New characterization of weighted inequalities involving superposition of Hardy integral operators

TL;DR

This paper provides a new characterization of weighted inequalities involving superpositions of Hardy integral operators, introducing a simpler discretization method and new weight conditions in both discrete and continuous settings.

Contribution

It offers a novel, more straightforward discretization approach and extends weight characterizations for Hardy operator inequalities, unifying previous results as special cases.

Findings

Developed a simpler discretization method for Hardy inequalities.

Established new weight characterizations in discrete and continuous forms.

Unified previous inequality characterizations as special cases.

Abstract

Let $1\leq p <\infty$ and $0 < q,r < \infty$. We characterize the 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(x)^p v(x) dx \bigg)^{\frac{1}{p}} \end{equation*} for all non-negative measurable functions $f$ 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 provide some new scales of weight characterizations of this inequality in both discrete and continuous forms and we obtain previous characterizations as the special case of the parameter.