Superposition operators and functions of bounded \(p\)-variation
An exhaustive standard reference on properties and applications of Nemytskii operators in Sobolev and related spaces is the well-known monograph by \textit{Th. Runst} and the third author [``Sobolev spaces of fractional order, Nemytskii operators and nonlinear partial differential equations''. Berlin: de Gruyter (1996; Zbl 0873.35001)]. On the other hand, apart from some degeneracy results, not much is known about Nemytskii operators in spaces of functions of bounded variation. In the present paper, the authors provide a thorough study of Nemytskii operators in the space \(\text{BV}^1_p(\mathbb R)\) of Lipschitz continuous functions whose first distributional derivatives belong to \(\text{BV}_p(\mathbb R)\) \((1\leq p\leq\infty)\). This is related to boundedness properties of Nemytskii operators in the Besov space \(B^s_{p,q}(\mathbb R)\) which are also considered in this paper.
homogeneous and inhomogeneous Besov spaces
scientific article; zbMATH DE number 5152580