On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group

Bonnefont, Michel; Chafäı, Djalil; Herry, Ronan

doi:10.5802/afst.1633

Cited by 5 publications

(5 citation statements)

References 19 publications

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“…Logarithmic Sobolev inequalities in the sub-Riemannian setting have been studied for a number of examples such as isotropic and non-isotropic Heisenberg groups. There are different approaches to study the inequality, we refer only to the most relevant publications [2,4,10,15,19,23,28,33,40,49].…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

Gordina

Luo

2022

Journal of Functional Analysis

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Section: Introductionmentioning

confidence: 99%

Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

Gordina

Luo

2022

Journal of Functional Analysis

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“…His proof is based on pointwise upper and lower heat kernel estimates, and a gradient estimate known as the Driver-Melcher inequality. Motivated by [23] M. Bonnefont, D. Chafaï and R. Herry in [10] used a random walk approximation to study the case n = 1. For n 1, W. Hebisch and B. Zegarlinski proved a logarithmic Sobolev inequality in [27] using the tensorization property of logarithmic Sobolev inequalities and a lifting to the product group first introduced by [19, Section 3].…”

Section: Introductionmentioning

confidence: 99%

“…The measure considered in [1,10,17,27] is the hypoelliptic heat kernel measure on H n ω which can be regarded as an analogue of the Gaussian measure on the Euclidean space. In a different direction, [3] obtained a dimension-dependent upper bound for the logarithmic Sobolev constant with respect to the invariant measure of a subelliptic generator using a generalized curvature-dimension condition as developed in [5].…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

Gordina¹,

Luo²

2021

Preprint

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We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. The argument relies on comparing logarithmic Sobolev constants for the threedimensional non-isotropic and isotropic Heisenberg groups, and tensorization of logarithmic Sobolev inequalities in the sub-Riemannian setting. Furthermore, we apply these results in an infinite-dimensional setting and prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modelled on an abstract Wiener space.Contents ω 4. Logarithmic Sobolev Inequalities on H n ω 4.1. Tensorization in the sub-Riemannian setting 4.2. From the product group to a non-isotropic Heisenberg group 5. The second approach: tensorization and lifting reversed 6. Logarithmic Sobolev inequalities on infinite-dimensional Heisenberg groups 6

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“…We remark that when the generalized Ricci curvature is strictly positive in the sense of [20], the global situation is simpler as the underlying measure is necessarily finite, and a global Poincaré as well as (a variant of) a log-Sobolev inequality, with explicit constants, were obtained by Baudoin-Bonnefont in [18]; however, it is not clear how to localize these estimates to geodesic balls. For gradient estimates on the heat-kernel on the Heisenberg group and its associated (global) Poincaré and log-Sobolev inequalities, see [9,24,36,47,57].…”

Section: Introductionmentioning

confidence: 99%

The Quasi Curvature‐Dimension Condition with Applications to Sub‐Riemannian Manifolds

Milman

2020

Comm Pure Appl Math

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We obtain the best known quantitative estimates for the Lp‐Poincaré and log‐Sobolev inequalities on domains in various sub‐Riemannian manifolds, including ideal Carnot groups and in particular ideal generalized H‐type Carnot groups and the Heisenberg groups, corank 1 Carnot groups, the Grushin plane, and various H‐type foliations, Sasakian and 3‐Sasakian manifolds. Moreover, this constitutes the first time that a quantitative estimate independent of the dimension is established on these spaces. For instance, the Li‐Yau / Zhong‐Yang spectral‐gap estimate holds on all Heisenberg groups of arbitrary dimension up to a factor of 4. We achieve this by introducing a quasi‐convex relaxation of the Lott‐Sturm‐Villani CD(K, N) condition we call the “quasi curvature‐dimension condition” QCD(Q, K, N). Our motivation stems from a recent interpolation inequality along Wasserstein geodesics in the ideal sub‐Riemannian setting due to Barilari and Rizzi. We show that on an ideal sub‐Riemannian manifold of dimension n, the measure contraction property MCP(K, N) implies QCD(Q, K, N) with Q = 2N − n ≥ 1, thereby verifying the latter property on the aforementioned ideal spaces; a result of Balogh‐Kristály‐Sipos is used instead to handle nonideal corank 1 Carnot groups. By extending the localization paradigm to completely general interpolation inequalities, we reduce the study of various analytic and geometric inequalities on QCD spaces to the one‐dimensional case. Consequently, we deduce that while (strictly) sub‐Riemannian manifolds do not satisfy any type of CD condition, many of them satisfy numerous functional inequalities with exactly the same quantitative dependence (up to a factor of Q) as their CD counterparts. © 2020 Wiley Periodicals LLC

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On logarithmic Sobolev inequalities for the heat kernel on the Heisenberg group

Cited by 5 publications

References 19 publications

Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

The Quasi Curvature‐Dimension Condition with Applications to Sub‐Riemannian Manifolds

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