Remarks on the Green's function of the linearized Monge-Amp\`ere operator

TL;DR

This paper establishes sharp bounds for the Green's function of the linearized Monge-Ampère operator under certain conditions, extending classical elliptic PDE results with affine invariance and applications to singularity removal.

Contribution

It provides the first affine invariant bounds for the Green's function of the linearized Monge-Ampère operator and explores its integrability and singularity properties.

Findings

Sharp bounds for Green's function under Hessian determinant conditions

L^p integrability of the gradient of Green's function in 2D

Removable singularity results for the linearized Monge-Ampère equation

Abstract

In this note, we obtain sharp bounds for the Green's function of the linearized Monge-Amp\`ere operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge-Amp\`ere measure satisfying a doubling condition. Our result is an affine invariant version of the classical result of Littman-Stampacchia-Weinberger for uniformly elliptic operators in divergence form. We also obtain the $L^{p}$ integrability for the gradient of the Green's function in two dimensions. As an application, we obtain a removable singularity result for the linearized Monge-Amp\`ere equation.