SUBSTITUTION OPERATORS IN THE SPACES OF FUNCTIONS OF BOUNDED VARIATION BV²_α(I)

Abstract

Abstract. The space BV 2 α (I) of all the real functions defined on interval I = [a, b] ⊂ R, which are of bounded second α-variation (in the sense De la Vallé Poussin) on I forms a Banach space. In this space we define an operator of substitution H generated by a function h : I × R −→ R, and prove, in particular, that if H maps BV 2 α (I) into itself and is globally Lipschitz or uniformly continuous, then h is an affine function with respect to the second variable.