scholarly journals BiLipschitz Decomposition of Lipschitz Maps between Carnot Groups

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Sean Li
Keyword(s):  
Carnot Group ◽  
Carnot Groups ◽  
Lipschitz Maps ◽  
Lipschitz Map

Abstract 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 B\Z can 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 admitting such a discretization.

scholarly journals A DEFINABLE -ADIC ANALOGUE OF KIRSZBRAUN’S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS

2015 ◽  
Vol 17 (1) ◽  
pp. 39-57 ◽  
Author(s):  
Raf Cluckers ◽  
Florent Martin
Keyword(s):  
Direct Application ◽  
Lipschitz Maps ◽  
Lipschitz Map ◽  
Lipschitz Extensions ◽  
Topological Closure

A direct application of Zorn’s lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun’s theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun’s theorem. In this paper, we prove in the $p$-adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable.

scholarly journals High Order Fefferman-Phong Type Inequalities in Carnot Groups and Regularity for Degenerate Elliptic Operators plus a Potential

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Pengcheng Niu ◽  
Kelei Zhang
Keyword(s):  
Euclidean Space ◽  
Vector Fields ◽  
Carnot Group ◽  
A Priori ◽  
Elliptic Operators ◽  
High Order ◽  
Carnot Groups ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic ◽  
Horizontal Vector

Let{X1,X2,…,Xm}be the basis of space of horizontal vector fields in a Carnot groupG=(Rn;∘) (m<n). We prove high order Fefferman-Phong type inequalities inG. As applications, we derive a prioriLp(G)estimates for the nondivergence degenerate elliptic operatorsL=-∑i,j=1maij(x)XiXj+V(x)withVMOcoefficients and a potentialVbelonging to an appropriate Stummel type class introduced in this paper. Some of our results are also new even for the usual Euclidean space.

LIPSCHITZ CONTINUITY FOR H-CONVEX FUNCTIONS IN CARNOT GROUPS

2006 ◽  
Vol 08 (01) ◽  
pp. 1-8 ◽  
Author(s):  
MINGBAO SUN ◽  
XIAOPING YANG
Keyword(s):  
Convex Functions ◽  
Lipschitz Continuity ◽  
Carnot Group ◽  
Carnot Groups ◽  
Lipschitz Continuous ◽  
Locally Lipschitz ◽  
Locally Lipschitz Continuous

For a Carnot group G of step two, we prove that H-convex functions are locally bounded from above. Therefore, H-convex functions on a Carnot group G of step two are locally Lipschitz continuous by using recent results by Magnani.

scholarly journals Maxwell’s Equations in Anisotropic Media and Carnot Groups as Variational Limits

2015 ◽  
Vol 15 (2) ◽  
Author(s):  
Annalisa Baldi ◽  
Bruno Franchi
Keyword(s):  
Dielectric Permittivity ◽  
Euclidean Space ◽  
Magnetic Permeability ◽  
Maxwell’S Equations ◽  
Maxwell's Equations ◽  
Carnot Group ◽  
Carnot Groups ◽  
Anisotropic Dielectric ◽  
Variational Functional ◽  
Simply Connected

AbstractLet G be a free Carnot group (i.e. a connected simply connected nilpotent stratified free Lie group) of step 2. In this paper, we prove that the variational functional generated by “intrinsic” Maxwell’s equations in G is the Γ-limit of a sequence of classical (i.e. Euclidean) variational functionals associated with strongly anisotropic dielectric permittivity and magnetic permeability in the Euclidean space.

scholarly journals Hausdorff measure of the singular set of quasiregular maps on Carnot groups

2003 ◽  
Vol 2003 (35) ◽  
pp. 2203-2220 ◽  
Author(s):  
Irina Markina
Keyword(s):  
Vector Space ◽  
Hausdorff Measure ◽  
Carnot Group ◽  
Quasiregular Mapping ◽  
Carnot Groups ◽  
Singular Set ◽  
Quasiregular Mappings ◽  
Local Index ◽  
Dimensional Hausdorff Measure ◽  
Homogeneous Dimension

Recently, the theory of quasiregular mappings on Carnot groups has been developed intensively. Letνstand for the homogeneous dimension of a Carnot group and letmbe the index of the last vector space of the corresponding Lie algebra. We prove that the(ν−m−1)-dimensional Hausdorff measure of the image of the branch set of a quasiregular mapping on the Carnot group is positive. Some estimates of the local index of quasiregular mappings are also obtained.

