Bmo for Nondoubling Measures

1. Introduction. The Calderón-Zygmund theory of singular integrals has been traditionally considered with respect to a measure satisfying a doubling condition. Recently, Tolsa [T] and, independently, Nazarov, Treil, and Volberg [NTV] have shown that this standard doubling condition was not really necessary. Likewise, in the homogeneous spaces setting, functions of bounded mean oscillation, BMO, and its predual H 1 , the atomic Hardy space, play an important role in the theory of singular integrals. This note is an attempt to find good substitutes for the spaces BMO and H 1 when the underlying measure is nondoubling. Our hope was that we would have been able to prove some results of Tolsa, Nazarov, Treil, and Volberg, via BMO-H 1 interpolation, but in this respect we were unsuccessful. Let µ be a nonnegative Radon measure on R n. A function f ∈ L 1 loc (µ) is said to belong to BMO(µ) if the inequality