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 $\psi\in SH_m(\Omega)$ 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--Amp\`ere 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 $(dd^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.