Global H\"older estimates for 2D linearized Monge-Amp\`ere equations with right-hand side in divergence form

We establish global H"older estimates for solutions to inhomogeneous linearized Monge-Amp`ere equations in two dimensions with the right hand side being the divergence of a bounded vector field. These equations arise in the semi-geostrophic equations in meteorology and in the approximation of convex functionals subject to a convexity constraint using fourth order Abreu type equations. Our estimates hold under natural assumptions on the domain, boundary data and Monge-Amp`ere measure being bounded away from zero and infinity. They are an affine invariant and degenerate version of global H"older estimates by Murthy-Stampacchia and Trudinger for second order elliptic equations in divergence form.