The operator of composition in Slobodeckij spaces

Abstract

The so-called Riesz class Ap = Ap(a, b) was introduced by Riesz in [5] in the following way: A function u defined in the not necessarily bounded open interval (a, b), belongs to the class Ap with 1 < p < ∞ if and only if u is absolutely continuous in the interval (a, b) and its derivative u′ belongs to the space Lp(a, b). In the same paper, the following characterization of the class Ap was proved: A function u defined in the interval (a, b) belongs to the class Ap if and only if there exists a constant K > 0 such that for any system {(ai, bi) ⊂ (a, b)} of pairwise disjoint bounded intervals we have