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 $p_1,p_2,q_1,q_2\in (0,\infty )$, we find necessary and sufficient conditions on nonnegative measurable functions $u_1,u_2,v_1,v_2$ on $(0,\infty )$ for which there exists a positive constant $c$ such that the inequality \begin{align*} &\bigg(\int_0^{\infty} \bigg(\int_0^t f(s)^{p_2} v_2(s)^{p_2} ds \bigg)^{\frac{q_2}{p_2}} u_2(t)^{q_2} dt \bigg)^{\frac{1}{q_2}} \notag \\ & \hspace{3cm} \leq c \bigg(\int_0^{\infty} \bigg(\int_t^{\infty} f(s)^{p_1} v_1(s)^{p_1} ds \bigg)^{\frac{q_1}{p_1}} u_1(t)^{q_1} dt \bigg)^{\frac{1}{q_1}} \end{align*} holds for every non-negative measurable function $f$ on $(0,\infty )$. 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 ${\mathrm{Cop}}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro spaces ${\mathrm{Ces}}_{p_2,q_2}(u_2,v_2)$, and also between spaces $S^q(w)$ equipped with the norm $\Vert f{\Vert }_{S^q(w)}=({\int }_0^{\infty }[f^{\ast \ast }(t)-f^{\ast }(t){]}^{q}w(t)dt{)}^{1/q}$ and classical Lorentz spaces of type $\Lambda$.