Weighted Inequalities for Discrete Iterated Hardy Operators

Abstract

We characterize a three-weight inequality for an iterated discrete Hardy-type operator. In the case when the domain space is a weighted space \(\ell ^{p}\) with \(p\in (0,1]\), we develop characterizations which enable us to reduce the problem to another one with \(p=1\). This, in turn, makes it possible to establish an equivalence of the weighted discrete inequality to an appropriate inequality for iterated Hardy-type operators acting on measurable functions defined on \({\mathbb {R}}\), for all cases of involved positive exponents.