This chapter is devoted to the Martin boundary of a fine domain U in the Greenian domain \(\Omega \subset {\mathbb R}^N\) and the integral representation of invariant functions on U. In Sect. 3.1, we define a topology on the cone \({\mathcal {S}}_+(U)\) of positive finely superharmonic functions on U for which it has a compact base \(\textbf{B}\) . We establish the integral representation in \({\mathcal {S}}_+(U)\) by means of extreme points of \(\textbf{B}\) and prove that any point of the set \({{\,\textrm{Ext}\,}}(\textbf{B})\) of extreme points of \(\textbf{B}\) is of the form \(\alpha G_U(\cdot ,y)\) for some \(\alpha >0\) and some \(y\in U\) , or an invariant function (called a minimal invariant function). We prove that the sets \({{\,\textrm{Ext}\,}}_p(\textbf{B})\) of extreme fine potentials in \(\textbf{B}\) and \({{\,\textrm{Ext}\,}}_i(\textbf{B})\) of minimal invariant functions in \(\textbf{B}\) are Borel subsets of \(\textbf{B}\) and that every fine potential, resp. invariant function, is uniquely represented by a measure \(\mu \) on \(\textbf{B}\) supported by \({{\,\textrm{Ext}\,}}_p(\textbf{B})\) , resp. \({{\,\textrm{Ext}\,}}_i(\textbf{B})\) .

The Martin Boundary of a Fine Domain

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发表日期：

2025年

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Mohamed El Kadiri,
Bent Fuglede

摘要

This chapter is devoted to the Martin boundary of a fine domain U in the Greenian domain \(\Omega \subset {\mathbb R}^N\) and the integral representation of invariant functions on U. In Sect. 3.1, we define a topology on the cone \({\mathcal {S}}_+(U)\) of positive finely superharmonic functions on U for which it has a compact base \(\textbf{B}\) . We establish the integral representation in \({\mathcal {S}}_+(U)\) by means of extreme points of \(\textbf{B}\) and prove that any point of the set \({{\,\textrm{Ext}\,}}(\textbf{B})\) of extreme points of \(\textbf{B}\) is of the form \(\alpha G_U(\cdot ,y)\) for some \(\alpha >0\) and some \(y\in U\) , or an invariant function (called a minimal invariant function). We prove that the sets \({{\,\textrm{Ext}\,}}_p(\textbf{B})\) of extreme fine potentials in \(\textbf{B}\) and \({{\,\textrm{Ext}\,}}_i(\textbf{B})\) of minimal invariant functions in \(\textbf{B}\) are Borel subsets of \(\textbf{B}\) and that every fine potential, resp. invariant function, is uniquely represented by a measure \(\mu \) on \(\textbf{B}\) supported by \({{\,\textrm{Ext}\,}}_p(\textbf{B})\) , resp. \({{\,\textrm{Ext}\,}}_i(\textbf{B})\) .