Displaying similar documents to “Invertible Carnot Groups”

On Conditions for Unrectifiability of a Metric Space

Piotr Hajłasz, Soheil Malekzadeh (2015)

Analysis and Geometry in Metric Spaces

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We find necessary and sufficient conditions for a Lipschitz map f : E ⊂ ℝk → X into a metric space to satisfy ℋk(f(E)) = 0. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups.

Inverse Function Theorems and Jacobians over Metric Spaces

Luca Granieri (2014)

Analysis and Geometry in Metric Spaces

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We present inversion results for Lipschitz maps f : Ω ⊂ ℝN → (Y, d) and stability of inversion for uniformly convergent sequences. These results are based on the Area Formula and on the l.s.c. of metric Jacobians.

Extension of Lipschitz functions defined on metric subspaces of homogeneous type.

Alexander Brudnyi, Yuri Brudnyi (2006)

Revista Matemática Complutense

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If a metric subspace Mº of an arbitrary metric space M carries a doubling measure μ, then there is a simultaneous linear extension of all Lipschitz functions on Mº ranged in a Banach space to those on M. Moreover, the norm of this linear operator is controlled by logarithm of the doubling constant of μ.

On the nonexistence of bilipschitz parametrizations and geometric problems about A-weights.

Stephen Semmes (1996)

Revista Matemática Iberoamericana

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How can one recognize when a metric space is bilipschitz equivalent to an Euclidean space? One should not take the abstraction of metric spaces too seriously here; subsets of R are already quite interesting. It is easy to generate geometric conditions which are necessary for bilipschitz equivalence, but it is not clear that such conditions should ever be sufficient. The main point of this paper is that the optimistic conjectures about the existence of bilipschitz parametrizations are...

Infinitesimal Structure of Differentiability Spaces, and Metric Differentiation

Jeff Cheeger, Bruce Kleiner, Andrea Schioppa (2016)

Analysis and Geometry in Metric Spaces

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We prove metric differentiation for differentiability spaces in the sense of Cheeger [10, 14, 27]. As corollarieswe give a new proof of one of the main results of [14], a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith [27] is equal to 1, and new nonembeddability results.

BiLipschitz Decomposition of Lipschitz Maps between Carnot Groups

Sean Li (2015)

Analysis and Geometry in Metric Spaces

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Let f : G → H be a Lipschitz map between two Carnot groups. We show that if B is a ball of G, then there exists a subset Z ⊂ B, whose image in H under f has small Hausdorff content, such that BZcan be decomposed into a controlled number of pieces, the restriction of f on each of which is quantitatively biLipschitz. This extends a result of [14], which proved the same result, but with the restriction that G has an appropriate discretization. We provide an example of a Carnot group not...

On the Lifshits Constant for Hyperspaces

K. Leśniak (2007)

Bulletin of the Polish Academy of Sciences. Mathematics

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The Lifshits theorem states that any k-uniformly Lipschitz map with a bounded orbit on a complete metric space X has a fixed point provided k < ϰ(X) where ϰ(X) is the so-called Lifshits constant of X. For many spaces we have ϰ(X) > 1. It is interesting whether we can use the Lifshits theorem in the theory of iterated function systems. Therefore we investigate the value of the Lifshits constant for several classes of hyperspaces.

Definitions of Sobolev classes on metric spaces

Bruno Franchi, Piotr Hajłasz, Pekka Koskela (1999)

Annales de l'institut Fourier

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There have been recent attempts to develop the theory of Sobolev spaces W 1 , p on metric spaces that do not admit any differentiable structure. We prove that certain definitions are equivalent. We also define the spaces in the limiting case p = 1 .

Spaces of Lipschitz functions on metric spaces

Diethard Pallaschke, Dieter Pumplün (2015)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper the universal properties of spaces of Lipschitz functions, defined over metric spaces, are investigated.