Hardy inequalities on the positive integer lattice: a dimensional transition

Abstract

We determine exactly in which dimensions the sharp Hardy coefficient on the positive integer lattice agrees with that on the positive Euclidean cone, with Dirichlet conditions on every coordinate face. Equality holds in dimensions one and two, whereas the discrete coefficient is strictly smaller in every dimension at least three. An exact transformation identifies the lattice problem with a weighted Dirichlet problem on a cube. A tangential integration identity evaluates the defect of an explicit critical profile and proves the dimensional transition. The continuous coefficient remains the lower edge of the essential spectrum of the transformed operator; below that edge, the sharp discrete coefficient is a simple eigenvalue with a positive eigenfunction. In dimension three, an explicit supersolution places the coefficient strictly between twelve and forty-nine quarters. A representation on a product of three-dimensional spheres relates the spectral threshold to the local geometry of the transformed problem.