Dirichlet-like space and capacity in complex analysis in several variables

TL;DR

This paper introduces a complex-analytic Dirichlet-like space on Kähler manifolds, establishing its capacity properties and exploring its potential-theoretic features using pluripotential theory.

Contribution

It defines a new space W* related to pluripotential theory, proving classical Dirichlet space results in this complex setting and analyzing its capacity and topological properties.

Findings

Functions in W* are defined up to a pluripolar set

The capacity associated to W* tests pluripolar sets and is a Choquet capacity

W* is not reflexive and smooth functions are not dense in it

Abstract

For a Kahler manifold X, we study a space of test functions W* which is a complex version of H1. We prove for W* the classical results of the theory of Dirichlet spaces: the functions in W* are defined up to a pluripolar set and the functional capacity associated to W* tests the pluripolar sets. This functional capacity is a Choquet capacity. The space W* is not reflexive and the smooth functions are not dense in it for the strong topology. So the classical tools of potential theory do not apply here. We use instead pluripotential theory and Dirichlet spaces associated to a current.