Coercive Inequalities in Higher-Dimensional Anisotropic Heisenberg Group

Dagher, Esther Bou; Zegarliński, Bogusław

doi:10.48550/arxiv.2105.02593

Cited by 3 publications

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“…It is interesting that by our technique we can provide new examples of probability measures for which Poincaré or even Log-Sobolev inequalities can be satisfied. Moreover we are able to provide classes of examples going far beyond the handsome existing results of [19], [21], [12], [13], [15]. In particular, we have results for all type 2 Carnot groups as well some others general cases (possibly satisfying certain technical conditions for homogeneous norms).…”

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

confidence: 83%

“…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%

“…Unfortunately, for smooth homogeneous norms, ∇N vanishes in some directions, so we cannot have η growing to infinity with the distance, and the pointwise strategy doesn't work. However, in some case one can overcome this problem by other arguments and get Poincaré inequality (but not Log-Sobolev), [21], [12], [13], [15].…”

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

confidence: 99%

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

confidence: 83%

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

confidence: 99%

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

confidence: 99%

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Coercive Inequalities on Carnot Groups: Taming Singularities

Dagher¹,

Zegarliński²

2021

Preprint

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“…The subject of the logarithmic inequalities has been extensively investigated and it is impossible to give a reasonably complete review of the literature here. We refer to surveys [AB00,GZ03], to works on relations to other inequalities [BL00,MF93], coercive inequalities on Carnot groups [Bou21,BZ21a,BZ21b,BZ21c,HZ09], Nash [Nas58, Bec98, BDS20, OS18] and weighted Nash [BBGM12] inequalities. There are works on the fractional Laplacian [Bec12] as well as on the logarithmic Sobolev inequalities for the fractional Laplacian [KRS20] in the setting of Folland and Stein ([FS82]) homogeneous groups.…”

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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“…where η(d) is a function having a suitable growth at infinity. Those methods were applied, for example, in the works [10,12,11] in the setting of Carnot groups. In [19], W. Hebisch and B.…”

Section: Introductionmentioning

confidence: 99%

Note on the $q$-Logarithmic Sobolev and $p$-Talagrand Inequalities on Carnot Groups

Dagher¹

2021

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In the setting of Carnot groups, we prove the q−Logarithmic Sobolev inequality for probability measures as a function of the Carnot-Carathéodory distance. As an application, we use the Hamilton-Jacobi equation in the setting of Carnot groups to prove the p−Talagrand inequality and hypercontractivity. Contents 1. Introduction 1 2. The Carnot Group and the Hamilton-Jacobi Equation 4 3. q−Logarithmic Sobolev Inequality 6 4. p−Talagrand inequality for dµ = e −U (d) Z dλ 10 5. Hypercontractivty of Hamilton-Jacobi solutions for dµ = e −U (d) Z dλ 13 References 14

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Coercive Inequalities in Higher-Dimensional Anisotropic Heisenberg Group

Cited by 3 publications

References 0 publications

Coercive Inequalities on Carnot Groups: Taming Singularities

Coercive Inequalities on Carnot Groups: Taming Singularities

Logarithmic Sobolev inequalities on Lie groups

Note on the $q$-Logarithmic Sobolev and $p$-Talagrand Inequalities on Carnot Groups

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