Alexander-Taylor's inequality for capacities in complex Sobolev spaces

TL;DR

This paper establishes a precise inequality relating Alexander-Taylor capacity and a functional capacity within complex Sobolev spaces on compact Kähler manifolds, advancing the understanding of capacities in complex geometry.

Contribution

It proves a sharp inequality connecting two types of capacities in complex Sobolev spaces, extending previous definitions by Dinh, Sibony, and Vigny.

Findings

Established a sharp inequality between capacities

Extended capacity theory in complex Sobolev spaces

Enhanced understanding of geometric inequalities in Kähler manifolds

Abstract

We prove a sharp inequality between the Alexander-Taylor capacity and the functional capacity in a complex Sobolev space on a compact K\"ahler manifold. The latter space and capacity were introduced by Dinh, Sibony and Vigny.