Green-function gradient estimates for linearized Monge–Ampère equations

For the linearized Monge–Ampère operator in n≥3 dimensions, seek an L¹ bound C h^β on the spatial gradient of its Dirichlet Green’s function over a section of height h, uniformly in the pole. The convex C³ potential has Hessian determinant between positive constants, and the nested sections satisfy compact-containment hypotheses.

Why it matters and what counts as progress

Why it matters: The estimate is an analytic ingredient for higher-dimensional regularity; later Hölder theorems and applications go beyond this scoped target.

What counts as progress: Prove the uniform Green-gradient estimate under only the stated determinant and section hypotheses, with explicit dependence of constants.

How the problem developed

4 Jun 2015 · Partial result [1]
Nam Q. Le Proved Lᵖ integrability of the Green-function gradient in two dimensions under stated hypotheses on the Monge–Ampère measure.
16 Sept 2026 · AI contribution [2]
GPT-6 Astra, with Guoqing Cui, Chong Gu, Nam Q. Le, Ling Wang, and Bin Zhou The authors report that GPT-6 Astra first produced the argument for Proposition 1.2, an L¹ Green-gradient estimate with decay uniform in the pole; the paper uses it as a key ingredient.
1 Oct 2026 · Reported claim [2]
Guoqing Cui, Chong Gu, Nam Q. Le, Ling Wang, and Bin Zhou Released global Hölder estimates in dimensions n≥3 under two-sided Hessian-determinant bounds and a bounded divergence-form vector field, using the Green-gradient estimate as an ingredient.

AI contribution

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.

AI involvementAI-centered

ImportanceNotable

Technical verificationFormal (human)

Human understandingFull