An optimal fractional Hardy inequality on the discrete half-line

TL;DR

This paper establishes an optimal Hardy inequality for the fractional Laplacian on the discrete half-line, providing sharp constants and new Hardy-weights, with applications to fractional Schrödinger equations.

Contribution

It introduces an optimal Hardy-weight for fractional Laplacian on the discrete half-line, improving existing bounds and enabling new unique continuation results.

Findings

Derived an optimal Hardy-weight $W^{ ext{op}}_{\sigma}$ for $\sigma o (0,1]$.

Provided sharp constants for fractional Hardy inequalities on $ $.

Applied results to unique continuation for fractional Schrödinger equations.

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 $W^{\mathrm{op}}_{\sigma}$ for exponents $\sigma\in\left(0,1\right]$. 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 $W^{\mathrm{op}}_{1}$ 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 Schr\"odinger equation.