Angular limits of holomorphic functions which satisfy an integrability condition (Q1207654)

Let \(\beta\in(-1,1)\), let \(2/(1-\beta)\leq p<\infty\), let \(p'\) denote the Hölder conjugate of \(p\), and let \(\gamma\) be an open arc of the unit circle. It is shown that, if \(f\) is a holomorphic function on the unit disc such that: (i) \((1-| z|)^ \beta \log^ + | f(z)|\) is \(L^ p\)- integrable on the sector \(\{r\zeta\): \(0<r<1\), \(\zeta\in\gamma\}\), and (ii) the subset of \(\gamma\) where \(f\) has an infinite asymptotic value has \(\sigma\)-finite \((2-(1+\beta)p')\)-dimensional Hausdorff measure, then \(f\) has finite angular limits on a subset of \(\gamma\) of positive linear measure. In fact, ``better-than-angular'' limits are shown to exist.