Sobolev inequalities and regularity of the linearized complex Monge-Ampère and Hessian equations

Let u u be a smooth, strictly k k -plurisubharmonic function on a bounded domain Ω ∈ C n Cⁿ with 2 ≤ k ≤ n 2 k n. The purpose of this paper is to study the regularity of solution to the linearized complex Monge-Ampère and Hessian equations when the complex k k -Hessian H k u Hₖu of u u is bounded from above and below. We first establish an estimate of Green’s functions associated to the linearized equations. Then we prove a class of new Sobolev inequalities. With these inequalities, we use Moser’s iteration to investigate the a priori estimates of Hessian equations and their linearized equations, as well as the Kähler scalar curvature equation. In particular, we obtain the Harnack inequality for the linearized complex Monge-Ampère and Hessian equations under an extra integrability condition on the coefficients.