Embedding theorems for star-invariant subspaces generated by smooth inner functions
type: Item
instance of: scholarly article
title: Embedding theorems for star-invariant subspaces generated by smooth inner functions
zbMATH Open document ID: 0911.30026
author: Konstantin M. Dyakonov
published in: Journal of Functional Analysis
publication date: 1999-04-27T00:00:00Z
review text: Let \(\mathbb D\) denote the unit disc in the complex plane, \(\mathbb T\) its boundary, and \(m\) the normalized arclength measure on \(\mathbb T\). Let \(H^p\) \((0<p\leq \infty) \) stand for the usual Hardy space of holomorphic functions on \(\mathbb D\). One also treat \(H^p\) as a subspace of \(L^p(\mathbb T,m)\). Suppose \(\theta\) is an inner function (i.e., \(\theta\) is a bounded holomorphic function with \(| \theta| =1\) a.e. on \(\mathbb T\). For \(1\leq p\leq \infty\), set \(K_\theta^p :=H^p\cap \theta \bar z \bar{H^p}\): a star-invariant subspace of \(H^p\). For \(0<\alpha<1\), one says that an inner function is \(\alpha\)-regular if \(\theta' \in N\): the Nevanlinna class in \(\mathbb D\) and if there is \(C>0\) such that \(\int_{\mathbb T} | \theta'(\xi)| ^\alpha| \xi-z| ^{-1}dm(\xi)\leq\) \(C/(1-| z|)^\alpha\). The author treats embedding theorems of the form \(K_\theta^p \subset L^q(\mu)\), where \(\mu\) is a measure on \(\mathbb D\). In particular, assuming that \(\theta\) is \(\alpha\)-regular, he characterizes the \(\mu\)'s such that every \(f\in K_\theta^1\) satisfies \(\big(\int_{\mathbb D}| f| ^qd\mu\big)^{1/q}\leq\) \(C\int_{\mathbb T}| f| | \theta'| ^\alpha| dz| \).
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 1211645
zbMATH Keywords: inner function
zbMATH Keywords: invariant subspace
zbMATH Keywords: Carleson measure
zbMATH Keywords: embedding theorem
zbMATH Keywords: Hardy spaces
reviewed by: Kôzô Yabuta
MaRDI profile type: Publication
full work available at URL: jfan.1998.3276
OpenAlex ID: W2039039265
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DOI: 10.1006/JFAN.1998.3276
description: scientific article; zbMATH DE number 1211645