Geometric properties of boundary sections of solutions to the Monge--Ampère equation and applications

Abstract

In this paper, we establish several geometric properties of boundary sections of convex solutions to the Monge-Ampère equations: the engulfing and separating properties and volume estimates. As applications, we prove a covering lemma of Besicovitch type, a covering theorem and a strong type $p-p$ estimate for the maximal function corresponding to boundary sections. Moreover, we show that the Monge-Ampère setting forms a space of homogeneous type.

24 pages

References in corpus (3)

• Interior estimates for solutions of Abreu's equation
• A localization property at the boundary for the Monge-Ampere equation
• Global second derivative estimates for the second boundary value problem of the prescribed affine mean curvature and Abreu's equations