Harnack inequality for the linearized parabolic Monge-Ampere equation

In this paper we prove the Harnack inequality for nonnegative solutions of the linearized parabolic Monge-Ampere equation u(t) - tr((D(2)phi(x))(-1) D(2)u) = 0 on parabolic sections associated with phi(x), under the assumption that the Monge-Ampere measure generated by phi satisfies the doubling condition on sections and the uniform continuity condition with respect to Lebesgue meas ure. The theory established is invariant under the group AT(n) x AT(1), whe re AT(n) denotes the group of all invertible affine transformations on R-n.