Fatou Theorems and Maximal Functions for Eigenfunctions of the Laplace-Beltrami Operator in a Bidisk

摘要

Karpelevic determined the Martin boundary X for the eigenfunctions of the Laplace-Beltrami operator in a bidisk. Any f is an element of L(1)(X) gives rise to an eigenfunction, by integration against a kernel. It is shown that a.a. values of f can be recovered as the limit of the corresponding normalized eigenfunction along certain geodetic curves. This answers a question of Koranyi and extends results due to Linden. The proof uses a maximal function in three dimensions, which is not invariant under translation. A theorem on weakly restricted convergence for the square root of the product Poisson kernel is also proved.