An optimal fractional Hardy inequality on the discrete half-line

In the context of Hardy inequalities for the fractional Laplacian $(-{\mathrm{\Delta }}_{\mathbb{N}}{)}^{\sigma }$ on the discrete half-line $\mathbb{N}$, we provide an optimal Hardy-weight $W_{\sigma }^{\mathrm{o}\mathrm{p}}$ for exponents $\sigma \in (0,1]$. As a consequence, we provide an estimate of the sharp constant in the fractional Hardy inequality with the classical Hardy-weight $n^{-2\sigma }$ on $\mathbb{N}$. It turns out that for $\sigma =1$ the Hardy-weight $W_1^{\mathrm{o}\mathrm{p}}$ is pointwise larger than the optimal Hardy-weight obtained by Keller–Pinchover–Pogorzelski near infinity. As an application of our main result, we obtain unique continuation results at infinity for the solutions of some fractional Schrödinger equation.