inequality holds for the symmetric decreasing rearrangement, $Sf$.