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

Abstract.

Given a Radon measure $\mu$ on ${\mathbb R}^d$, 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, Calderón-Zygmund operators which are bounded on $L^2(\mu)$ are also bounded from $L^\infty(\mu)$ 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 $L^\infty(\mu)$ into the new BMO space and from its predual $H^1$ into $L^1(\mu)$ must be bounded on $L^p(\mu)$, $1< p< infty$. 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 $L^2(\mu)$ with functions of the new BMO are bounded on $L^p(\mu), 1< p < \infty$.