La. Caffarelli et Ce. Gutierrez, REAL ANALYSIS RELATED TO THE MONGE-AMPERE EQUATION, Transactions of the American Mathematical Society, 348(3), 1996, pp. 1075-1092
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.