Superposition operators on Dirichlet spaces
type: Item
instance of: scholarly article
title: Superposition operators on Dirichlet spaces
zbMATH Open document ID: 1067.31010
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2005-03-21T00:00:00Z
review text: Let \((\mathcal E, \mathcal D)\) be a strongly local, regular symmetric Dirichlet form. A function \(K\) is said to operate on \(\mathcal D\), if \(K\circ u \in \mathcal D\) for all \(u\in\mathcal D\). By the very definition of Dirichlet forms all normal contractions operate on \(\mathcal D\) and satisfy \(\mathcal E(K\circ u,K\circ u) \leq M^2\cdot\mathcal E(u,u)\); note that normal contractions are Lipschitz continuous. The aim of the paper is to show the converse of this assertion, namely, that every \(K\) that operates on \(\mathcal D\) is already (locally) Lipschitz continuous. In the context of Sobolev spaces \(W_p^1(U)\), \(1\leq p < \infty\), over a bounded domain \(U\subset\mathbb R^d\) this question has been answered in the affirmative [see e.g. \textit{J. Appell} and \textit{P. Zabreiko}, Nonlinear superposition operators. Cambridge Univ. Press (1990; Zbl 0701.47041) for a survey]. The Dirichlet form context refers to the particular case of \(\mathcal D = W^1_2\). The main theorem of this paper states that a Borel measurable \(K:\mathbb R \to \mathbb R\) such that \(K\circ u\in\mathcal D\) for all \(u\in \mathcal D\cap L^\infty\) and such that \(K(0)=0\) is locally Lipschitz continuous. If \(K\) operates on \(\mathcal D\), it is globally Lipschitz continuous. The proofs rely on the co-area formula for condenser potentials.
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 60J45
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 47H30
zbMATH DE Number: 2148489
zbMATH Keywords: Dirichlet space
zbMATH Keywords: local Dirichlet form
zbMATH Keywords: Lipschitz continuity
zbMATH Keywords: co-area formula
author: Patrick J. Fitzsimmons
reviewed by: R. L. Schilling
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113246670
OpenAlex ID: W1964962681
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DOI: 10.2748/TMJ/1113246670
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description: scientific article; zbMATH DE number 2148489