Additional Smoothness Phenomena for Analytic Functions

Abstract

We study the influence of geometric properties of a domain on the smoothness of Holder class analytic functions defined on it. The case of the disk is covered by the classical results of Hardy and Littlewood. We consider a domain G with an inward cusp boundary point ξ)(this means that meas \(U_\xi \cap ({\mathbb{C}}\backslash G)\)/ meas \(U_\xi \to 0\) as meas\(U_\xi \to 0\), whereUξ is a neighborhood of ξ). Three zones are distinguished near such a point: the outer zone, where high smoothness occurs, the boundary zone, where the smoothness is “standard,” and the intermediate zone, where the smoothness decays steadily from high to standard one. A sharp geometric description of these zones is given. Bibliography: 7 titles.