$q$-Poincar{é} inequalities on Carnot Groups with filiform type Lie algebra

Chatzakou, Marianna; Federico, Serena; Zegarliński, Bogusław

doi:10.48550/arxiv.2007.04689

Cited by 7 publications

(19 citation statements)

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“…More general versions of nonlocal Poincaré inequalities of this kind were proved in [9] in the setting of a topological measure space endowed with a family of sets which play the role of unit balls and satisfy suitable assumptions. Recently in [8] nonlocal L p -Poincaré inequalities were obtained on Carnot groups of Engel type in the case when the density of the measure depends on a homogeneous norm of the group; we note however that the case p " 2 is always excluded.…”

Section: Nonlocal Poincaré Inequalitymentioning

confidence: 83%

Local and nonlocal Poincaré inequalities on Lie groups

Bruno¹,

Peloso²,

Vallarino³

2021

Preprint

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We prove a local L p -Poincaré inequality, 1 ď p ă 8, on noncompact Lie groups endowed with a sub-Riemannian structure. We show that the constant involved grows at most exponentially with respect to the radius of the ball, and that if the group is nondoubling, then its growth is indeed, in general, exponential. We also prove a nonlocal L 2 -Poincaré inequality with respect to suitable finite measures on the group.

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Section: Nonlocal Poincaré Inequalitymentioning

confidence: 83%

Local and nonlocal Poincaré inequalities on Lie groups

Bruno¹,

Peloso²,

Vallarino³

2021

Preprint

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“…[AS94]. We can also refer to the recent work on the Poincaré inequality on filiform type stratified groups [CFZ21].…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Sobolev inequalities on Lie groups

Chatzakou¹,

Kassymov²,

Ruzhansky³

2021

Preprint

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In this paper we show a number of logarithmic inequalities on several classes of Lie groups: log-Sobolev inequalities on general Lie groups, log-Sobolev (weighted and unweighted), log-Gagliardo-Nirenberg and log-Caffarelli-Kohn-Nirenberg inequalities on graded Lie groups. Furthermore, on stratified groups, we show that one of the obtained inequalities is equivalent to a Grosstype log-Sobolev inequality with the horizontal gradient. As a result, we obtain the Gross log-Sobolev inequality on general stratified groups but, very interestingly, with the Gaussian measure on the first stratum of the group. Moreover, our methods also yield weighted versions of the Gross log-Sobolev inequality. In particular, we also obtain new weighted Gross-type log-Sobolev inequalities on R n for arbitrary choices of homogeneous quasi-norms. As another consequence we derive the Nash inequalities on graded groups and an example application to the decay rate for the heat equations for sub-Laplacians on stratified groups. We also obtain weighted versions of log-Sobolev and Nash inequalities for general Lie groups.

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“…When V (x) diverges to infinity with a distance d(x, 0) → ∞, such bounds have a multitude applications to prove coercive inequalities, see e.g. [31], [2], [19], [21], [22], [32], [15]. In particular, what we are after, is an interesting and very challenging problem how to get coercive inequalities for probability measures on Carnot group.…”

Section: Introduction: Coercive Inequalities Problem On Nilpotent Lie...mentioning

confidence: 99%

“…We know that, if we switch to smooth homogeneous norms, generally we cannot get Log-Sobolev inequality, [19] . But there are examples in the literature of Poincaré and weaker log β inequalities in such setup, see [21], [22] for Heisenberg group, [12], [13], for type -2 and [15] for filiform type groups. Later in his paper we provide some general constructions which work for more extensive classes of Carnot groups.…”

Section: Introduction: Coercive Inequalities Problem On Nilpotent Lie...mentioning

confidence: 99%

Coercive Inequalities on Carnot Groups: Taming Singularities

Dagher¹,

Zegarliński²

2021

Preprint

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In the setting of Carnot groups, we propose an approach of taming singularities to get coercive inequalities. That is, we develop a technique to introduce natural singularities in the energy function U in order to force one of the coercivity conditions. In particular, we explore explicit constructions of probability measures on Carnot groups which secure Poincaré and even Logarithmic Sobolev inequalities.

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$q$-Poincar{é} inequalities on Carnot Groups with filiform type Lie algebra

Cited by 7 publications

References 0 publications

Local and nonlocal Poincaré inequalities on Lie groups

Local and nonlocal Poincaré inequalities on Lie groups

Logarithmic Sobolev inequalities on Lie groups

Coercive Inequalities on Carnot Groups: Taming Singularities

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