New bounds and continuity for rearrangements on BMO and VMO.

We prove explicit bounds on the mean oscillation for two important rearrangements on $\mathbb{R}^n$. Specifically, we sharpen a classical inequality of Bennett-DeVore-Sharpley for the decreasing rearrangement, and establish that the symmetric decreasing rearrangement is bounded on BMO, as well. We demonstrate by example that these rearrangements are not continuous on BMO. Furthermore, we show that the decreasing rearrangement maps rearrangeable functions in VMO into VMO, and derive sufficient conditions for the rearrangements of convergent sequences in VMO to converge.