Extreme Points in the Hardy Class H1 of a Riemann Surface

Abstract

The theorems presented here extend known results on the set of extreme points of the unit ball of the Hardy class H1 of a disk to the situation of an arbitrary Riemann surface. Several new results are obtained. The initial motivation for this work was provided by the theorem of de Leeuw and Rudin [2, p. 471] characterizing the extreme points in the case ol a disk. Careful scrutiny of the proof of that theorem yields one necessary and one sufficient condition for being an extreme point in H1 of an arbitrary surface (Theorems 1 and 4 below). The material presented here on compact bordered surfaces is closely related to the beautiful results of Gamelin and Voichick [4] and the results of Forelli [3].For a subharmonic function u, which has a harmonic majorant on the Riemann surface R, Mu will denote the least harmonic majorant of u.