Frostman shifts of inner functions
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 \({\mathcal M}\). As the main goal of the article, the authors present some other nontrivial examples of singular inner functions in \({\mathcal M}\). The following theorem is a base for the construction of some examples. Theorem 1. Let \(I\) be an inner function. Then the following assertions are equivalent. (a) \(I\in{\mathcal M}\). (b) For every \(0<\varepsilon<1\), there exists \(c=c(\varepsilon,I)<1\) such that the set \(\{z\in{\mathbb D}:\,\varepsilon<| I(z)| <1-\varepsilon\}\) does not contain any pseudohyperbolic disc of radius bigger than \(c\). (c) For every \(0<\varepsilon<1\), there exist \(\eta\in(0,1)\) and \(\delta\in(0,1)\) such that for any \(z\in{\mathbb D}\) with \(\varepsilon<| I(z)| <1-\varepsilon\), one can find \(w\in{\mathbb D}\) with \(\rho(z,w)<\delta\) such that \((1-| w| ^2) | I'(w)| >\eta\). (d) For any trivial point \(m\in M(H^\infty)\), that is \(P(m)=\{m\}\), one has either \(| I(m)| =1\) or \(I(m)=0\). (e) For any \(m\in G\), either \(I\circ L_m\) is an inner function or \(I\circ L_m\equiv 0\). Here \(L_m\) is the Hoffman map corresponding to \(m\). Several other interesting results are also obtained in the article. In particular, some examples of continuous singular inner functions in \({\mathcal M}\) are given. It is proven that any Blaschke product whose zero set lies in a Stolz angle belongs to the class \({\mathcal M}\). At the end of the article, some interesting observations and questions are presented. The bibliography contains several useful links to the results obtained in this area. The article is highly recommended to the specialists in complex analysis, especially in the theory of bounded analytic functions that are described well in [\textit{J. B. Garnett}, Bounded analytic functions. (Pure and Applied Mathematics, 96. New York etc.: Academic Press) (1981; Zbl 0469.30024)].
scientific article; zbMATH DE number 2112869
scientific article