We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterise all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on a given interval, and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. As an application, we prove a new Rellich inequality.

中文翻译：

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关于线性二阶椭圆算子的最优 Hardy 权重族

我们使用一维归约为亚临界线性二阶椭圆算子构建最佳哈代权重族。更准确地说，我们首先在给定的区间上刻画关于一维亚临界 Sturm-Liouville 算子的所有最优 Hardy 权重，然后应用这个结果来获得更高维的一般线性二阶椭圆算子的最优 Hardy 不等式族. 作为应用，我们证明了一个新的 Rellich 不等式。