Boundary Harnack inequality for the linearized Monge-Amp\`ere equations and applications

In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Amp`ere equations under natural assumptions on the domain, Monge-Amp`ere measures and boundary data. Our results are boundary versions of Caffarelli and Guti'errez’s interior Harnack inequality for the linearized Monge-Amp`ere equations. As an application, we obtain sharp upper bound and global $L^p$-integrability for the Green’s function of the linearized Monge-Amp`ere operator.