Another approach to weighted inequalities for a superposition of Copson and Hardy operators

TL;DR

This paper develops new weighted inequalities for a superposition of Copson and Hardy operators using reduction and discretization methods, providing a comprehensive solution to a complex integral inequality involving multiple weight functions.

Contribution

It introduces a novel approach combining reduction techniques and discretization to solve weighted inequalities for superpositions of Copson and Hardy operators.

Findings

Derived necessary and sufficient conditions for the inequality to hold.

Extended the theory of weighted inequalities for integral operators.

Provided a framework applicable to various weight functions and parameters.

Abstract

In this paper, we present a solution to the inequality $$ \bigg( \int_0^{\infty} \bigg( \int_x^{\infty} \bigg( \int_0^t h \bigg)^q w(t)\,dt \bigg)^{r / q} u(x)\,ds \bigg)^{1/r}\leq C \, \bigg( \int_0^{\infty} h^p v \bigg)^{1 / p}, \quad h \in {\mathfrak M}^+(0,\infty), $$ using a combination of reduction techniques and discretization. Here $1 \le p < \infty$, $0 < q ,\, r < \infty$ and $u,\,v,\,w$ are weight functions on $(0,\infty)$.