The Persson--Stepanov theorem revisited

TL;DR

This paper presents a new, elementary proof of the Persson--Stepanov theorem on Hardy inequalities with weights, extending its applicability to the full parameter range including the critical case p=1.

Contribution

The authors develop a novel proof technique that avoids duality and discretization, allowing the theorem's extension to the full parameter spectrum.

Findings

Extended the theorem to include the critical case p=1

Provided a proof that avoids duality and discretization

Enhanced understanding of weighted Hardy inequalities

Abstract

We develop a new proof of the result of L.-E.~Persson and V.D.~Stepanov \cite[Theorems 1 and 3]{Per:02}, which provides a characterization of a Hardy integral inequality involving two weights, and which can be applied to an effective treatment of the geometric mean operator. Our approach enables us to extend their result to the full range of parameters, in particular involving the critical case $p=1$, which was excluded in the original work. Our proof avoids all duality steps and discretization techniques and uses solely elementary means.