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

Bibliographie

[1] [1] Globevnik, J., Holomorphic functions which are highly nonintegrable at the boundary. Israel J. Math. 115(2000), 195–203. Google Scholar

[2] [2] Jakóbczak, P., The exceptional sets for functions from the Bergman space. J. Port. Math. 50(1993), no. 1, 115–128. Google Scholar

[3] [3] Jakóbczak, P., The exceptional sets for holomorphic functions in Hartogs domains. Complex Variables Theory Appl. 32(1997), no. 1, 89–97. Google Scholar

[4] [4] Jakóbczak, P., Highly nonintegrable functions in the unit ball. Israel J. Math 97(1997), 175–181. Google Scholar

[5] [5] Jakóbczak, P., Exceptional sets of slices for functions from the Bergman space in the ball. Canad. Math. Bull. 44(2001), no. 2, 150–159. Google Scholar

[6] [6] Kot, P., Description of simple exceptional sets in the unit ball. Czechoslovak Math. J. 54(129)(2004), no. 1, 55–63. Google Scholar

[7] [7] Kot, P., Maximum sets of semicontinuous functions. Potential Anal. 23(2005), no. 4, 323–356. Google Scholar

[8] [8] Kot, P., Exceptional sets in Hartogs domains. Canad. Math. Bull 48(2005), no. 4, 580–586. Google Scholar

[9] [9] Kot, P., Exceptional sets in convex domains. J. Convex Anal. 12(2005), no. 2, 351–364. Google Scholar

[10] [10] Pommerenke, C., Boundary Behavior of Conformal Maps. Grundlehren der Mathematischen Wissenschaften 299, Springer-Verlag, Berlin, 1992. Google Scholar