Hardy-type inequalities on a half-space in the Heisenberg group

Liu, Hengxing; Li, Jingwen

doi:10.1186/1029-242x-2013-291

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…Let τ p −1 (q) = p −1 q, q ∈ H n , be the left translation w.r.t. the Heisenberg group law (18). By left-invariance of the sub-Riemannian structure on H n , it holds that for any straight line γ : R → H n contained in D p and such that γ(0) = p the curve η = τ p −1 • γ is such that η(0) = 0 and η(t) ∈ D 0 = {z = 0} for any t ∈ R. In particular, m(p) coincides with the maximal length of such η intersected with τ p −1 (C Σ ).…”

Section: Full Hardy Constant On Homogeneous Conesmentioning

confidence: 99%

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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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Section: Full Hardy Constant On Homogeneous Conesmentioning

confidence: 99%

“…On the other hand, for the center Z = {x = y = 0} = C {(0,0,1)} we have α {(0,0,1)} = 0 and ρ {(0,0,1)} = 2π. For Hardy inequalities in the half space we refer to [18,19] and in more general convex domains we refer to [16,23].…”

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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“…Moreover, the sharp constants can be attained if they are strictly less than

\left(\frac{N}{2}\right)^{2}

. The interested reader is referred to, for instance, [2, 5–7, 11, 29, 32, 39, 45, 47] for more information and results on the Hardy type inequalities with boundary singularities.…”

Section: Introductionmentioning

confidence: 99%

Some Hardy identities on half‐spaces

Duy

Lam

Phi

2021

Mathematische Nachrichten

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We prove some Hardy identities on the half‐space double-struckR+N$\mathbb {R}_{+}^{N}$. Our equalities imply correponding versions of the Hardy type inequalities with exact remainder terms on double-struckR+N$\mathbb {R}_{+}^{N}$. These equalities give straightforward understandings of the optimal constants as well as the nonexistence of nontrivial optimizers for various Hardy type inequalities on half‐spaces. These identities also provide the “virtual” ground state in the sense of Frank and Seiringer [13] for several Hardy type inequalities on double-struckR+N$\mathbb {R}_{+}^{N}$.

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Picone’s identity for biharmonic operators on Heisenberg group and its applications

Dwivedi

Tyagi

2016

Nonlinear Differ. Equ. Appl.

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In this article we prove the nonlinear analogue of Picone's identity for p−biharmonic operator. As an application of our result we show that the Morse index of the zero solution to a p−biharmonic boundary value problem is 0. We also prove a Hardy type inequality and Sturmian comparison principle. We also show the strict monotonicity of the principle eigenvalue and linear relationship between the solutions of a system of singular p-biharmonic system.

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Hardy-type inequalities on a half-space in the Heisenberg group

Abstract: We prove some Hardy-type inequalities on half-spaces for Kohn's sub-Laplacian in the Heisenberg group. Furthermore, the constants we obtained are sharp.

Cited by 4 publications

References 8 publications

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

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

Some Hardy identities on half‐spaces

Picone’s identity for biharmonic operators on Heisenberg group and its applications

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