An optimal fractional Hardy inequality on the discrete half-line

In the context of Hardy inequalities for the fractional Laplacian (-Δ N)σ ( - Δ N ) σ on the discrete half-line N N , we provide an optimal Hardy-weight Wᵒᵖσ W σ op for exponents σ ∈ ( 0,1] 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-2σ n - 2 σ on N N . It turns out that for σ =1 σ = 1 the Hardy-weight Wᵒᵖ₁ 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.