Vanishing Mean Oscillation and Continuity of Rearrangements

Abstract

We study the decreasing rearrangement of functions in VMO, and show that for rearrangeable functions, the mapping f -> f* preserves vanishing mean oscillation. Moreover, as a map on BMO, while bounded, it is not continuous, but continuity holds at points in VMO (under certain conditions). This also applies to the symmetric decreasing rearrangement. Many examples are included to illustrate the results.

arXiv comment21 pages, 4 figures. This paper comprises parts of our previous article arXiv:2011.09111v1 (specifically Theorem 1.3, and Sections 3 and 6), which were cut from the subsequent version arXiv:2011.09111v2