La. Caffarelli et Ce. Gutierrez, PROPERTIES OF THE SOLUTIONS OF THE LINEARIZED MONGE-AMPERE EQUATION, American journal of mathematics, 119(2), 1997, pp. 423-465
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.