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A mean oscillation inequality. (English) Zbl 0572.46025
It is proved that \(\| f^*\|_{BMO}\leq \| f\|_{BMO},\) where \(f^*\) is the decreasing rearrangement of a function \(f\in BMO([0,1])\). The proof is given for the case \(p=1\) of the norm: \[ \| f\|_{BMO}=\sup \{(1/| J| \int_{J}| f-f_ J|^ p)^{1/p}:\quad J\quad an\quad interval\quad in\quad [0,1]\}. \] A more general inequality is stated, which includes the above for all \(1\leq p<\infty\) as a special case, but which is based on the same argument as the \(p=1\) case; the rising sum lemma for an interval. For symmetric decreasing rearrangement on the circle, the sharp inequality above fails, but other versions of the norm are not considered (such as the Möbius invariant norm).

MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D10 Inequalities involving derivatives and differential and integral operators
46E15 Banach spaces of continuous, differentiable or analytic functions
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[1] Colin Bennett, Ronald A. DeVore, and Robert Sharpley, Weak-\?^\infty and BMO, Ann. of Math. (2) 113 (1981), no. 3, 601 – 611. · Zbl 0465.42015 · doi:10.2307/2006999 · doi.org
[2] G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, 1967. · Zbl 0634.26008
[3] F. Riesz, Sur un théorème de maximum de Mm. Hardy et Littlewood, London Math. Soc. 7 (1932), 10-13. · JFM 58.0261.02
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