Optimal discrete Hardy-Rellich-Birman inequalities

TL;DR

This paper establishes optimal weights for discrete Hardy-Rellich-Birman inequalities on the half-line, improving known results and confirming conjectures for various powers of the discrete Laplacian.

Contribution

It provides explicit optimal weights for inequalities involving powers of the discrete Laplacian, including new proofs, improvements, and the resolution of a conjecture.

Findings

Derived explicit optimal weights for $ ext{Rellich}$ and $ ext{Birman}$ inequalities.

Improved upon previously known weights for $ ext{Rellich}$ inequalities.

Confirmed a conjecture for $ ext{higher powers}$ of the discrete Laplacian.

Abstract

We prove sufficient conditions on a parameter sequence to determine optimal weights in inequalities for an integer power $\ell$ of the discrete Laplacian on the half-line. By a concrete choice of the parameter sequence, we obtain explicit optimal discrete Rellich ($\ell=2$) and Birman ($\ell\geq3$) weights. For $\ell=1$, we rediscover the optimal Hardy weight of Keller-Pinchover-Pogorzelski. For $\ell=2$, we improve upon the best known Rellich weights due to Gerhat-Krej\v{c}i\v{r}\'{i}k-\v{S}tampach and Huang-Ye. For $\ell\geq3$, our main result proves a conjecture by Gerhat-Krej\v{c}i\v{r}\'{i}k-\v{S}tampach and improves the discrete analogue of the classical Birman weight due to Huang-Ye to the optimal.