Inner functions in \(Q_K\) type spaces
type: Item
instance of: scholarly article
title: Inner functions in \(Q_K\) type spaces
zbMATH Open document ID: 1244.30074
DOI: 10.1155/2011/765101
author: Congli Yang
published in: Journal of Function Spaces and Applications
publication date: 2012-01-16T00:00:00Z
review text: Let \({\mathcal H}(\mathbb D)\) be the algebra of all analytic functions in the unit disc \(\mathbb D\) of the complex plane, let \(g(z,a) = - \log|\phi_a(z)|\) be the Green function of \(\mathbb D\), where \(\phi_a(z) : = (a - z)/(1 - \overline{a}z)\) denotes the automorphism of \(\mathbb D\) which interchanges the origin and the point \(a \in \mathbb D\), let \(K : [0,\infty) \to [0,\infty)\) be an arbitrary function, and let \(0 < p < \infty\) and \(-2 < q < \infty\). The space \(Q_K(p,q)\) consists of those \(f \in {\mathcal H}(\mathbb D) \) such that \[ \sup_{a \in \mathbb D}\int_{\mathbb D}|f'(z)|^p(1 - |z|^2)^qK(g(z,a))dA(z) < \infty, \tag{1} \] where \(dA\) is the element of the Lebesgue area on \(\mathbb D\). The space \(Q_K(p,q)\) was introduced by \textit{H. Wulan} and \textit{J. Zhou} [J. Funct. Spaces Appl. 4, No. 1, 73--84 (2006; Zbl 1098.30027)] and it is strongly motivated by the \(Q_p\) theory and the studies on the \(Q_K\) and \(F(p,q,s)\) spaces which are different natural generalizations of \(Q_p\). The definition of \(Q_K(p,q)\), however, generalizes the concept of both of these spaces. Indeed, \(Q_K(2,0) = Q_K\) and \(Q_K(p,q) = F(p,q,s)\) when \(K(t) = t^s\). In the paper under review the inner functions in \(Q_K(p,q)\) and \(Q_{K,0}(p,q)\) are studied. \(Q_{K,0}(p,q)\) denotes the subspace of \(Q_{K}(p,q)\) that consist of those functions \(f \in {\mathcal H}(\mathbb D)\) for which the integral expression in (1) tends to zero as \(a\) goes to the unit circle \(\mathbb T\). For certain values of \(p\) and \(q\), and assuming that the weight \(K\) satisfies some conditions, Theorem 2 gives a characterization for singular inner functions \(S\) belonging to \(Q_K(p,q)\). The proof, given in Section 3, distinguishes between atomic cases and the non-atomic case. Another main result of the paper on inner functions is Theorem 6 which gives a complete characterization of inner function in \(Q_K(p,p-2)\). Specifically, provided \(p \geq 1\) and that th…
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 30J05
Mathematics Subject Classification ID: 30J10
Mathematics Subject Classification ID: 30H10
Mathematics Subject Classification ID: 30B10
zbMATH DE Number: 5997920
zbMATH Keywords: inner function
zbMATH Keywords: Blaschke product
zbMATH Keywords: \(F(p
zbMATH Keywords: q
zbMATH Keywords: s)\)
zbMATH Keywords: \(Q_K\) space
zbMATH Keywords: \(Q_K (p
zbMATH Keywords: q)\)
reviewed by: Fernando Pérez-González
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2011/765101
OpenAlex ID: W2149018625
description: scientific article; zbMATH DE number 5997920