Dirichlet's problem on Green lines and some convergence theorems for Denjoy's domains
The author proves a Fatou type theorem and the convergence of almost all Green lines in the Martin boundary for Denjoy domains in the plane. These are open subsets of \(\mathbb{C}\) with nonpolar boundary strictly included in \(\mathbb{R}\). For the corresponding sets in higher dimensions we prove convergence along the Green lines in the sense of \(L^ 1\) for every bounded harmonic function. The main difficulty is that we do not know the geometric behaviour of the Green lines (for example, non tangential approach at the boundary).
scientific article; zbMATH DE number 549769