Relative Finite Energy Classes for Complex Hessian Equations with Prescribed Singularities

Abstract

Let $\Omega \subset {\mathbb{C}}^n$ be a bounded $m$-hyperconvex domain and let $\psi \in S{H}_m(\Omega )$ be a fixed negative $m$-subharmonic function. In this paper we introduce a relative finite energy class ${\mathcal{E}}_{m,\psi }(\Omega ),$ which may be viewed as a Hessian analogue of the relative energy classes appearing in the pluripotential theory of complex Monge--Ampère equations. We develop a systematic pluripotential theory in this setting. More precisely, we introduce a relative Hessian capacity associated with the prescribed singularity type $\psi$, construct relative mixed Hessian products, and establish their fundamental properties. We prove a monotone convergence theorem and a Bedford--Taylor type continuity theorem for Hessian measures in the class ${\mathcal{E}}_{m,\psi }(\Omega )$. A central result of the paper is a relative comparison principle, which yields uniqueness of solutions to complex Hessian equations with prescribed singularities. As an application, we establish an existence and uniqueness theorem for the equation $(d{d}^{c}u{)}^m\wedge {\beta }^{n-m}=\mu$ for a large class of positive Radon measures that do not charge $m$-polar sets. The results obtained here provide a relative finite energy framework for complex Hessian equations and extend several fundamental aspects of Cegrell's theory to the setting of prescribed singularity types.