An extremal problem for generalized Lelong numbers

TL;DR

This paper investigates bounds on plurisubharmonic functions near singularities based on generalized Lelong numbers, formulating and solving an extremal problem to understand their behavior.

Contribution

It introduces an extremal problem for generalized Lelong numbers and provides explicit solutions in specific cases, advancing the understanding of plurisubharmonic function bounds.

Findings

Explicit solutions to the extremal problem in certain cases

Pointwise bounds on plurisubharmonic functions near singularities

Enhanced understanding of generalized Lelong numbers' role

Abstract

We look for pointwise bounds on a plurisubharmonic function near its singularity point, given the value of its generalized Lelong number with respect to a plurisubharmonic weight. To this end, an extremal problem is considered. In certain cases, the problem is solved explicitly.