Optimal fractional discrete Hardy inequalities on the half-line

Abstract

We consider a Toeplitz realisation of the fractional discrete Laplacian $(-\Delta {)}^{\alpha }$ on the half-line $\mathbb{N}$ as a compression of the full-line fractional discrete Laplacian to ${\ell }^2(\mathbb{N})$. For all $\alpha >0$, we prove that the fractional Hardy inequality $$(-\Delta {)}^{\alpha }\ge \frac{4^{\alpha }{\Gamma }^2(\alpha +1/2)}{\pi }\frac{\Gamma (2\cdot -1)}{\Gamma (2\cdot -1+2\alpha )}$$ holds on ${\ell }^2(\mathbb{N})$ and is optimal in a strong sense. In particular, we show that the inequality cannot be improved and equality is not attained by any nonzero element of ${\ell }^2(\mathbb{N})$. As a consequence, we deduce a fractional generalisation of the discrete Birman inequality.