Weighted inequalities involving two Hardy operators

TL;DR

This paper establishes necessary and sufficient conditions for weighted inequalities involving two Hardy operators, using a new discretization method to extend previous results.

Contribution

It introduces a novel discretization approach to characterize weighted inequalities for two Hardy operators, overcoming earlier limitations.

Findings

Derived explicit conditions for weights ensuring inequality holds.

Extended the class of admissible weights beyond previous results.

Provided a new proof technique using discretization.

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.