Sharp Regularity Results on Second Derivatives of Solutions to the Monge-Ampere Equation with VMO Type Data

We establish interior estimates for L(p)-norms, Orlicz norms, and mean oscillation of second derivatives of solutions to the Monge-Ampere equation det D(2)u = f(x) with zero boundary value, where f(x) is strictly positive, bounded, and satisfies a VMO-type condition. These estimates develop the regularity theory of the Monge-Ampere equation in VMO-type spaces. Our Orlicz estimates also sharpen Caffarelli's celebrated W(2,p)-estimates. (C) 2008 Wiley Periodicals, Inc.