On the John-Strömberg characterization of BMO for nondoubling measures

A well known result proved by F. John for 0 < λ < 1/2 and by J.-O. Strömberg for λ = 1/2 states that for any measure ω 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 nondou- bling 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 func- tion.