The Best Approach for Boundary Limits

Abstract

In [5], J. L. Doob showed that for positive harmonic functions on the unit disk, there is no coarsest approach neighborhood system, i.e. filter, for which a Fatou boundary limit theorem holds when the filter is copied by rotation at all points of the unit circle. In [4]the authors used the principal result from [3] to show that there is a coarsest filter when the problem is suitably normalized. The normalization assigns to each positive harmonic function a zero set; this is a set of boundary points at which the function must vanish. Known limits such as those provided by the Lebesgue Differentiation Theorem or the Fine Limit Theorem force consistency in this assignment. The zero sets on the boundary are then used in constructing approach neighborhoods which are level sets in the disk. These neighborhoods form the coarsest filters for which a Fatou boundary limit theorem holds and the required zero limits achieved. The construction is new for the unit disk, but it is also valid for very general settings. It shows that any limit theorem for positive harmonic functions can be replaced with one which is at least as good (in terms of the coarseness of the filters) where the approach neighborhoods are generated by level sets of harmonic functions.