Tangential limits of Blaschke products and functions of bounded mean oscillation (Q1075463)

Let \(\Gamma\) : [0,\(\pi\) ]\(\to \{| z| \leq 1\}\) be a Jordan arc such that Arg \(\Gamma\) (t)\(=t\) and \(| \Gamma (t)|\) is strictly decreasing on [0,\(\pi\) ] with \(\Gamma (0)=1\). Let \(\Omega\) be the simply- connected region in the unit disk \(\Delta =\{| z| <1\}\) bounded by \(\Gamma\) and its conjugate \({\bar \Gamma}\). Set \(\eta \Omega =\{\eta z:\) \(z\in \Omega \}\), the rotate of \(\Omega\) by \(\eta \in C=\{| z| =1\}\). In this paper we are concerned with the existence of \(\eta\) \(\Pi\)-limits at a given point \(\eta\in C\) and off of a small exceptional set E of \(\eta\). We improve results of Frostman and Cargo where a condition is placed on the rate at which the zeros of a Blaschke product approach C and the exceptional set E is shown to be of capacity 0. We prove that E actually has Hausdorff measure 0. This result is shown to be sharp in several ways. A similar result holds for \(K_*(B)=K_ 2(B)\cap BMOA\) where \(K_ 2(B)\) is the orthogonal complement in \(H^ 2\) of the invariant subspace \(BH^ 2\) and BMOA is the space of analytic functions of bounded mean oscillation on \(\Delta\).