REAL ANALYSIS RELATED TO THE MONGE-AMPERE EQUATION

In this paper we consider a family of convex sets in R(n), F = {S(x, t )}, x is an element of R(n), t > 0, satisfying certain axioms of affin e invariance, and a Borel measure mu satisfying a doubling condition w ith respect to the family F. The axioms are modelled on the properties of the solutions of the real Monge-Ampere equation. The purpose of th e paper is to show a variant of the Calderon-Zygmund decomposition in terms of the members of F. This is achieved by showing first a Besicov itch-type covering lemma for the family F and then using the doubling property of the measure mu. The decomposition is motivated by the stud y of the properties of the linearized Monge-Ampere equation. We show c ertain application; to maximal functions, and we prove a John and Nire nberg-type inequality for functions with bounded mean oscillation with respect to F.