Weighted inequalities involving Hardy and Copson operators

Abstract

We characterize a four-weight inequality involving the Hardy operator and the Copson operator. More precisely, given p1,p2,q1,q2∈(0,∞), we find necessary and sufficient conditions on non-negative measurable functions u1,u2,v1,v2 on (0,∞) for which there exists a positive constant c such that the inequality(∫0∞(∫0tf(s)p2v2(s)p2ds)q2p2u2(t)q2dt)1q2≤c(∫0∞(∫t∞f(s)p1v1(s)p1ds)q1p1u1(t)q1dt)1q1 holds for every non-negative measurable function f on (0,∞). The proof is based on discretizing and antidiscretizing techniques. The principal innovation consists in development of a new method which carefully avoids duality techniques and therefore enables us to obtain the characterization in previously unavailable situations, solving thereby a long-standing open problem. We then apply the characterization of the inequality to the establishing of criteria for embeddings between weighted Copson spaces Copp1,q1(u1,v1) and weighted Cesàro spaces Cesp2,q2(u2,v2), and also between spaces Sq(w) equipped with the norm ‖f‖Sq(w)=(∫0∞[f⁎⁎(t)−f⁎(t)]qw(t)dt)1/q and classical Lorentz spaces of type Λ.