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p-Radial Exceptional Sets and Conformal Mappings

Published online by Cambridge University Press:  20 November 2018

Piotr Kot
Piotr Kot*
Affiliation:
Politechnika Krakowska, Instytut Matematyki, ul. Warszawska 24, 31-155 Kraków, Poland e-mail: pkot@usk.pk.edu.pl
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Abstract

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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 $f\,\in \,\mathbb{O}\left( \mathbb{D} \right)$ such that

$${{\int_{\mathbb{D}\backslash \left[ 0,\,1 \right]E}{\left| f \right|}}^{p}}\,d{{\mathfrak{L}}^{2}}\,<\,\infty \,\text{and}\,E\,=\,{{E}^{p}}\left( f \right)\,=\,\{\,z\,\in \,\partial \mathbb{D}\,:\,\int _{0}^{1}\,{{\left| f\left( tz \right) \right|}^{p}}\,dt\,=\,\infty \}.$$

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

AMS subject classification: 30B3030E25boundary behaviour of holomorphic functionsexceptional sets
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 4 , 01 December 2007 , pp. 579 - 587
Copyright
Copyright © Canadian Mathematical Society 2007

References

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