Vanishing mean oscillation and continuity of rearrangements. (English) Zbl 1531.42043
Summary: We study the decreasing rearrangement of functions in VMO, and show that for rearrangeable functions, the mapping \(f \mapsto f^\ast\) 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.
MSC:
| 42B35 | Function spaces arising in harmonic analysis |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 49Q20 | Variational problems in a geometric measure-theoretic setting |
Keywords:
vanishing mean oscillation; monotone rearrangement; symmetric decreasing rearrangement; continuity of nonlinear operatorsReferences:
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