Dirichlet Spaces In Balls And Half-spaces of $\R^n$

TL;DR

This paper characterizes Dirichlet spaces in higher-dimensional balls and half-spaces, extending classical 2-D results to more complex geometries with identical norm descriptions.

Contribution

It provides new characterizations of Dirichlet norms in higher dimensions, generalizing classical 2-D results to balls and half-spaces in ^n.

Findings

Identical Dirichlet norm characterizations as in the classical 2-D case

Extension of Dirichlet space theory to higher-dimensional geometries

Unified framework for Dirichlet spaces in balls and half-spaces

Abstract

The present paper studies the Dirichlet spaces in balls and upper-half Euclidean spaces. As main results, we give identical characterizations of the Dirichlet norms in the respective contexts as for the classical 2-D disc case proved by Douglas and Ahlfors.