An optimal fractional Hardy inequality on the discrete half-line

Abstract In the context of Hardy inequalities for the fractional Laplacian (- ₍) ^ (- Δ N) σ on the discrete half-line N N, we provide an optimal Hardy-weight W^op W σ op for exponents (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^-2 n - 2 σ on N N. It turns out that for =1 σ = 1 the Hardy-weight W^op₁ 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.