Résumé : We study the existence of non-trivial lower bounds for positive powers of the discrete Dirichlet Laplacian on the half line. Unlike in the continuous setting where both $-\Delta$ and $(-\Delta)^2$ admit a Hardy-type inequality, their discrete analogues exhibit a different behaviour. While the discrete Laplacian is subcritical, its square is critical and the threshold where the criticality of $(-\Delta)^\alpha$ first appears turns out to be $\alpha=3 / 2$. We provide corresponding (non-optimal) Hardy-type inequalities in the subcritical regime. Moreover, for the critical exponent $\alpha=2$, we employ a remainder factorisation strategy to derive a discrete Rellich inequality on a suitable subspace (with a weight improving upon the classical Rellich weight). Based on joint work with D. Krejčiřík and F. Štampach.

Keywords : Discrete Laplacian; criticality; Hardy inequality