Fe b 20 00 BMO , H 1 , AND CALDERÓN - ZYGMUND OPERATORS FOR NON DOUBLING MEASURES

Given a Radon measure μ on R, 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(μ) when μ is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming μ doubling. For instance, Calderón-Zygmund operators which are bounded on L(μ) are also bounded from L(μ) into the new BMO space. Moreover, this space also satisfies a John-Nirenberg inequality, and its predual is an atomic space H. Using a sharp maximal operator it is shown that operators which are bounded from L(μ) into the new BMO space and from its predual H into L(μ) must be bounded on L(μ), 1 < p <∞. 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 Calderón-Zygmund operators bounded on L(μ) with functions of the new BMO are bounded on L(μ), 1 < p <∞. Date: February 17, 2000. 1991 Mathematics Subject Classification. Primary 42B20; Secondary 42B30.