Geodesics in the Heisenberg Group

Hajłasz, Piotr; Zimmerman, Scott

doi:10.1515/agms-2015-0020

Cited by 15 publications

(17 citation statements)

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“…Since the formula (4.1) for the distance holds also in higher Heisenberg groups H N (as proved in [13]), the same proof works also in that case; however, for simplicity we presented the proof for H 1 .…”

Section: Comments and Extensionsmentioning

confidence: 73%

Bourgain–Brezis–Mironescu Approach in Metric Spaces with Euclidean Tangents

Górny

2022

J Geom Anal

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In the setting of metric measure spaces satisfying the doubling condition and the (1, p)-Poincaré inequality, we prove a metric analogue of the Bourgain–Brezis–Mironescu formula for functions in the Sobolev space $$W^{1,p}(X,d,\nu )$$ W 1 , p ( X , d , ν ) , under the assumption that for $$\nu $$ ν -a.e. point the tangent space in the Gromov–Hausdorff sense is Euclidean with fixed dimension N.

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Section: Comments and Extensionsmentioning

confidence: 73%

Bourgain–Brezis–Mironescu Approach in Metric Spaces with Euclidean Tangents

Górny

2022

J Geom Anal

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“…A key assumption will be that the unit ball with respect to a distance d is symmetric in the horizontal directions, as is clearly the case for the distance d ∞ . This is known for the CC distance only in a few cases, such as Heisenberg groups, see [25] or [15,Corollary 3.2].…”

Section: Carnot Groupsmentioning

confidence: 99%

Local and nonlocal 1-Laplacian in Carnot groups

Górny

2022

Ann. Fenn. Math.

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We formulate and study the nonlocal and local least gradient problem, which is the Dirichlet problem for the 1-Laplace operator, in the non-Euclidean setting of Carnot groups. We study the passage from the nonlocal problem to the local problem as the range of the interaction goes to zero. During this procedure, we prove a total variation estimate of independent interest and give an existence result for the local problem.

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“…We choose to work primarily with the box distance due to the fact that the unit ball with respect to this distance is symmetric in the horizontal directions; this will play a key role in the proofs in Section 4. This is known for the CC distance only in a few cases, such as Heisenberg groups, see [19] or [10,Corollary 3.2].…”

Section: Preliminariesmentioning

confidence: 99%