A Plessner Decomposition Along Transverse Curves

Abstract

A classical theorem of Plessner [6] asserts that any holomorphic function f on the unit disk partitions the unit circle, modulo a null set, into two disjoint pieces such that at each point of the first piece, f has a non-tangential limit, and at each point of the second piece, the cluster set of f in any Stolz angle is the entire plane. Higher dimensional versions of this result were first obtained by Calderon [2], who considered holomorphic functions on Cartesian products of half-planes. In this setting, an exact analogue of the one-dimensional result is obtained, in which the circle is replaced by the distinguished boundary, and the Stolz angles are replaced by products of cones in the coordinate half-planes. The ideas of Calderon were further developed by Rudin [8, pp. 79-83], who considered holomorphic and invariant harmonic functions in the ball of Cn. In this case, the circle is replaced by the unit sphere, and the Stolz angles are replaced by the approach regions of Korányi [4].