Vanishing Mean Oscillation and Continuity of Rearrangements

TL;DR

This paper investigates how the decreasing rearrangement operator affects functions in VMO and BMO spaces, showing preservation of vanishing mean oscillation and conditions for continuity.

Contribution

It demonstrates that the rearrangement preserves VMO and clarifies the continuity properties of the rearrangement operator on BMO and VMO functions.

Findings

Rearrangement preserves vanishing mean oscillation in VMO.

The rearrangement operator is not continuous on BMO but is continuous at VMO points.

Includes numerous examples illustrating the theoretical results.

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.