Abstract

This paper is devoted to weighted Hardy type inequalities. Using the Bessel functions, we prove one-dimensional inequalities and give some remarks on extensions of the one-dimensional inequalities to \(n\)-dimensional convex domains with finite inner radius. Constants in those inequalities depend on the roots of parametric Lamb equation for the Bessel function and turn out to be sharp in some particular cases. We establish the inequalities in \(L^{p}\) spaces, with \(p\in[1,\infty)\). Also new \(L_{2}\)- inequality with sharp constants is obtained.

中文翻译：

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加权Hardy型不等式和参数Lamb方程

摘要

本文致力于加权Hardy型不等式。使用贝塞尔函数，我们证明了一维不等式，并对一维不等式扩展到具有有限内半径的\（n \）维凸域进行了一些说明。这些不等式中的常数取决于贝塞尔函数的参数Lamb方程的根，在某些特定情况下会变得很尖锐。我们用\（p \ in [1，\ infty）\）在\（L ^ {p} \）空间建立不等式。还获得新的\（L_ {2} \） -具有尖锐常数的不等式。