New local characterizations of the weighted energy class $\mathcal{E}_{\chi,\mathrm{loc}}(\Omega)$

TL;DR

This paper introduces new local characterizations and control methods for the weighted energy class _{ ext{loc}}(\u03a9) of plurisubharmonic functions, improving understanding of their energy and Monge--Ampre measure behavior.

Contribution

It establishes explicit local boundedness of weighted energy and weakens conditions for class membership, expanding the framework for analyzing singularities of plurisubharmonic functions.

Findings

Proves local boundedness of weighted Monge--Ampère energy under certain conditions.

Shows that domination of Monge--Ampère measures implies class membership.

Provides a more flexible framework for functions with singularities.

Abstract

Let \(\Omega\subset\mathbb{C}^n\) be a hyperconvex domain and let \(\chi:\mathbb{R}^-\to\mathbb{R}^+\) be a decreasing function. This note studies the local weighted energy class \(\mathcal{E}_{\chi,\mathrm{loc}}(\Omega)\) introduced in \cite{HHQ13}. We establish two main results on local membership in this class. First, we prove a new local boundedness property for the weighted Monge--Amp\`ere energy: if \(u\in\mathrm{PSH}^-(\Omega)\) admits suitable local majorants in \(\mathcal{E}_{\chi,\mathrm{loc}}\) near the boundary of every relatively compact hyperconvex subdomain \(D\Subset\Omega\), then the weighted energy \(\int_K \chi(u)(dd^c u)^n\) remains locally finite for every compact set \(K\subset D\). This gives the first explicit local control of the energy functional and is new even in the unweighted setting. Second, we obtain a substantial improvement concerning the local control of the Monge--Amp\`ere measure. We show that if, in addition to the boundary condition, \((dd^c u)^n\) is locally dominated by \((dd^c w)^n\) for some \(w\in\mathcal{E}_{\chi,\mathrm{loc}}(D)\) inside \(D\), then \(u\in\mathcal{E}_{\chi,\mathrm{loc}}(D)\). This domination condition is strictly weaker than the previous requirement of local finiteness of the weighted energy, thereby significantly enlarging the class of admissible functions. Our results extend and refine the local theory developed in \cite{Q24,Q25} and provide a more flexible framework for plurisubharmonic functions with possible singularities on compact subsets.