Differentiability properties of the autonomous composition operator in Sobolev spaces
type: Item
instance of: scholarly article
title: Differentiability properties of the autonomous composition operator in Sobolev spaces
zbMATH Open document ID: 0896.47051
DOI: 10.4171/ZAA/782
author: Massimo Lanza de Cristoforis
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1998-09-23T00:00:00Z
review text: The author studies the autonomous composition operator, which takes a pair of functions \((f,g)\) into its composite function \(f\circ g\). He assumes that \(f\) and \(g\) belong to Sobolev spaces defined on open subsets of \(\mathbb{R}^n\), and he concentrates on the case in which the space for \(g\) is a Banach algebra. He gives a sufficient condition in order that the composition maps bounded sets to bounded sets, and exploits the density of the polynomial functions in the space for \(f\) in order to prove that for suitable Sobolev exponents of the spaces for \(f\) and \(g\), the composition is continuous and differentiable with continuity up to order \(r\), with \(r\geq 1\). Then he shows the optimality of such conditions by means of theorems of `inverse' type.
Mathematics Subject Classification ID: 47H30
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 1068014
zbMATH Keywords: theorems of inverse type
zbMATH Keywords: autonomous composition operator
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: Sobolev exponents
reviewed by: Jürgen Appell
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/782
OpenAlex ID: W2048732057
description: scientific article; zbMATH DE number 1068014