Interior $C^{1,α}$ estimates for the linearized Monge--Ampère equation in two dimensions

Abstract

We prove an interior $C^{1,α}$ estimate for solutions of the homogeneous linearized Monge--Ampère equation in dimension two under the assumption \[ 0<λ\leq \det D^2φ\leqΛ<+\infty. \] No continuity assumption on the Monge--Ampère density is required. Our result is an affine-invariant analogue of the classical Morrey--Nirenberg $C^{1,α}$ estimate in two dimensions. The core of the proof is the partial Legendre transform. After the transform, the first derivatives of the solution are quotients of adjoint solutions for a uniformly elliptic non-divergence form equation. Bauman&#39;s Harnack inequality gives the Hölder control of the quotient, while the Jacobian identity of the partial Legendre transform and a Caccioppoli estimate give its local boundedness. As an application, we prove a Liouville theorem for entire solutions with at most linear growth.