Weighted inequalities for iterated Copson integral operators

TL;DR

This paper characterizes all weight functions for which a specific iterated Copson integral inequality holds, solving a long-standing open problem using a new discretization approach.

Contribution

It introduces a constructive approximation method with a novel discretization principle to solve the weighted inequality problem for iterated Copson operators.

Findings

Complete characterization of weight functions for the inequality.

Development of a new discretization principle for analysis.

Solution to a long-standing open problem in weighted inequalities.

Abstract

We solve a long-standing open problem in theory of weighted inequalities concerning iterated Copson operators. We use a constructive approximation method based on a new discretization principle that is developed here. In result, we characterize all weight functions $w,v,u$ on $(0,\infty)$ for which there exists a constant $C$ such that the inequality $$ \left(\int_0^{\infty}\left(\int_t^\infty \left(\int_s^{\infty}h(y)\,\text{d}y\right)^mu(s) \,\text{d}s\right)^{\frac{q}{m}}w(t)\,\text{d}t\right)^{\frac{1}{q}} \le C \left(\int_0^{\infty}h(t)^pv(t)\,\text{d}t\right)^{\frac{1}{p}} $$ holds for every non-negative measurable function $h$ on $(0,\infty)$, where $p,q$ and $m$ are positive parameters. We assume that $p\geq 1$ but otherwise $p,q$ and $m$ are unrestricted.