POGORELOV ESTIMATES FOR THE MONGE-AMPÈRE EQUATIONS.

In this paper, we study the Pogorelov estimate for the Monge-Ampère equation detD2u = f(x) under the assumption f1/n-1 ∈ C1,1(Ω̄). When n ≥ 3, we improve the Pogorelov estimate (w-u)α|D²u| ≤ C by Błocki [Bull. Austral. Math. Soc. 68 (2003), pp. 81-92] from α = n - 1 to all α > 1. Some applications of the Pogorelov estimate are discussed. [ABSTRACT FROM AUTHOR]