Optimal Hardy weights on the Euclidean lattice

Abstract

We investigate the large-distance asymptotics of optimal Hardy weights on ${\mathbb{Z}}^d$, $d\ge 3$, via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar $\frac{(d-2{)}^2}{4}|x{|}^{-2}$ as $|x|\to \infty$. We prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients on ${\mathbb{Z}}^d$: (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 $|x{|}^{-4}$-scaling for Rellich weights on ${\mathbb{Z}}^d$. Analogous results are also new in the continuum setting. The proofs leverage Green's function estimates rooted in homogenization theory.

arXiv comment28 pages. v2: result on asymptotic expansion of annealed Green's function has been split off as arXiv:2107.11583 [math.AP]

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