An optimal fractional Hardy inequality on the discrete half-line

Abstract In the context of Hardy inequalities for the fractional Laplacian $$(-\Delta _{\mathbb {N}})^{\sigma }$$ ( - Δ N ) σ on the discrete half-line $$\mathbb {N}$$ N , we provide an optimal Hardy-weight $$W^{\textrm{op}}_{\sigma }$$ W σ op for exponents $$\sigma \in \left( 0,1\right] $$ σ ∈ 0 , 1 . As a consequence, we provide an estimate of the sharp constant in the fractional Hardy inequality with the classical Hardy-weight $$n^{-2\sigma }$$ n - 2 σ on $$\mathbb {N}$$ N . It turns out that for $$\sigma =1$$ σ = 1 the Hardy-weight $$W^{\textrm{op}}_{1}$$ W 1 op 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.