Optimal Hardy weights on the Euclidean lattice

Keller, Matthias; Lemm, Marius

doi:10.1090/tran/8939

Optimal Hardy weights on the Euclidean lattice
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by Matthias Keller and Marius Lemm

Trans. Amer. Math. Soc. 376 (2023), 6033-6062

DOI: https://doi.org/10.1090/tran/8939

Published electronically: June 22, 2023

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Abstract:

We investigate the large-distance asymptotics of optimal Hardy weights on $\mathbb Z^d$, $d\geq 3$, 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.

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