Local potential and Hölder estimates for the linearized Monge-Ampère equation

Abstract: In this paper, we establish local potential estimates and H\"older estimates for solutions of linearized Monge-Amp`ere equations with the right-hand side being a signed measure, under suitable assumptions on the data. In particular, the interior H\"older estimate holds for an inhomogeneous linearized Monge-Amp`ere equation with right-hand side being the nonnegative divergence of a bounded vector field in all dimensions. As an application, we give a new approach for the interior estimate of the singular Abreu equation.