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Optimal Hardy Inequality for Fractional Laplacians on the Lattice

We study the fractional Hardy inequality on the integer lattice. We prove null-criticality of the Hardy weight and hence optimality of the constant. More specifically, we present a family of Hardy weights with respect to a parameter and show that below a certain threshold the Hardy weight is positive critical while above the threshold it is subcritical. In particular, the Hardy weight at the threshold is optimal in the sense that any larger weight would fail to be a Hardy weight and the Hardy inequality does not allow for a minimizer. A crucial ingredient in our proof is an asymptotic expansion of the fractional discrete Riesz kernel.