A Green's function approach to linearized Monge-Amp\`ere equations in divergence form and application to singular Abreu type equations

In this paper, we establish local and global regularity estimates for linearized Monge-Ampère equations in divergence form via critical Lorentz space estimates for the Green's function of the linearized Monge-Ampère operator and its gradient.

These estimates hold under suitable conditions on the data and the convex Monge-Ampère potential is assumed to have Hessian determinant bounded between two positive constants.

As an application, we obtain the solvability in all dimensions of the second boundary value problem for a class of singular fourth-order Abreu type equations that arise from the approximation analysis of variational problems subject to convexity constraints.