Interior Estimates for Monge-Ampère Equation in Terms of Modulus of Continuity

Abstract: We investigate the Monge-Amp`ere equation subject to zero boundary value and with a positive right-hand side unction assumed to be continuous or essentially bounded. Interior estimates of the solution's first and second derivatives are obtained in terms of moduli of continuity. We explicate how the estimates depend on various quantities but have them independent of the solution's modulus of convexity. Our main theorem has many useful consequences. One of them is the nonlinear dependence between the H\"older seminorms of the solution and of the right-side function, which confirms the results of Figalli, Jhaveri and Mooney (J. Func. Anal. 2016). Our technique is in part inspired by Jian and Wang (SIAM J. Math. Anal. 2007) which includes using a sequence of so-called sections.