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

Abstract

We obtain optimal Hardy inequalities for fractional powers of the $p$-Laplacian ${\Delta }_p^{\sigma }$ on $\mathbb{Z}$, $p\in (1,\infty )$, $\sigma \in (0,1/p)$. Asymptotically, the optimal weights behave like $|x{|}^{-p\sigma }$ as $|x|\to \infty$. 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.

arXiv comment33 pages