Optimal discrete Hardy-Rellich-Birman inequalities

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.