Abstract This paper is devoted to Hardy type inequalities with remainders for compactly supported smooth functions on open sets in the Euclidean space. We establish new inequalities with weight functions depending on the distance function to the boundary of the domain. One-dimensional L 1 and L p inequalities and their multidimensional analogues are proved. We consider spatial inequalities in open convex domains with the finite inner radius. Constants in these inequalities depend on the roots of parametric Lamb equation for the Bessel function and turn out to be sharp in some particular cases.

中文翻译：

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Hardy 型不等式和参数 Lamb 方程

摘要 本文致力于欧几里得空间中开集上紧支撑光滑函数的带余数的哈代型不等式。我们根据到域边界的距离函数建立具有权重函数的新不等式。证明了一维L 1 和L p 不等式及其多维类似物。我们考虑具有有限内半径的开放凸域中的空间不等式。这些不等式中的常数取决于 Bessel 函数的参数 Lamb 方程的根，并且在某些特定情况下变得尖锐。