Real analysis related to the Monge-Ampère equation

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Abstract
In this paper we consider a family of convex sets in R n \mathbf {R}^{n} , F = { S ( x , t ) } \mathcal {F}= \{S(x,t)\} , x ∈ R n x\in \mathbf {R}^{n} , t > 0 t>0 , satisfying certain axioms of affine invariance, and a Borel measure μ \mu satisfying a doubling condition with respect to the family F . \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 F . \mathcal {F}. This is achieved by showing first a Besicovitch-type covering lemma for the family F \mathcal {F} and then using the doubling property of the measure μ . \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 F . \mathcal {F}.

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