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

Let $u$ be a smooth, strictly $k$-plurisubharmonic function on a bounded domain $\Omega\in\mathbb C^n$ with $2\leq k\leq n$. The purpose of this paper is to study the regularity of solution to the linearized complex Monge-Amp\`ere and Hessian equations when the complex $k$-Hessian $H_k[u]$ of $u$ is bounded from above and below. We first establish some estimates 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 K\"ahler scalar curvature equation. In particular, we obtain the Harnack inequality for the linearized complex Monge-Amp\`ere and Hessian equations under an extra integrability condition on the coefficients. The approach works in both real and complex case.