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

Ruzhansky, Michael; Сураган, Дурвудхан

doi:10.1016/j.aim.2017.07.020

Cited by 72 publications

(75 citation statements)

References 39 publications

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“…Hence we felt we had to restrict ourselves basically to Rellich and higher-order Hardy inequality references only and thus we refer, for instance, to [1], [2], [4], [5], [6, Ch. 6], [7], [8], [9], [11], [12], [17], [18], [19], [20], [22], [23], [24], [25], [26], [27], [34], [41], [46], [47], [49], [51], [54], [55], [57], [60], [61], [62], and the extensive literature cited therein. For the case of Hardy-type inequalities we only refer to the standard monographs such as, [6], [44], [45], and [53].…”

Section: )mentioning

confidence: 99%

Factorizations and Hardy–Rellich-type inequalities

Gesztesy

Littlejohn

2018

Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis

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Dedicated with great pleasure to Helge Holden on the occasion of his 60th birthday.Abstract. The principal aim of this note is to illustrate how factorizations of singular, even-order partial differential operators yield an elementary approach to classical inequalities of Hardy-Rellich-type. More precisly, introducing the two-parameter n-dimensional homogeneous scalar differential expressions T α,β := −∆ + α|x| −2 x · ∇ + β|x| −2 , α, β ∈ R, x ∈ R n \{0}, n ∈ N, n ≥ 2, and its formal adjoint, denoted by T + α,β , we show that nonnegativity of T + α,β T α,β on C ∞ 0 (R n \{0}) implies the fundamental inequality,

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Section: )mentioning

confidence: 99%

Factorizations and Hardy–Rellich-type inequalities

Gesztesy

Littlejohn

2018

Non-Linear Partial Differential Equations, Mathematical Physics, and Stochastic Analysis

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“…The flaw of the argument in [26] is the implicit assumption that, as in the Euclidean setting, the horizontal unit vector field ∇ H δ coincides with the generator of the dilations We also mention that in [25] a radial Hardy inequality with sharp constant n 2 is obtained for any homogenous norm · (and thus also for δ), by replacing the integrated quantity on the l.h.s. of (8) with | d d p u| 2 , which takes into account non-horizontal directions.…”

Section: Introductionmentioning

confidence: 99%

Hardy-Type Inequalities for the Carnot–Carathéodory Distance in the Heisenberg Group

Franceschi

Prandi

2020

J Geom Anal

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In this paper we study various Hardy inequalities in the Heisenberg group H n , w.r.t. the Carnot-Carathéodory distance δ from the origin. We firstly show that the optimal constant for the Hardy inequality is strictly smaller than n 2 = (Q − 2) 2 /4, where Q is the homogenous dimension. Then, we prove that, independently of n, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along ∇ H δ. This is in stark contrast with the Euclidean case, where the radial Hardy inequality is equivalent to the standard one, and has the same constant.Motivated by these results, we consider Hardy inequalities for non-radial directions, i.e., directions tangent to the Carnot-Carathéodory balls. In particular, we show that the associated constant is bounded on homogeneous cones CΣ with base Σ ⊂ S 2n , even when Σ degenerates to a point. This is a genuinely sub-Riemannian behavior, as such constant is well-known to explode for homogeneous cones in the Euclidean space.Date: April 2, 2019. 1 arXiv:1903.08486v2 [math.AP] 1 Apr 2019 d dλ λ . However, in the Heisenberg setting the latter is not horizontal. We fix this problem in Section 6, and show that this technique yields exactly the weighted Hardy inequality (5).

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“…[1] for stability of trace theorems, [4] for stability of Sobolev inequalities, etc. For more general Lie group discussions of above inequalities we refer to recent papers [9], [10] and [11] as well as references therein.…”

Section: Introductionmentioning

confidence: 99%

A note on stability of Hardy inequalities

Ruzhansky¹,

Сураган²

2018

Ann. Funct. Anal.

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In this note we formulate recent stability results for Hardy inequalities in the language of Folland and Stein's homogeneous groups. Consequently, we obtain remainder estimates for Rellich type inequalities on homogeneous groups. Main differences from the Euclidean results are that the obtained stability estimates hold for any homogeneous quasi-norm. Recently Sano and Takahashi obtained the improved versions of Hardy inequalities in their works [12], [13], [14] and [15]. The aim of this note is to formulate their results one of the largest classes of nilpotent Lie groups on R n , namely, homogeneous Lie groups since obtained results give new insights even for the Abelian groups in term of arbitrariness of homogeneous quasi-norm.

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

Cited by 72 publications

References 39 publications

Factorizations and Hardy–Rellich-type inequalities

Factorizations and Hardy–Rellich-type inequalities

Hardy-Type Inequalities for the Carnot–Carathéodory Distance in the Heisenberg Group

A note on stability of Hardy inequalities

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