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2002/2003 On the John-Strömberg characterization of BMO for nondoubling measures.
A. K. Lerner
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Real Anal. Exchange 28(2): 649-660 (2002/2003).


A well known result proved by F. John for \(0<\la<1/2\) and by J. O. Strömberg for \(\lambda=1/2\) states that \[\|f\|_{BMO(\o)}\asymp\sup_Q\inf_{c\in {\mathbb R}} \inf\{\a>0: \o\{x\in Q:|f(x)-c|>\a\}<\la\o(Q)\}\] 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 Str\"omberg'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-Str\"omberg 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-Str\"omberg maximal function.


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A. K. Lerner. "On the John-Strömberg characterization of BMO for nondoubling measures.." Real Anal. Exchange 28 (2) 649 - 660, 2002/2003.


Published: 2002/2003
First available in Project Euclid: 20 July 2007

zbMATH: 1044.42018
MathSciNet: MR2010347

Primary: 42B25 , 46E30

Keywords: $BMO$ , nondoubling measures , rearrangements

Rights: Copyright © 2002 Michigan State University Press

Vol.28 • No. 2 • 2002/2003
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