On the John-Strömberg characterization of BMO for nondoubling measures.

Abstract

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.