Weighted Inequalities for a Superposition of the Copson Operator and the Hardy Operator

Abstract

We study a three-weight inequality for the superposition of the Hardy operator 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 used to equivalently turn this inequality into the one in which the Hardy and Copson operators swap their positions. We focus on characterizing those triples of weight functions $(u,v,w)$ for which this inequality holds for all nonnegative measurable functions $f$ with a constant independent of $f$. We use a new type of approach based on an innovative method of discretization which enables us to avoid duality techniques and therefore to remove various restrictions that appear in earlier work. This paper is dedicated to Professor Stefan Samko on the occasion of his 80th birthday.