PROPERTIES OF THE SOLUTIONS OF THE LINEARIZED MONGE-AMPERE EQUATION

Let phi: R(n) --> R be a function strictly convex and smooth, and mu = det D-2 phi is the Monge-Ampere generated by phi. Given x epsilon R(n ) and t > 0, let S(x,t) = {y epsilon R(n): phi(y) < phi(x) + del phi(x ) . (y - x) + t}. The purpose of this paper is to study the properties of the solutions of the linearized Monge-Ampere equation given by a(i j)(x)D(ij)u = 0 where the coefficients a(ij)(x) are the cofactors of t he matrix D-2 phi(x). It is assumed that mu satisfies a doubling condi tion on the sets S(x,t) and a uniform continuity condition at every sc ale with respect to Lebesgue measure. We establish that the distributi on functions of nonnegative solutions u at altitude t decay like a neg ative power of t and prove an invariant Harnack's inequality on the se ctions S(x,t). All the estimates are independent of the regularity of phi and depend only on the constants in the hypotheses made on the mea sure mu.