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arXiv · 2205.07221

Hardy and Rellich inequality on lattices

Shubham Gupta
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Abstract

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 $\Delta^m$ and $\nabla(\Delta^m)$ 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$.

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Shubham Gupta. 2022-05-15. Hardy and Rellich inequality on lattices. https://doi.org/10.1007/s00526-022-02407-0

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