$p$-Radial Exceptional Sets and Conformal Mappings

http://dx.doi.org/10.4153/CMB-2007-055-6
Canad. Math. Bull. 50(2007), 579-587

  Published:2007-12-01
 Printed: Dec 2007

For $p>0$ and for a given set $E$ of type $G_{\delta}$ in the boundary of the unit disc $\partial\mathbb D$ we construct a holomorphic function $f\in\mathbb O(\mathbb D)$ such that \[ \int_{\mathbb D\setminus[0,1]E}|ft|^{p}\,d\mathfrak{L}^{2}<\infty\] and\[ E=E^{p}(f)=\Bigl\{ z\in\partial\mathbb D:\int_{0}^{1}|f(tz)|^{p}\,dt=\infty\Bigr\} .\] In particular if a set $E$ has a measure equal to zero, then a function $f$ is constructed as integrable with power $p$ on the unit disc $\mathbb D$.