On the $$W^{2,1+\varepsilon }$$ W 2 , 1 + ε -estimates for the Monge-Ampère equation and related real analysis

Abstract

We build upon the techniques introduced by De Philippis and Figalli regarding \(W^{2,1+\varepsilon }\) bounds for the Monge-Ampere operator, to improve the recent \(A_\infty \) estimates for \(\Vert D^2 \varphi \Vert \) to \(A_2\) ones. Also, we prove a \((1,2)-\)Poincare inequality and weak \((q,p)-\)Poincare inequalities associated to the Monge-Ampere quasi-metric structure. In turn, these Poincare inequalities are used to prove Harnack’s inequality for non-negative solutions to the linearized Monge-Ampere under minimal geometric assumptions.