Optimal Hardy Inequalities for Fractional $p$-Laplacians on the Integers

We obtain optimal Hardy inequalities for fractional powers of the $p$-Laplacian $Δ_p^σ$ on $\mathbb{Z}$, $p\in (1,\infty)$, $σ\in (0, 1/p)$. Asymptotically, the optimal weights behave like $|x|^{-pσ}$ as $|x|\to \infty$. We show a similar result for the Riesz fractional $p$-Laplacian. In an append