p-Radial Exceptional Sets and Conformal Mappings

Kot, Piotr

Canadian mathematical bulletin, Tome 50 (2007) no. 4, pp. 579-587

Résumé

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 $\mathbb{D}$ .

Détail

DOI : 10.4153/CMB-2007-055-6

Mots-clés : AMS subject classification: 30B30, 30E25, boundary behaviour of holomorphic functions, exceptional sets