An Improved Discrete Hardy Inequality

For a given subcritical discrete Schrödinger operator H on a weighted infinite graph X, we construct a Hardy-weight w which is optimal in the following sense. The operator H − λw is subcritical in X for all λ < 1, null-critical in X for λ = 1, and supercritical near any neighborhood of infinity in X for any λ > 1. Our results rely on a criticality theory for Schrödinger operators on general weighted graphs.

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“…[14]). For all finitely supported functions ϕ : N 0 → R with ϕ(0) = 0 we have ∞ n=0 |ϕ(n) − ϕ(n + 1)| 2 ≥ ∞ n=1 w(n)|ϕ(n)| 2 (7.1)…”

mentioning

confidence: 99%

Optimal Hardy inequalities for Schrödinger operators on graphs

Keller

Pinchover

Pogorzelski

2018

Commun. Math. Phys.

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“…If q 0 satisfies some Hardy inequality, see e.g. the discussion in [10,11], or the graph has positive Cheeger constant, see e.g. [1], then also certain V without a fixed sign induce a nonnegative form q V .…”

Section: Setup and Main Resultsmentioning

confidence: 99%

A note on the surjectivity of operators on vector bundles over discrete spaces

Koberstein

Schmidt

2019

Arch. Math.

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“…arising from the supersolution construction of n is optimal and can be explicitly computed (confer [16]) to be…”

Section: The Function θ(T) = T αmentioning

confidence: 99%

“…Let us explicate all components of this inequality. First, the Hardy weight

w : N \to false(0, \infty false), w = \frac{Δ (n^{1 / 2})}{n^{1 / 2}}

arising from the supersolution construction of

n

is optimal and can be explicitly computed (confer [16]) to be

\begin{matrix} true \\ right w (k) = 2 - {(1 + \frac{1}{k})}^{1 / 2} - {(1 - \frac{1}{k})}^{1 / 2} = \sum_{l = 1}^{\infty} (\frac{0pt}{4 l}) \frac{1}{false(4 l - 1 false) 2^{4 l - 1}} \frac{1}{k^{2 l}} > \frac{1}{4 k^{2}} \end{matrix}

for

k ⩾ 2

, and

w false(1 false) = 2 - \sqrt{2} > 1 / 4

. Furthermore, Corollary 4.1 provides a constant for the Rellich‐type inequality by

\begin{matrix} true \\ right γ = {(\frac{1 - D^{- α / 2}}{1 - D^{- 1 / 2}})}^{2} = {(\frac{1 - 2^{- α / 2}}{1 - 2^{- 1 / 2}})}^{2} . \end{matrix}

It turns out that for any

0 < α < 1

, we have

0 < γ < 1

.…”

Section: The Function θFalse(tfalse)=tαmentioning

confidence: 99%

From Hardy to Rellich inequalities on graphs

Keller

Pinchover

Pogorzelski

2020

Proc. London Math. Soc.

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We show how to deduce Rellich inequalities from Hardy inequalities on infinite graphs. Specifically, the obtained Rellich inequality gives an upper bound on a function by the Laplacian of the function in terms of weighted norms. These weights involve the Hardy weight and a function which satisfies an eikonal inequality. The results are proven first for Laplacians and are extended to Schrödinger operators afterwards.

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An Improved Discrete Hardy Inequality

Abstract: We improve the classical discrete Hardy inequality

Cited by 30 publications

References 4 publications

Optimal Hardy inequalities for Schrödinger operators on graphs

Optimal Hardy inequalities for Schrödinger operators on graphs

A note on the surjectivity of operators on vector bundles over discrete spaces

From Hardy to Rellich inequalities on graphs

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