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On the John-Strömberg characterization of BMO for nondoubling measures. (English) Zbl 1044.42018

Summary: A well-known result proved by F. John for \(0<\lambda< 1/2\) and by J.-O. Strömberg for \(\lambda=1/2\) states that \[ \| f\|_{\text{BMO} (\omega)} \asymp\sup_Q \inf_{c\in\mathbb{R}} \Bigl\{ \alpha>0:\omega\biggl\{x\in Q:\bigl| f(x)-c\bigr|>\alpha \biggr\} <\lambda\omega (Q)\Bigr\} \] for any measure \(\omega\) satisfying the doubling condition. In this note we extend this result to all absolutely continuous measures. In particular, we show that Strömberg’s “1/2-phenomenon” still holds in the nondoubling case. An important role in our analysis is played by a weighted rearrangement inequality, relating any measurable function and its John-Strömberg maximal function. This inequality was proved earlier by the author in the doubling case; here we show that actually it holds for all weights. Also we refine a result due to B. Jawerth and A. Torchinsky, concerning pointwise estimates for the John-Strömberg maximal function.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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