Optimal Hardy weights on the Euclidean lattice

Abstract

We investigate the large-distance asymptotics of optimal Hardy weights on Z d \mathbb Z^d , d ≥ 3 d\geq 3 , via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar ( d − 2 ) 2 4 | x | − 2 \frac {(d-2)^2}{4}|x|^{-2} as | x | → ∞ |x|\to \infty . We prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients on Z d \mathbb Z^d : (1) averages over large sectors have inverse-square scaling, (2) for ergodic coefficients, there is a pointwise inverse-square upper bound on moments, and (3) for i.i.d. coefficients, there is a matching inverse-square lower bound on moments. The results imply | x | − 4 |x|^{-4} -scaling for Rellich weights on Z d \mathbb Z^d . Analogous results are also new in the continuum setting. The proofs leverage Green’s function estimates rooted in homogenization theory.