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1
Content available remote

Geodesics in the Heisenberg Group

100%
Hajłasz P., Zimmerman S.
Analysis and Geometry in Metric Spaces
|
2015
|
tom 3
|
nr 1
EN
We provide a new and elementary proof for the structure of geodesics in the Heisenberg group Hn. The proof is based on a new isoperimetric inequality for closed curves in R2n.We also prove that the Carnot- Carathéodory metric is real analytic away from the center of the group.
2
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The Lusin Theorem and Horizontal Graphs in the Heisenberg Group

100%
Hajłasz P., Mirra J.
Analysis and Geometry in Metric Spaces
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2013
|
tom 1
295-301
EN
In this paper we prove that every collection of measurable functions fα , |α| = m, coincides a.e. withmth order derivatives of a function g ∈ Cm−1 whose derivatives of order m − 1 may have any modulus of continuity weaker than that of a Lipschitz function. This is a stronger version of earlier results of Lusin, Moonens-Pfeffer and Francos. As an application we construct surfaces in the Heisenberg group with tangent spaces being horizontal a.e.
3
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Heisenberg Hausdorff Dimension of Besicovitch Sets

100%
Venieri L.
Analysis and Geometry in Metric Spaces
|
2014
|
tom 2
|
nr 1
EN
We consider (bounded) Besicovitch sets in the Heisenberg group and prove that Lp estimates for the Kakeya maximal function imply lower bounds for their Heisenberg Hausdorff dimension.
4
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Some Results on Maps That Factor through a Tree

88%
Züst R.
Analysis and Geometry in Metric Spaces
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2015
|
tom 3
|
nr 1
EN
We give a necessary and sufficient condition for a map deffned on a simply-connected quasi-convex metric space to factor through a tree. In case the target is the Euclidean plane and the map is Hölder continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over winding number functions. This in particular shows that if the target is the Heisenberg group equipped with the Carnot-Carathéodory metric and the Hölder exponent of the map is bigger than 2/3, the map factors through a tree.
5
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Blow-up of regular submanifolds in Heisenberg groups and applications

75%
Magnani V.
Open Mathematics
|
2006
|
tom 4
|
nr 1
82-109
EN
We obtain a blow-up theorem for regular submanifolds in the Heisenberg group, where intrinsic dilations are used. Main consequence of this result is an explicit formula for the density of (p+1)-dimensional spherical Hausdorff measure restricted to a p-dimensional submanifold with respect to the Riemannian surface measure. We explicitly compute this formula in some simple examples and we present a lower semicontinuity result for the spherical Hausdorff measure with respect to the weak convergence of currents. Another application is the proof of an intrinsic coarea formula for vector-valued mappings on the Heisenberg group.
6
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Some remarks on Bochner-Riesz means

75%
Thangavelu S.
Colloquium Mathematicum
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2000
|
tom 83
|
nr 2
217-230
EN
We study $L^p$ norm convergence of Bochner-Riesz means $S_R^δ f$ associated with certain non-negative differential operators. When the kernel $S_R^m(x,y)$ satisfies a weak estimate for large values of m we prove $L^p$ norm convergence of $S_R^δ f$ for δ > n|1/p-1/2|, 1 < p < ∞, where n is the dimension of the underlying manifold.
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