Optimal Hardy weights for the fractional discrete Dirichlet Laplacian on the half-line

For positive powers of the discrete Dirichlet Laplacian $(-{\mathrm{\Delta }}_{\mathrm{D}}{)}^{\sigma }$, non-trivial Hardy weights exist if and only if $0<\sigma <3\mathrm{/}2$. For $0<\sigma \le 1$, an explicit optimal Hardy weight for $(-{\mathrm{\Delta }}_{\mathrm{D}}{)}^{\sigma }$ was obtained by Das and de la Fuente-Fernández. Their ground state argument does not apply when $1<\sigma <3\mathrm{/}2$, because the off-diagonal matrix entries of $(-{\mathrm{\Delta }}_{\mathrm{D}}{)}^{\sigma }$ are no longer all non-positive. Nevertheless, we show that their result is true for the entire subcriticality range $0<\sigma <3\mathrm{/}2$. For $1<\sigma <3\mathrm{/}2$, our argument combines the factorisation $$(-{\mathrm{\Delta }}_{\mathrm{D}}{)}^{\sigma }={\mathrm{\nabla }}_{\mathrm{D}}^{\ast }(-{\mathrm{\Delta }}_{\mathrm{N}}{)}^{\sigma -1}{\mathrm{\nabla }}_{\mathrm{D}},$$ where ${\mathrm{\nabla }}_{\mathrm{D}}$ and $-{\mathrm{\Delta }}_{\mathrm{N}}$ are the discrete Dirichlet gradient and the discrete Neumann Laplacian, respectively, with a suitable two-term ground state representation. In addition, we use these arguments to derive another explicit family of optimal Hardy weights for any $\sigma \in (0,3\mathrm{/}2)$.