Weighted Hardy and Rellich inequality on Carnot groups

Jin, Yongyang; Shen, Shoufeng

doi:10.1007/s00013-011-0220-y

Cited by 19 publications

(13 citation statements)

References 32 publications

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“…In the case of the Heisenberg group (1.2) was proved for p = 2 by Garofalo and Lanconelli [GL90], see also D'Ambrosio [D'A05], and its extension to p = 2 was obtained by Niu, Zhang and Wang [NZW01]. Further extensions appeared by Danielli, Garofalo and Phuc [DGP11] on groups of Heisenberg type, on polarisable groups by Goldstein and Kombe [GK08], and on Carnot groups by Jin and Shen [JS11] and Lian [Lia13], together with certain weighted versions.…”

Section: Introductionmentioning

confidence: 97%

Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

Ruzhansky

Сураган

2017

Advances in Mathematics

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We give sharp remainder terms of Lp and weighted Hardy and Rellich inequalities on one of most general subclasses of nilpotent Lie groups, namely the class of homogeneous groups. As consequences, we obtain analogues of the generalised classical Hardy and Rellich inequalities and the uncertainty principle on homogeneous groups. We also prove higher order inequalities of Hardy–Rellich type, all with sharp constants. A number of identities are derived including weighted and higher order types

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Section: Introductionmentioning

confidence: 97%

Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

Ruzhansky

Сураган

2017

Advances in Mathematics

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“…A global scaling invariant version of (1.3) which we do not mention here was proved in [33]. Recently, there have been enormous works to generalize the Hardy inequality (1.1) to many different settings such as the fractional Hardy inequalities [23-25, 30, 45, 54], the Hardy inequalities on Heisenberg groups [15,17,27,47], on polarizable groups [28], on Carnot groups [34,39], on stratified groups [14,48] and on more general homogeneous groups [49][50][51], etc. The Hardy inequalities on Riemannian manifold (M, g) was initially studied by Carron [13] in the weighted L 2 -form under some geometric assumption on the weighted function ρ.…”

Section: Introductionmentioning

confidence: 99%

New sharp Hardy and Rellich type inequalities on Cartan–Hadamard manifolds and their improvements

Nguyen¹

2019

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) for radial derivations (i.e., the derivation along the geodesic curve) on Cartan-Hadamard manifolds. By Gauss lemma, our new Hardy inequality are stronger than the classical one. We also established the improvements of these inequalities in terms of sectional curvature of underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequality in hyperbolic space H n . Especially, we show that our new Rellich inequality is indeed stronger the classical one in hyperbolic space H n . for any n ≥ 2 (see, e.g., [14]). The constant (n − 1) n /n n is the best constant in (1.2). It also was shown in [14] that (1.2) is equivalent to the critical case of the Sobolev-Lorentz inequality. It is remarked that (1.2) is not invariant under the scaling as (1.1). A scaling invariant version of (1.2) (nowaday called the critical Hardy inequality) was recently established by Ioku and Ishiwata [25],Again, the constant (n − 1) n /n n is sharp. The inequality (1.3) in bounded domains was also discussed in [25]. It is surprise that the critical Hardy inequality (1.3) is equivalent to the Hardy inequality (1.1) in larger dimension spaces (see [45]). We refer the readers to [26] for a global scaling invariant version of (1.3) which we do not mention here.Recently, there is an enourmous work to generalize the Hardy inequality to many different settings. For examples, the fractional Hardy inequality was established in [17-19, 38, 47] and references therein. The Hardy inequality also was proved on group structure, e.g., on Heisenberg groups [8,10,21,40], on polarisable groups [22], on Carnot groups [28,33], on stratified groups [7,41], and on more general homogeneous groups [42][43][44]. The Hardy inequality on Riemannian manifold (M, g) was studied by Carron [6] in the weighted L 2 −form under some geometric assumption on the weighted function ρ. More precisely, he proved the following inequalitywhere ρ is a nonnegative function on M such that |∇ g ρ| g = 1, ∆ g ρ ≥ γ/ρ, and dV g , ∇ g , ∆ g and |·| g denote the volume element, gradient, Laplace-Beltrami operator and the length of a vector field with respect to the Riemannian metric g on M respectively, and the set ρ −1 (0) is a compact set of zero capacity. Under the same hypotheses on the function ρ, Kombe andÖzaydin [29] extended the result of Carron to the case p = 2 and they presented an application to the hyperbolic space H n with ρ being the distance function from the origin point 0. We refer the reader to [9] and the references therein for more results in this direction. The sharp Hardy inequality on Cartan-Hadamard manifolds (M, g) was recently obtained by Yang, Su and Kong [48]

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“…This problem was investigated by many authors. See [1,6,9,10,13,14,15,20,21,22,23,27,28,36], to name just a few. See also the books [24,31] that are by now standard references on Hardy inequalities.…”

Section: Introductionmentioning

confidence: 99%

Hardy and Hardy–Rellich type inequalities with Bessel pairs

Lam¹

2018

Ann. Acad. Sci. Fenn. Math.

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Weighted Hardy and Rellich inequality on Carnot groups

Abstract: We prove some weighted Hardy and Rellich inequalities on general Carnot groups with weights associated to the norm constructed by Folland's fundamental solution of the Kohn sub-Laplacian. Mathematics Subject Classification (2000). 26D10, 22E30, 43A80.

Cited by 19 publications

References 32 publications

Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

Hardy and Rellich inequalities, identities, and sharp remainders on homogeneous groups

New sharp Hardy and Rellich type inequalities on Cartan–Hadamard manifolds and their improvements

Hardy and Hardy–Rellich type inequalities with Bessel pairs

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