BMO, H1, and Calderon-Zygmund operators for non doubling measures

Given a Radon measure mu on R-d, which may be non doubling, we introduce a space of type BMO with respect to this measure. It is shown that many properties which hold for the classical space BMO(mu) when mu, is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming mu doubling. For instance, Calderon-Zygmund operators Which are bounded on L-2(mu) are also bounded from L-infinity(mu) into the new BMO space. Moreover, this space also satisfies a John-Nirenberg inequality, and its predual is an atomic space H-1. Using a sharp maximal operator it is shown that operators which are bounded from L-infinity(mu) into the new BMO space and from its predual H-1 into L-1(mu) must be bounded on L-p(mu), 1 < p < infinity. From this result one can obtain a new proof of the T(1) theorem for the Cauchy transform for non doubling measures. Finally, it is proved that commutators of Calderon-Zygmund operators bounded on L-2(mu) with functions of the new BMO are bounded on L-p(mu), 1 < p < infinity.