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 that hold when $\mu$ is doubling remain valid for the space BMO introduced in this paper, without assuming $\mu$ doubling. For instance, Calderon-Zygmund operators which are bounded in $L^2$ are bounded from $L^\infty$ into the new BMO space. Moreover, a John-Nirenberg inequality is satisfied, and the predual of BMO is an atomic space $H^1$. Using a sharp maximal function it is proved that operators bounded from $L^\infty$ into BMO and from $H^1$ into $L^1$ are also bounded on $L^p$, $1<p<\infty$. This result gives a new proof of the T(1) theorem for the Cauchy transform with non doubling measures. Finally, a result about commutators is obtained.