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We study a three-weight inequality for the superposition of the Hardyoperator and the Copson operator, namely \begin{equation*}  \bigg(\int_a^b \bigg(\int_t^b \bigg(\int_a^s f(\tau)^p v(\tau) \,d\tau\bigg)^\frac{q}{p} u(s) \,ds \bigg)^{\frac{r}{q}} w(t) \,dt\bigg)^{\frac{1}{r}} \leq C \int_a^b f(t)\,dt, \end{equation*} in which $(a,b)$is any nontrivial interval, $q,r$ are positive real parameters and $p\in(0,1]$.A simple change of variables can be used to obtain any weighted $L^p$-norm with$p\ge1$ on the right-hand side. Another simple change of variables can be usedto equivalently turn this inequality into the one in which the Hardy and Copsonoperators swap their positions. We focus on characterizing those triples ofweight functions $(u,v,w)$ for which this inequality holds for all nonnegativemeasurable functions $f$ with a constant independent of $f$.  We use a new type of approach based on an innovative method of discretizationwhich enables us to avoid duality techniques and therefore to remove variousrestrictions that appear in earlier work.  This paper is dedicated to Professor Stefan Samko on the occasion of his 80thbirthday.

Weighted inequalities for a superposition of the Copson operator and the Hardy operator