The local Dirichlet integral and applications

TL;DR

This paper investigates the local Dirichlet integral of distance functions within harmonic Dirichlet spaces, providing estimates, conditions for polar sets, and insights into cyclicity, advancing understanding of function behavior in these spaces.

Contribution

It introduces new estimates for the local Dirichlet integral and offers criteria for polar sets and cyclicity in harmonic Dirichlet spaces, enhancing theoretical understanding.

Findings

Derived bounds for the local Dirichlet integral of distance functions

Established sufficient conditions for sets to be polar

Analyzed cyclicity properties in harmonic Dirichlet spaces

Abstract

We study the local Dirichlet integral of distance functions and their behavior within the harmonic Dirichlet space. We provide estimates for the local Dirichlet integral of distance functions, which allow us to study their membership in the algebra of multipliers of the Dirichlet space. We give sufficient condition for a closed subset of the unit circle to be polar and we also examine cyclicity in the harmonic Dirichlet spaces.