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A well known result proved by F. John for 0 < λ < 1/2 and by J.-O. Strömberg for λ = 1/2 states that ‖f‖BMO(ω) sup Q inf c∈R inf{α > 0 : ω{x ∈ Q : |f(x)− c| > α} < λω(Q)} 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 nondoubling case. An important role in our analysis is played by a weighted rearrangement inequality, relating any measurable function and its JohnStrömberg 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.

On the John-Strömberg Characterization of <em>BMO</em> for Nondoubling Measures