REMOVABLE SINGULARITES FOR ANALYTIC FUNCTIONS OF ZYGMUND CLASS

It follows from a theorem of Dolzhenko, that if a compact set in the plane has zero Hausdorff measure with respect to the measure function h(r) = r2 log(1/r), then it is removable for analytic functions of the Zygmund class. In this paper it is shown that there is a compact set N in the plane such that N has zero Hausdorff measure with respect to each measure function h(r) that is o r2(log(1/r)) 12 and N is non–removable for some analytic function of the Zygmund class. Some related results of a real–variable nature are obtained. 1