Comparison principles for Monge-Amp\`ere measures on pluripolar sets

TL;DR

This paper develops comparison principles for Monge-Ampère measures on pluripolar sets, introducing a new singularity comparison based on Bedford--Taylor capacity, with applications to uniqueness and characterization results.

Contribution

It introduces a novel singularity comparison for plurisubharmonic functions and establishes comparison principles on pluripolar sets within Cegrell classes.

Findings

Established comparison principles for Monge-Ampère measures on pluripolar sets.

Characterized the singularity relation via auxiliary functions in the energy class.

Proved a uniqueness theorem for the Monge-Ampère equation in this context.

Abstract

In this paper, we introduce a notion of singularity comparison for plurisubharmonic functions based on the Bedford--Taylor capacity. We establish comparison principles for the complex Monge--Amp\`ere operator on pluripolar sets in the Cegrell classes. As applications, we obtain a characterization of this relation via auxiliary functions in the energy class and prove a corresponding uniqueness result for the Monge--Amp\`ere equation.