Dirichlet's problem on Green lines and some convergence theorems for Denjoy's domains (Q1319327)

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).