Weighted inequalities involving two Hardy operators

Abstract

We find necessary and sufficient conditions on weights $u_1,u_2,v_1,v_2$, i.e. measurable, positive, and finite, a.e. on $(a,b)$, for which there exists a positive constant $C$ such that for given $0<p_1,q_1,p_2,q_2<\infty$ the inequality \begin{equation*} \begin{split} \bigg(\int_a^b \bigg(\int_a^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}}& \\ & \hspace{-3cm}\le C \bigg(\int_a^b \bigg(\int_a^t 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{split} \end{equation*} holds for every non-negative, measurable function $f$ on $(a,b)$, where $0\le a<b\le \infty$. The proof is based on a recently developed discretization method that enables us to overcome the restrictions of the earlier results.