Mean oscillation bounds on rearrangements

Abstract

We use geometric arguments to prove explicit bounds on the mean oscillation for two important rearrangements on R n {\mathbb {R}^n} . For the decreasing rearrangement f ∗ f^* of a rearrangeable function f f of bounded mean oscillation (BMO) on cubes, we improve a classical inequality of Bennett–DeVore–Sharpley, ‖ f ∗ ‖ BMO ( R + ) ≤ C n ‖ f ‖ BMO ( R n ) \|f^*\|_{{\operatorname {BMO}}(\mathbb {R}_+)}\leq C_n \|f\|_{{\operatorname {BMO}}(\mathbb {R}^n)} , by showing the growth of C n C_n in the dimension n n is not exponential but at most of the order of n \sqrt {n} . This is achieved by comparing cubes to a family of rectangles for which one can prove a dimension-free Calderón–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, S f Sf .