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

$$\begin{aligned} \begin{aligned}&\bigg (\int _a^b \bigg (\int _a^t f(s)^{p_2} v_2(s)^{p_2} ds\bigg )^{\frac{q_2}{p_2}} u_2(t)^{q_2} dt \bigg )^{\frac{1}{q_2}}\\&\quad \le C \bigg (\int _a^b \bigg (\int _a^t f(s)^{p_1} v_1(s)^{p_1} ds\bigg )^{\frac{q_1}{p_1}} u_1(t)^{q_1} dt \bigg )^{\frac{1}{q_1}} \end{aligned} \end{aligned}$$

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.