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Distortion of boundary sets under inner functions and applications

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dc.affiliation.dptoUC3M. Departamento de Matemáticases
dc.affiliation.grupoinvUC3M. Grupo de Investigación: Análisis Aplicadoes
dc.contributor.authorFernández, José L.
dc.contributor.authorPestana, Domingo
dc.date.accessioned2010-01-20T13:33:22Z
dc.date.available2010-01-20T13:33:22Z
dc.date.issued1992
dc.description10 pages, no figures.-- MSC2000 codes: 30C85, 30D50.
dc.descriptionMR#: MR1183352 (93k:30014)
dc.descriptionZbl#: Zbl 0765.30011
dc.description.abstractAn 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].
dc.description.sponsorshipThe first author was supported in part by a grant from CICYT, Ministerio de Educación y Ciencia, Spain.
dc.description.statusPublicado
dc.format.mimetypeapplication/pdf
dc.identifier.bibliographicCitationIndiana University Mathematics Journal, 1992, vol. 41, n. 2, p. 439-448
dc.identifier.doi10.1512/iumj.1992.41.41025
dc.identifier.issn0022-2518
dc.identifier.urihttp://hdl.handle.net/10016/6557
dc.language.isoeng
dc.publisherIndiana University, Department of Mathematics
dc.relation.publisherversionhttp://www.iumj.indiana.edu/
dc.relation.publisherversionhttp://dx.doi.org/10.1512/iumj.1992.41.41025
dc.rights© Indiana University Mathematics Journal
dc.rights.accessRightsopen access
dc.subject.ecienciaMatemáticas
dc.subject.otherInner function
dc.subject.otherBorel subset
dc.subject.otherHausdorff dimension
dc.titleDistortion of boundary sets under inner functions and applications
dc.typeresearch article*
dc.type.reviewPeerReviewed
dspace.entity.typePublication
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