The boundary behaviour of harmonic univalent maps
Let H denote the set of complex-valued functions f that are harmonic, orientation-preserving and univalent in \(\Delta =\{z\in {\mathbb{C}}:\) \(| z| <1\}\). Each \(f\in H\) can be written \(f=g+\bar h\) where g and h are analytic in \(\Delta\). The study of H was initiated by \textit{J. G. Clunie} and \textit{T. Sheil-Small} [Ann. Acad. Sci. Fenn., Ser. A I 9, 3-35 (1984; Zbl 0539.30006)]. The authors prove that there is a constant \(\lambda_ 0>0\) such that if \(f\in H\) and \(f=g+\bar h\) as above, then \(g,h\in H^{\lambda}\) and \(f\in h^{\lambda}\) for \(0<\lambda <\lambda_ 0\). Here \(H^{\lambda}\) denotes the Hardy space of analytic functions and \(h^{\lambda}\) denotes the Hardy space of harmonic functions. The authors also show that the integrals \[ \int^{1}_{0}| g'(re^{i\theta})| dr,\quad \int^{1}_{0}| h'(re^{i\theta})| dr\quad and\quad \int^{1}_{0}| \partial f/\partial r| dr \] converge for almost all \(\theta\) and the boundary function \(f(e^{i\theta})\equiv \lim_{r\to 1^-}f(re^{i\theta})\) exists for almost all \(\theta\). The paper contains a number of results concerning the boundary behavior of \(f\in H\). In particular, the case is studied where f extends continuously on an arc of \(\partial \Delta\) and is constant on that arc. Facts are obtained about the correspondence under \(f\in H\) between \(\partial \Delta\) and the prime ends of f(\(\Delta)\).
scientific article; zbMATH DE number 4120479