标题： 重新审视Persson--Stepanov定理 显示英文标题

标题： The Persson--Stepanov theorem revisited

摘要： 我们开发了L.-E.~Persson和V.D.~Stepanov的结论\cite[Theorems 1 and 3]{Per:02}的新证明，该结论提供了涉及两个权重的Hardy积分不等式的特征，并且可以用于对几何平均算子的有效处理。 我们的方法使我们能够将他们的结果扩展到完整的参数范围，特别是涉及原始工作中被排除的关键情况$p=1$。 我们的证明避免了所有对偶步骤和离散化技术，仅使用基本手段。 显示英文摘要

摘要： 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.