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Remarks on the Green's function of the linearized Monge-Ampère operator

In this note, we obtain sharp bounds for the Green's function of the linearized Monge-Ampère operators associated to convex functions with either Hessian determinant bounded away from zero and infinity or Monge-Ampère 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-Ampère equation.