AI 中文总结
本文针对具有给定奇点的复 Hessian 方程，引入相对有限能量类并建立相对比较原理，从而证明解的唯一性及一大类测度下方程的存在唯一性。
AI 中文摘要
设 $\Omega\subset \mathbb C^n$ 为有界 $m$-超凸域，且 $\psi\in SH_m(\Omega)$ 为固定的负 $m$-次调和函数。本文引入相对有限能量类 $\mathcal E_{m,\psi}(\Omega)$，它可视为复 Monge–Ampère 方程多重位势理论中出现的相对能量类的 Hessian 类比。我们在此框架下发展了系统的多重位势理论。具体而言，我们引入与给定奇点类型 $\psi$ 相关的相对 Hessian 容量，构造相对混合 Hessian 乘积，并建立其基本性质。我们证明了单调收敛定理以及类 $\mathcal E_{m,\psi}(\Omega)$ 中 Hessian 测度的 Bedford–Taylor 型连续性定理。本文的一个核心结果是相对比较原理，它给出了具有给定奇点的复 Hessian 方程解的唯一性。作为应用，我们为一大类不承载 $m$-极集的正 Radon 测度建立了方程 $(dd^c u)^m\wedge\beta^{n-m} =\mu $ 的存在唯一性定理。所得结果为复 Hessian 方程提供了相对有限能量框架，并将 Cegrell 理论的若干基本方面推广到给定奇点类型的情形。
英文摘要
Let $Ω\subset \mathbb C^n$ be a bounded $m$-hyperconvex domain and let $ψ\in SH_m(Ω)$ be a fixed negative $m$-subharmonic function. In this paper we introduce a relative finite energy class $\mathcal E_{m,ψ}(Ω),$ 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 $ψ$, 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 $\mathcal E_{m,ψ}(Ω)$. 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 $(dd^c u)^m\wedgeβ^{n-m} =μ$ 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.
