Weighted inequalities involving iteration of two Hardy integral operators

TL;DR

This paper provides a characterization of weighted inequalities involving the iteration of Hardy integral operators, introducing a new discretization method and analyzing the inequality in both discrete and continuous settings.

Contribution

It introduces a simpler discretization approach for characterizing weighted inequalities involving iterated Hardy operators, extending analysis to both discrete and continuous cases.

Findings

Established necessary and sufficient conditions for the inequality.

Developed a more straightforward discretization method.

Provided dual discrete and continuous characterizations.

Abstract

Let $1\leq p <\infty$ and $0 < q,r < \infty$. We characterize validity of the inequality for the composition of the Hardy operator, \begin{equation*} \bigg(\int_a^b \bigg(\int_a^x \bigg(\int_a^t f(s)ds \bigg)^q u(t) dt \bigg)^{\frac{r}{q}} w(x) dx \bigg)^{\frac{1}{r}} \leq C \bigg(\int_a^b f^p(x) v(x) dx \bigg)^{\frac{1}{p}} \end{equation*} for all non-negative measurable functions on $(a,b)$, $-\infty \leq a < b \leq \infty$. We construct a more straightforward discretization method than those previously presented in the literature, and we characterize this inequality in both discrete and continuous forms.