Analytic functions smooth up to the boundary
type: Item
instance of: scholarly article
title: Analytic functions smooth up to the boundary
zbMATH Open document ID: 0656.30029
published in: Lecture Notes in Mathematics
publication date: 1993-06-05T00:00:00Z
review text: The principal theme of the book is the inner-outer factorization in classes of functions analytic in the open unit disc \({\mathbb{D}}\) and ``smoothly'' extendable to its boundary \({\mathbb{T}}\). It is a detailed account of the impressive progress achieved by the author in the investigation of the multiplicative structure of the above classes. In what follows \(I_ f\) will denote the inner part of a function \(f\in H^ 1({\mathbb{D}})\) (the Hardy class). A set \(X\subset H^ 1({\mathbb{D}})\) is said to possess the F-property (F-p.) if \(I^{-1}\cdot f\in X\) whenever \(f\in X\) and I is an inner function dividing \(I_ f\). It is known that many natural classes X possess the F-p. (e.g. \(\{f\in H^ 1:\) \(f^{(n)}\in H^ p\}\) (n\(\geq 0\), \(p\in (1,+\infty))\), \(\{f\in H^ 1:\) \(f\in Lip_{\alpha}({\mathbb{D}})\}\) \((0<\alpha <1))\). One of the main results by the author presented in the first chapter states that for every continuity modulus \(\omega\) and \(n\in {\mathbb{Z}}_+\) the class \[ \{f\in H^ 1:\quad \forall z,\zeta \in {\mathbb{D}},\quad | f^{(n)}(z)- f^{(n)}(\zeta)| \leq C_ f\omega (| z-\zeta |)\} \] possesses the F-p.. Classes with ``varying boundary smoothness'' are also considered. Some new classes not possessing the F-p. are exhibited. The second chapter treats the problem of description of moduli \(| f| /{\mathbb{T}}\) for f's belonging to a ``smooth'' class \(X\subset H^ 1\). General results obtained by the author imply quite a few previously known and new facts and, in particular, the famous (though never published) Carleson-Jacobs theorem (f outer, \(| f| /{\mathbb{T}}\in C^{\alpha}({\mathbb{T}})\Rightarrow f| {\mathbb{D}}\in C^{\alpha /2}({\mathbb{D}}\cup {\mathbb{T}})\) for an arbitrary positive \(\alpha\) with the usual Zygmund-like interpretation of \(C^{\alpha}\) for an integer \(\alpha)\). The third chapter is devoted to the description of zero sets. It contains the following remarkable result. Let \(\omega\) be an arbitrary continuity mod…
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 193001
zbMATH Keywords: closed ideal
zbMATH Keywords: Nevanlinna class
zbMATH Keywords: inner function
author: N. A. Shirokov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0082810
OpenAlex ID: W1507513928
DOI: 10.1007/BFB0082810
description: scientific article; zbMATH DE number 193001