Sequences of composition operators in spaces of functions of bounded \(\Phi\)-variation
The spaces of functions of bounded variation were studied from diverse viewpoints. Here, the authors study conditions on a sequence of functions \(h_n : \langle c,d\rangle \to \langle a,b\rangle\) \((h_n : \mathbb R \to \mathbb R)\), \(n = 1, 2, \dots,\) under which the sequences of operators of right composition \(f\to f\circ h_n\) (of operators of left composition \(f\to h_n \circ f\), respectively) on the space \( BV_{\Phi} \langle a,b\rangle \) is pointwise bounded with respect to \({\Psi}\)-variation (i.e, for any function \(f\) in the class \( BV_{\Phi} \langle a,b\rangle \), the sequence of \({\Psi}\)-variations \(\{V_{\Psi} (\langle c,d \rangle : f\circ h_n)\}_{n= 1}^{\infty }\), \(\{V_{\Psi}(\langle a,b \rangle : h_n \circ f )\}_{n = 1}^{\infty }\) is bounded. The author gives a brief history of the problem.
scientific article; zbMATH DE number 5628161
scientific article