# Weighted inequalities for discrete iterated Hardy operators

## TL;DR

This paper characterizes a three-weight inequality for an iterated discrete Hardy operator, reducing complex cases to simpler ones and establishing equivalences with continuous Hardy inequalities across various exponents.

## Contribution

It provides new characterizations for weighted inequalities of iterated discrete Hardy operators, simplifying analysis by reducing to the case p=1 and linking to continuous Hardy inequalities.

## Key findings

- Reduced discrete inequality to the case p=1.
- Established equivalence with continuous Hardy inequalities.
- Provided comprehensive characterizations for weighted inequalities.

## Abstract

We characterize a three-weight inequality for an iterated discrete Hardy-type operator. In the case when the domain space is a weighted space $\ell^p$ with $p\in(0,1]$, we develop characterizations which enable us to reduce the problem to another one with $p=1$. This, in turn, makes it possible to establish an equivalence of the weighted discrete inequality to an appropriate inequality for iterated Hardy-type operators acting on measurable functions defined on $\mathbb{R}$, for all cases of involved positive exponents.