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Global Hölder estimates for linearized Monge–Ampère equations in divergence form with applications to dual semigeostrophic equations and periodic homogenization

Guoqing Cui, Chong Gu, Nam Q. Le, Ling Wang, Bin Zhou

For n≥3, Proposition 1.2 bounds the spatial L¹ norm of the gradient of the Dirichlet Green’s function for the linearized Monge–Ampère operator by C h^β, uniformly in its pole on a section of height h. The convex C³ potential has Hessian determinant between positive constants; the nested sections satisfy the paper’s compact-containment hypotheses.

Sources for this record

What was already known

The authors’ earlier unconditional divergence-form Hölder estimates were restricted to dimension two. A suitable uniform Green-gradient integrability estimate was the missing ingredient in their higher-dimensional program.

AI involvement AI-centered

The five authors state that the argument of Proposition 1.2 was first produced through repeated interactions with GPT-6 Astra on 16 September 2026. They then checked it in detail and independently verified it before completing the rest of their program without further AI assistance.

Importance Notable

Supplies the specific analytic estimate that the authors use to complete higher-dimensional regularity arguments. The later Hölder theorems and applications retain human credit.

Limitations

The retained AI contribution is Proposition 1.2, not the entire paper. Constants depend on dimension, determinant bounds, the reference section height and geometry. The section of doubled height is compactly contained in the prescribed larger section, itself compactly contained in the domain. Human authors completed the remaining regularity results and applications.

Technical verification Formal (human)

The named human authors explicitly report checking the argument of Proposition 1.2 in detail and verifying its proof independently. This supports complete source-reported human checking of this proposition.

Independent checking

The authors explicitly report detailed checking and independent verification of the model-produced Proposition 1.2 argument. They are the paper’s authors, not an outside review panel.

Checker
Guoqing Cui, Chong Gu, Nam Q. Le, Ling Wang and Bin Zhou
Scope
Proposition 1.2 only: uniform L¹ Green-gradient decay in the stated nested Monge–Ampère sections.
Basis
Reported by the linked checker
Evidence source

Human understanding Full

The authors explain the estimate’s mechanism in Remark 1.3 and Section 3: Green-function level bands localize to sections whose Laplacian mass controls the gradient integral. They use the checked ingredient to complete their program.

Correction and publication history

2026-10-20 · publication

Added the historical AI-origin Green-gradient estimate, with the authors’ completed proof checking and subsequent human work separately credited.

Read this published revision