Fatou type theorems. Maximal functions and approach regions
The topic of this book is Fatou theorems, or the study of boundary behaviour of holomorphic or harmonic functions. For instance it is well known that under certain growth conditions a harmonic function in the unit disc has radial, and nontangential boundary limits almost everywhere. It is also known that if one approaches the boundary along the rotates of a fixed curve which is tangential to the boundary of the unit disc, then there are harmonic functions that do not have a limit along these curves. \textit{A. Nagel} and \textit{E. M. Stein} [Adv. Math. 54, 83-106 (1984; Zbl 0546.42017)], showed that there are regions strictly larger than any nontangential region, in which the convergence still holds almost everywhere. These regions contain a tangential sequence, but do not contain a tangential curve. This book gives an interesting and thorough introduction to the results and methods used in this field in the classical context, and also extends the result of Nagel and Stein to spaces where the boundary does not admit a group acting on it. Among these are NTA domains in \(\mathbb{R}^{n}\) and pseudoconvex domains of finite type in \(\mathbb{C}^2\). The method used here relies on a process of discretization and using an earlier result for trees by \textit{N. Arcozzi, F. Di Biase} and \textit{R. Urbanke} [J. Reine Angew. Math. 472, 157-175 (1996; Zbl 0839.31012)].
scientific article; zbMATH DE number 1099335