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

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.