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

Abstract

For positive powers of the discrete Dirichlet Laplacian $(-Δ_{\mathrm{D}})^σ$, non-trivial Hardy weights exist if and only if $0<σ<3/2$. For $0<σ\leq 1$, an explicit optimal Hardy weight for $(-Δ_{\mathrm{D}})^σ$ was obtained by Das and de la Fuente-Fernández. Their ground state argument does not apply when $1<σ<3/2$, because the off-diagonal matrix entries of $(-Δ_{\mathrm{D}})^σ$ are no longer all non-positive. Nevertheless, we show that their result is true for the entire subcriticality range $0<σ<3/2$. For $1<σ<3/2$, our argument combines the factorisation $$(-Δ_{\mathrm{D}})^σ=\nabla_{\mathrm{D}}^{*}(-Δ_{\mathrm{N}})^{σ-1}\nabla_{\mathrm{D}},$$ where $\nabla_{\mathrm{D}}$ and $-Δ_{\mathrm{N}}$ 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 $σ\in(0,3/2)$.