An optimal fractional Hardy inequality on the discrete half-line

Abstract

In the context of Hardy inequalities for the fractional Laplacian $(-{\Delta }_{\mathbb{N}}{)}^{\sigma }$ on the discrete half-line $\mathbb{N}$, we provide an optimal Hardy-weight $W_{\sigma }^{\mathrm{op}}$ for exponents $\sigma \in \left(0,1\right]$. As a consequence, we provide 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 $W_1^{\mathrm{op}}$ 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.

26 pages v2: We provide the sharp constant in the fractional Hardy inequality with the classical Hardy-weight $n^{-2\sigma }$

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• Hardy and Rellich inequality on lattices
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