Superposition operators on Dirichlet spaces
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.
scientific article; zbMATH DE number 2148489
scientific article