Gradient Estimates for the Heat Semigroup on H-Type Groups

Hu, Jun-Qi; Li, Hongquan

doi:10.1007/s11118-010-9173-1

Cited by 18 publications

(22 citation statements)

References 23 publications

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“…However, current methods for proving (LSI) in this setting produce a constant c > 1 2 (see [3, Sections 1.2 and 6.1] in conjunction with [7, Theorem 1.6]), although we do not know whether this is sharp. The situation for H-type groups is similar [8,9,21,27].…”

Section: Statement Of Resultsmentioning

confidence: 95%

“…It is an open problem to determine which stratified Lie groups satisfy the logarithmic Sobolev inequality (LSI), and we hope this paper may provide additional motivation for further work on this difficult question. The current state of the art, as far as we are aware, is that (LSI) is true for H-type groups ( [9,21]; see Example 4.3 below for definitions and references), and of course in the "step 1" Euclidean case (Example 4.1). In all other stratified Lie groups, including all those of step ≥ 3, it is apparently unknown whether (LSI) holds or not.…”

Section: Statement Of Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on stratified Lie groups

Eldredge

2018

Nonlinear Analysis

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On a stratified Lie group G equipped with hypoelliptic heat kernel measure, we study the behavior of the dilation semigroup on L p spaces of log-subharmonic functions. We consider a notion of strong hypercontractivity and a strong logarithmic Sobolev inequality, and show that these properties are equivalent for any group G. Moreover, if G satisfies a classical logarithmic Sobolev inequality, then both properties hold. This extends similar results obtained by Graczyk, Kemp and Loeb in the Euclidean setting.

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Section: Statement Of Resultsmentioning

confidence: 95%

Section: Statement Of Resultsmentioning

confidence: 99%

Strong hypercontractivity and strong logarithmic Sobolev inequalities for log-subharmonic functions on stratified Lie groups

Eldredge

2018

Nonlinear Analysis

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“…Dungey [39,40] obtained (GLY ∞ ) on Riemannian covering manifolds with polynomial growth. On Heisenberg type groups, Driver and Melcher [38] and Hu and Li [67] obtained a Bakry-Émery type inequality, which implies (GLY ∞ ). Zhang [113] obtained Yau's gradient estimate (K = 0) on Riemannian manifolds of non-negative Ricci curvature modulo a small perturbation.…”

Section: Background and Main Resultsmentioning

confidence: 99%

“…These two gradient estimates are fundamental tools in geometric analysis and related fields, and there have been many efforts afterwards to generalise them to different settings, see for instance [34,38,39,40,44,52,67,73,82,90,91,95,113,114,115]. Let us review some of these generalisations.…”

Section: Background and Main Resultsmentioning

confidence: 99%

Gradient estimates for heat kernels and harmonic functions

Coulhon

Jiang

Koskela

et al. 2020

Journal of Functional Analysis

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Let (X, d, µ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a "carré du champ". Assume that (X, d, µ, E ) supports a scale-invariant L 2 -Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p ∈ (2, ∞]: (i) (G p ): L p -estimate for the gradient of the associated heat semigroup; (ii) (RH p ): L p -reverse Hölder inequality for the gradients of harmonic functions; (iii) (R p ): L p -boundedness of the Riesz transform (p < ∞); (iv) (GBE): a generalised Bakry-Émery condition. We show that, for p ∈ (2, ∞), (i), (ii) (iii) are equivalent, while for p = ∞, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the L 2 -Poincaré inequality. Our result gives a characterisation of Li-Yau's gradient estimate of heat kernels for p = ∞, while for p ∈ (2, ∞) it is a substantial improvement as well as a generalisation of earlier results by Auscher-Coulhon-Duong-Hofmann [7] and Auscher-Coulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and sub-Riemannian manifolds as well as to non-smooth spaces, and to degenerate elliptic/parabolic equations in these settings.

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“…We note there is another distance, called Carnot-Carathéodory distance on H n and this distance plays an important role in the study of hypoelliptic heat kernel (see [1,9,12,13,15,16,17]). Since d cc (ξ) ≥ d(ξ) for all ξ ∈ H n (see [18], Lemma 5.1), inequality ( However, the constant in (1.3) is not sharp.…”

Section: Introductionmentioning

confidence: 99%