Real analysis related to the Monge-Ampère equation

In this paper we consider a family of convex sets in $\mathbf {R}^{n}$, $\mathcal {F}= \{S(x,t)\}$, $x\in \mathbf {R}^{n}$, $t>0$, satisfying certain axioms of affine invariance, and a Borel measure $\mu$ satisfying a doubling condition with respect to the family $\mathcal {F}.$ The axioms are modelled on the properties of the solutions of the real Monge-Ampère equation. The purpose of the paper is to show a variant of the Calderón-Zygmund decomposition in terms of the members of $\mathcal {F}.$ This is achieved by showing first a Besicovitch-type covering lemma for the family $\mathcal {F}$ and then using the doubling property of the measure $\mu .$ The decomposition is motivated by the study of the properties of the linearized Monge-Ampère equation. We show certain applications to maximal functions, and we prove a John and Nirenberg-type inequality for functions with bounded mean oscillation with respect to $\mathcal {F}.$