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.   Given a Radon measure  on , 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  when  is a doubling measure remain valid for the space of type BMO introduced in this paper, without assuming  doubling. For instance, Caldern-Zygmund operators which are bounded on  are also bounded from  into the new BMO space. Moreover, this space also satisfies a John-Nirenberg inequality, and its predual is an atomic space . Using a sharp maximal operator it is shown that operators which are bounded from  into the new BMO space and from its predual  into  must be bounded on , . 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 Caldern-Zygmund operators bounded on  with functions of the new BMO are bounded on .

$BMO, H^1$, and Calderón-Zygmund operators for non doubling measures