Relative Finite Energy Classes for Complex Hessian Equations with Prescribed Singularities
Abstract
Let Ω⊂ ℂn be a bounded m-hyperconvex domain and let ψ∈ SHm(Ω) be a fixed negative m-subharmonic function. In this paper we introduce a relative finite energy class ℰm,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 ℰm,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 (ddc u)m∧βn−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.
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BibTeX
@misc{dat2026relative,
title = {{Relative Finite Energy Classes for Complex Hessian Equations with Prescribed Singularities}},
author = {Truong Dinh Dat},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12032},
eprint = {2609.12032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12032},
}
RIS
TY - UNPB
TI - Relative Finite Energy Classes for Complex Hessian Equations with Prescribed Singularities
AU - Truong Dinh Dat
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.12032
UR - https://arxiv.org/abs/2609.12032
AB - Let Ω⊂ ℂ^n be a bounded m-hyperconvex domain and let ψ∈ SH_m(Ω) be a fixed negative m-subharmonic function. In this paper we introduce a relative finite energy class 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 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∧β^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.
ER -