Properties of the solutions of the linearized Monge-Ampère equation

Let Φ: Rⁿ → R be a function strictly convex and smooth, and μ = detD²Φ is the Monge-Ampere generated by Φ. Given x ∈ Rⁿ and t > 0, let'S(x, t) = {y ∈ Rⁿ: Φ(y) < Φ(x) + ∇Φ(x) · (y - x) + t}. The purpose of this paper is to study the properties of the solutions of the linearized Monge-Ampère equation given by a_ij(x)D_iju = 0 where the coefficients a_ij(x) are the cofactors of the matrix D²Φ(x). It is assumed that μ satisfies a doubling condition on the sets S(x, t) and a uniform continuity condition at every scale with respect to Lebesgue measure. We establish that the distribution functions of nonnegative solutions u at altitude t decay like a negative power of t and prove an invariant Harnack's inequality on the sections S(x, t). All the estimates are independent of the regularity of Φ and depend only on the constants in the hypotheses made on the measure μ.