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

We obtain optimal Hardy inequalities for fractional powers of the $p$-Laplacian ${\mathrm{\Delta }}_p^{\sigma }$ on $\mathbb{Z}$, $p\in (1,\mathrm{\infty })$, $\sigma \in (0,1\mathrm{/}p)$. Asymptotically, the optimal weights behave like $\mathrm{\mid }x{\mathrm{\mid }}^{-p\sigma }$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$. We show a similar result for the Riesz fractional $p$-Laplacian. In an appendix, we show that for general $p$-Laplace operators on locally summable graphs, $p$-null-criticality implies $p$-optimality at infinity.