On existence conditions for the Stieltjes integral

V. I. Matsaev ; M. Z. Solomyak

Sbornik. Mathematics, Tome 17 (1972) no. 4, pp. 515-527

Résumé

A modification of the definition of the Stieltjes integral $\int_0^1f\,dg$ is proposed, and it is shown that this integral exists if $g\in\operatorname{Lip}\alpha$, $f\in W_1^{1-\alpha}$, and $0\alpha1$ ($W_1^{1-\alpha}$ is the Sobolev–Slobodetskii class. It is shown that this integral defines a general form of a linear functional on $W_1^{1-\alpha}$ and on the class $\operatorname{Lip}_0\alpha$ of functions $g$ for which $g(x)-g(y)=o(|x-y|^\alpha)$. Applications to the integration of abstract functions and to the theory of double operator integrals are given. Bibliography: 8 titles.