Frostman shifts of inner functions
type: Item
instance of: scholarly article
title: Frostman shifts of inner functions
zbMATH Open document ID: 1064.30029
DOI: 10.1007/BF02787765
author: Raymond Mortini
author: Artur Nicolau
published in: Journal d'Analyse Mathématique
publication date: 2004-11-01T00:00:00Z
review text: Let \(M(H^\infty)\) denote the maximal ideal space of \(H^\infty\). The Gleason part of a point \(m\in M(H^\infty)\) is defined as \(P(m)=\{\widetilde m\in M(H^\infty):\rho(m, \widetilde m)<1\}\). Here \(\rho\) is a pseudohyperbolic distance: \[ \rho(m,\widetilde m)=\sup\{| f(m)| :\, f\in H^\infty, \| f\| _\infty\leq 1, f(\widetilde m)=0\} \] (we are identifying \(f\) with its Gelfand transform, that is, \(f(m)= m(f)\)). Due to the fundamental result of K. Hoffman, either \(P(m)\) is a singleton or \(P(m)\) is an analytic disc. Moreover, for every \(m\in M(H^\infty)\), there exists a continuous map \(L_m\) from \({\mathbb D}\) onto \(P(m)\) with \(L_m(0)=m\) such that \(f\circ L_m\in H^\infty\) whenever \(f\in H^\infty\). The Hoffman map \(L_m\) has the form \[ L_m(z)=\lim_\alpha{z+ z_\alpha\over{1+\overline{z_\alpha}z}}, \] where \(\{z_\alpha\}\) is any net of points in \({\mathbb D}\) converging to \(m\) in the weak-*-topology of \(M(H^\infty)\). The set of points in \(M(H^\infty)\) with nontrivial Gleason part is denoted by \(G\). A Blaschke product \(B\) with zero sequence \(\{z_n\}\) is called a Carleson-Newman Blaschke product if the measure \[ \nu=\sum_{n=1}^\infty(1-| z_n| ^2)\delta_{z_n} \] is a Carleson measure. The class \({\mathcal P}\) consists of all Carleson-Newman Blaschke products \(B\) for which \(\tau_\alpha\circ B\) is also a Carleson-Newman Blaschke product for any \(\alpha\in{\mathbb D}\). (Here \(\tau_\alpha\) denotes the automorphism of \({\mathbb D}\) sending \(\alpha\) to \(0\): \(\tau_\alpha(z)=(\alpha-z)/(1-\overline\alpha z)\).) The main object of investigation of the article is the class \({\mathcal M}\) of inner functions \(I\) for which \(\tau_\alpha\circ I\) is a Carleson-Newman Blaschke product for any \(\alpha\in{\mathbb D}\setminus\{0\}\). This class is more wide than \({\mathcal P}\). For example, the singular inner function \(S(z)=\exp((1+z)/(1-z))\), that corresponds to the Dirac measure at the point \(1\), belongs to \({\mathc…
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 46E30
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 2112869
zbMATH Keywords: inner function
zbMATH Keywords: singular inner function
zbMATH Keywords: Frostman shift
zbMATH Keywords: Carleson-Newman Blaschke product
zbMATH Keywords: Hoffman map
zbMATH Keywords: Carleson measure
zbMATH Keywords: porous set
reviewed by: Alexander Tovstolis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2112869