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$p$-Radial Exceptional Sets and Conformal Mappings

Published:2007-12-01
Printed: Dec 2007
• Piotr Kot
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Abstract

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 $\mathbb D$.
 Keywords: boundary behaviour of holomorphic functions, exceptional sets
 MSC Classifications: 30B30 - Boundary behavior of power series, over-convergence 30E25 - Boundary value problems [See also 45Exx]

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