It is known that, under very general conditions, Blaschke products generate branched covering surfaces of the Riemann sphere. We are presenting here a method of finding fundamental domains of such coverings and we are studying the corresponding groups of covering transformations.
Let be the class of bounded analytic functions on , and let be the set of maximal ideals of the algebra , a compactification of . The relations between functions in and their cluster values on are studied. Let be the subset of over the point 1. A subset of is a “Fatou set” if every in has a limit at for almost every . The nontangential subset of is a Fatou set according to the Fatou theorem. There are many larger Fatou sets, for example the fine topology subset of but...
We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ 0,1. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves...
We study boundary properties of universal Taylor series. We prove that if f is a universal Taylor series on the open unit disk, then there exists a residual subset G of the unit circle such that f is unbounded on all radii with endpoints in G. We also study the effect of summability methods on universal Taylor series. In particular, we show that a Taylor series is universal if and only if its Cesàro means are universal.
Let be harmonic in the half-space , . We show that can have a fine limit at almost every point of the unit cubs in but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary.In it is known that the Hardy classes , , may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the integrability...