RT Journal Article
T1 Distortion of boundary sets under inner functions and applications
A1 Fernández, José L.
A1 Pestana, Domingo
AB An inner function is a bounded holomorphic function from the unit disc $\Delta$ of the complex plane such that the radial boundary values have modulus 1 a.e. . If $E$ is a Borel subset of $\partial\Delta$ we also define $f(E)=\{e\sp{i\theta}/\lim\sb{r\to 1} f(re\sp{i\theta})$ exists and belongs to $E\}$. Let $M\sb \alpha$, $\text{cap}\sb \alpha$ and dim denote respectively the $\alpha$-dimensional content, $\alpha$- dimensional capacity and the Hausdorff dimension. In relation to the available results the authors in this paper prove that if $f$ is inner, $f(0)=0$, and $E$ is a Borel subset of $\partial\Delta$ then $M\sb \alpha(f\sp{-1}(E)) \geq C\sb \alpha M\sb \alpha(E)$ and for $0\leq\alpha<1$, $\text{cap}\sb \alpha(f\sp{-1}(E)) \geq C\sb \alpha \text{cap}\sb \alpha(E)$. An immediate consequence of course is $\dim(f\sp{-1}(E))\geq \dim E$. They also give examples to show that the inequalities cannot be reversed [source: Zentralblatt MATH].
PB Indiana University, Department of Mathematics
SN 0022-2518
YR 1992
FD 1992
LK http://hdl.handle.net/10016/6557
UL http://hdl.handle.net/10016/6557
LA eng
NO 10 pages, no figures.-- MSC2000 codes: 30C85, 30D50.
NO MR#: MR1183352 (93k:30014)
NO Zbl#: Zbl 0765.30011
NO The first author was supported in part by a grant from CICYT, Ministerio de Educación y Ciencia, Spain.
DS e-Archivo
RD 27 abr. 2024