Let $H^{\infty }$ be the class of bounded analytic functions on $D:\left|z\right|\<1$, and let $\bar{D}$ be the set of maximal ideals of the algebra $H^{\infty }$, a compactification of $D$. The relations between functions in $H^{\infty }$ and their cluster values on $\bar{D}-D$ are studied. Let $D_1$ be the subset of $\bar{D}$ over the point 1. A subset $A$ of $D_1$ is a “Fatou set” if every $f$ in $H^{\infty }$ has a limit at $e^{i\theta }A$ for almost every $\theta$. The nontangential subset of $D_1$ is a Fatou set according to the Fatou theorem. There are many larger Fatou sets, for example the fine topology subset of $D_1$ but...

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 $u$ be harmonic in the half-space $R_{+}^{n+1}$, $n\ge 2$. We show that $u$ can have a fine limit at almost every point of the unit cubs in $R^n=\partial R_{+}^{n+1}$ 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 $R_{+}^2$ it is known that the Hardy classes $H^p$, $0\<p\<\infty$, may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the integrability...