Sequences of Composition Operators in Spaces of Functions of Bounded $\Phi$-Variation

O. E. Galkin

Matematičeskie zametki, Tome 85 (2009) no. 3, pp. 330-341

Résumé

The main results of the paper are contained in Theorems 1 and 2. Theorem 1 presents necessary and sufficient conditions for a sequence of functions $h_n\colon\langle c,d\rangle\to\langle a,b\rangle$, $n=1,2,\dots$, to have bounded sequences of $\Psi$-variations $\{V_\Psi(\langle c,d\rangle;f\circ h_n)\}_{n=1}^\infty$ evaluated for the compositions of an arbitrary function $f\colon\langle a,b\rangle\to\mathbb R$ with finite $\Phi$-variation and the functions $h_n$. In Theorem \ref{t2:u330}, the same is done for a sequence of functions $h_n\colon\mathbb R\to\mathbb R$, $n=1,2,\dots$, and the sequence of $\Psi$-variations $\{V_\Psi(\langle a,b\rangle;h_n\circ f)\}_{n=1}^\infty$.

Détail

Keywords: composition operator, $\varphi$-function, modulus of continuity, Lipschitz function, Hölder property.
Mots-clés : $\Phi$-variation