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arXiv · 2608.22180

Interior $C^{1,\alpha}$ estimates for the linearized Monge--Amp\`ere equation in two dimensions

Ling Wang·Bin Zhou
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Abstract

We prove an interior $C^{1,\alpha}$ estimate for solutions of the homogeneous linearized Monge--Amp\`ere equation in dimension two under the assumption \[ 0<\lambda\leq \det D^2\varphi\leq\Lambda<+\infty. \] No continuity assumption on the Monge--Amp\`ere density is required. Our result is an affine-invariant analogue of the classical Morrey--Nirenberg $C^{1,\alpha}$ 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's Harnack inequality gives the H\"older 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.

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Ling Wang, Bin Zhou. 2026-08-23. Interior $C^{1,\alpha}$ estimates for the linearized Monge--Amp\`ere equation in two dimensions. https://arxiv.org/abs/2608.22180

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