On minimal decay at infinity of Hardy-weights

Kovařík, Hynek; Pinchover, Yehuda

doi:10.1142/s0219199719500469

Cited by 16 publications

(17 citation statements)

References 13 publications

(28 reference statements)

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“…Moreover, we prove that the existence of a minimal positive Green function, with an additional assumption in the case of p > n, implies the subcriticality of Q ′ p,A,V . Finally, we extend the results in [20] and answer the question: how large can Hardy-weights be?…”

Section: Introductionsupporting

confidence: 58%

“…Criticality theory has applications in a number of areas of analysis. For example in spectral theory of Schrödinger operators [33], variational inequalities (like Hardy, Rellich, and Hardy-Sobolev-Maz'ya type inequalities) [20,39], and stochastic processes [41]. Among the applications in PDE we mention results concerning the large time behavior of the heat kernel [30], Liouville-type theorems [36], the behavior of the minimal positive Green function [31,33], and the asymptotic behavior of positive solutions near an isolated singularity [11].…”

Section: Introductionmentioning

confidence: 99%

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Positive solutions of the $\mathcal{A}$-Laplace equation with a potential

Hou¹,

Pinchover²,

Rasila³

2021

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In this paper, we study positive solutions of the quasilinear elliptic equation, where n ≥ 2, 1 < p < ∞, the divergence of A is the well known A-Laplace operator considered in the influential book of Heinonen, Kilpeläinen, and Martio, and the potential V belongs to a certain local Morrey space. The main aim of the paper is to extend criticality theory to the operator Q ′ p,A,V . In particular, we prove an Agmon-Allegretto-Piepenbrink (AAP) type theorem, establish the uniqueness and simplicity of the principal eigenvalue of Q ′ p,A,V in a domain ω ⋐ Ω and various characterizations of criticality. Furthermore, we also study positive solutions of the equation Q ′ p,A,V [u] = 0 of minimal growth at infinity in Ω, the existence of minimal positive Green function, and the minimal decay at infinity of Hardy-weights.

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Section: Introductionsupporting

confidence: 58%

Section: Introductionmentioning

confidence: 99%

Positive solutions of the $\mathcal{A}$-Laplace equation with a potential

Hou¹,

Pinchover²,

Rasila³

2021

Preprint

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“…We proceed to find a family of optimal Hardy-weights in Ω greater (in a neighborhood of ∞) than W class , the classical optimal weight obtained in [19] which satisfies…”

Section: Improved Optimal Hardy-weights In the Nonsymmetric Casementioning

confidence: 99%

“…We note that the original definition of optimal Hardy-weights in [12] includes the requirement that W is optimal at infinity. Recently it was proved in [19,Corollary 3.4] that the latter property follows, in fact, from the null-criticality of P − W with respect to W .…”

Section: Introductionmentioning

confidence: 98%

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On families of optimal Hardy-weights for linear second-order elliptic operators

Pinchover

Versano

2020

Journal of Functional Analysis

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We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator using a one-dimensional reduction. More precisely, we first characterize all optimal Hardy-weights with respect to one-dimensional subcritical Sturm-Liouville operators on (a, b), ∞ ≤ a < b ≤ ∞, and then apply this result to obtain families of optimal Hardy inequalities for general linear second-order elliptic operators in higher dimensions. As an application, we prove a new Rellich inequality.2000 Mathematics Subject Classification. Primary 35B09; Secondary 35J08, 35J20, 47J20, 49J40.

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