On interior \(C^2\)-estimates for the Monge-Ampère equation (Q1661009)

The author utilizes methods of his previous paper [J. Differ. Equations 256, No. 6, 1987--2022 (2014; Zbl 1287.35046)] based on a mean-value inequality for non-negative subsolutions to obtain new apriori interior and Dirichlet estimates for the strictly convex solutions of the Monge-Ampère equation with a more general condition on the Monge-Ampère measure \(\mu_{\varphi}\) (\(\mu_{\varphi}\in(DC)_{\phi}\), see Section 2.). Notice that this condition on \(\mu_{\varphi}\), as a special case, contains classical positive \(L^{\infty}\)-condition on the right-hand side of the Monge-Ampère equation \(\det D^{2}\varphi =f\). As a corollary of the author's methods, a new derivation of the classical Pogorelov estimates in dimension \(2\) is presented.