论文标题
Hardy和Rellich的不平等现象
Hardy and Rellich inequality on lattices
论文作者
论文摘要
在本文中，我们研究了离散的hardy和rellich不平等中尖锐常数的渐近行为，在晶格$ \ mathbb {z}^d $ as $ d \ rightarrow \ rightarrow \ infty $上的渐近行为。在此过程中，我们证明了对运营商的$δ^m $和$ \ nabla（δ^m）$的一些强壮型不平等，用于$ d $ dimensional torus上的非阴性整数$ m $。事实证明，离散Hardy和Rellich不平等的急剧常数分别为$ d $和$ d^2 $，为$ d \ rightarrow \ infty $。
In this paper, we study the asymptotic behaviour of the sharp constant in discrete Hardy and Rellich inequality on the lattice $\mathbb{Z}^d$ as $d \rightarrow \infty$. In the process, we proved some Hardy-type inequalities for the operators $Δ^m$ and $\nabla(Δ^m)$ for non-negative integers $m$ on a $d$ dimensional torus. It turns out that the sharp constant in discrete Hardy and Rellich inequality grows as $d$ and $d^2$ respectively as $ d \rightarrow \infty$.
