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Register a new userOn minimal decay at infinity of Hardy-weights
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We study the behaviour of Hardy-weights for a class of variational quasi-linear elliptic operators of $p$-Laplacian type. In particular, we obtain necessary sharp decay conditions at infinity on the Hardy-weights in terms of their integrability with respect to certain integral weights. Some of the results are extended also to nonsymmetric linear elliptic operators. Applications to various examples are discussed as well.
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We investigate the large-distance asymptotics of optimal Hardy weights on $mathbb Z^d$, $dgeq 3$, 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 $|x|toinfty$. 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 Greens function estimates rooted in homogenization theory.
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