The two-weight Hardy inequality: a new elementary and universal proof

TL;DR

This paper presents a new, elementary proof of the known criteria for the two-weight Hardy inequality, emphasizing simplicity and broad applicability compared to previous methods.

Contribution

The paper introduces a novel, elementary, and versatile proof technique for the two-weight Hardy inequality criteria, simplifying understanding and potential applications.

Findings

Elementary proof of Hardy inequality criteria

Versatile approach applicable to various weight functions

Simplifies previous complex proofs

Abstract

We give a~new proof of the known criteria for the inequality \begin{equation*} \left(\int_{0}^{\infty}\left(\int_{0}^{t}f\right)^{q}w(t)\,dt\right)^{\frac{1}{q}} \leq C \left(\int_{0}^{\infty}f^{p}v\right)^{\frac{1}{p}}. \end{equation*} The innovation is in the elementary nature of the proof and its versatility.