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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.
We obtain a trace Hardy inequality for the Euclidean space with a bounded cut $Sigmasubsetmathbb R^d$, $d ge 2$. In this novel geometric setting, the Hardy-type inequality non-typically holds also for $d = 2$. The respective Hardy weight is given in
For a general subcritical second-order elliptic operator $P$ in a domain $Omega subset mathbb{R}^n$ (or noncompact manifold), we construct Hardy-weight $W$ which is optimal in the following sense. The operator $P - lambda W$ is subcritical in $Omega$
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
We prove asymptotic completeness in the energy space for the nonlinear Schrodinger equation posed on hyperbolic space in the radial case, in space dimension at least 4, and for any energy-subcritical, defocusing, power nonlinearity. The proof is base
We show improved local energy decay for the wave equation on asymptotically Euclidean manifolds in odd dimensions in the short range case. The precise decay rate depends on the decay of the metric towards the Euclidean metric. We also give estimates