scholarly journals Nonexistence Results for Semilinear Equations in Carnot Groups

2013 ◽  
Vol 1 ◽  
pp. 130-146 ◽  
Author(s):  
Fausto Ferrari ◽  
Andrea Pinamonti
Keyword(s):  
Heisenberg Group ◽  
Carnot Group ◽  
Carnot Groups ◽  
Semilinear Equations ◽  
Engel Group ◽  
Large Step

Abstract In this paper, following [3], we provide some nonexistence results for semilinear equations in the the class of Carnot groups of type ★.This class, see [20], contains, in particular, all groups of step 2; like the Heisenberg group, and also Carnot groups of arbitrarly large step. Moreover, we prove some nonexistence results for semilinear equations in the Engel group, which is the simplest Carnot group that is not of type ★.

scholarly journals Space of signatures as inverse limits of Carnot groups

2021 ◽  
Author(s):  
Enrico Le Donne ◽  
Roger Zuest
Keyword(s):  
Metric Spaces ◽  
Carnot Group ◽  
Inverse System ◽  
Carnot Groups ◽  
Inverse Limits ◽  
Limit Space ◽  
Metric Tree ◽  
Geodesic Metric ◽  
Fixed Rank

We formalize the notion of limit of an inverse system of metric spaces with $1$-Lipschitz projections having unbounded fibers. The construction is applied to the sequence of free Carnot groups of fixed rank $n$ and increasing step. In this case, the limit space is in correspondence with the space of signatures of rectifiable paths in $\mathbb R^n$, as introduced by Chen. Hambly-Lyons's result on the uniqueness of signature implies that this space is a geodesic metric tree. As a particular consequence we deduce that every path in $\mathbb R^n$ can be approximated by projections of some geodesics in some Carnot group of rank $n$, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step.

scholarly journals On rectifiable measures in Carnot groups: representation

2021 ◽  
Vol 61 (1) ◽  
Author(s):  
Gioacchino Antonelli ◽  
Andrea Merlo
Keyword(s):  
Large Class ◽  
Hausdorff Measure ◽  
Carnot Group ◽  
Lipschitz Functions ◽  
Carnot Groups ◽  
Geodesic Sphere ◽  
Area Formula ◽  
Definition Of ◽  
Lipschitz Graphs ◽  
Almost All

AbstractThis paper deals with the theory of rectifiability in arbitrary Carnot groups, and in particular with the study of the notion of $$\mathscr {P}$$ P -rectifiable measure. First, we show that in arbitrary Carnot groups the natural infinitesimal definition of rectifiabile measure, i.e., the definition given in terms of the existence of flat tangent measures, is equivalent to the global definition given in terms of coverings with intrinsically differentiable graphs, i.e., graphs with flat Hausdorff tangents. In general we do not have the latter equivalence if we ask the covering to be made of intrinsically Lipschitz graphs. Second, we show a geometric area formula for the centered Hausdorff measure restricted to intrinsically differentiable graphs in arbitrary Carnot groups. The latter formula extends and strengthens other area formulae obtained in the literature in the context of Carnot groups. As an application, our analysis allows us to prove the intrinsic $$C^1$$ C 1 -rectifiability of almost all the preimages of a large class of Lipschitz functions between Carnot groups. In particular, from the latter result, we obtain that any geodesic sphere in a Carnot group equipped with an arbitrary left-invariant homogeneous distance is intrinsic $$C^1$$ C 1 -rectifiable.

scholarly journals Lipschitz and Bi-Lipschitz maps from PI spaces to Carnot groups

2020 ◽  
Vol 69 (5) ◽  
pp. 1685-1731
Author(s):  
Guy David ◽  
Kyle Kinneberg
Keyword(s):  
Carnot Groups ◽  
Lipschitz Maps

scholarly journals Sharp measure contraction property for generalized H-type Carnot groups

2018 ◽  
Vol 20 (06) ◽  
pp. 1750081 ◽  
Author(s):  
Davide Barilari ◽  
Luca Rizzi
Keyword(s):  
Absolute Continuity ◽  
Carnot Group ◽  
Optimal Synthesis ◽  
Carnot Groups ◽  
Contraction Property ◽  
Dimension Formula ◽  
Quadratic Cost ◽  
Measure Contraction Property ◽  
The Absolute

We prove that H-type Carnot groups of rank [Formula: see text] and dimension [Formula: see text] satisfy the [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in this paper, and for which we compute explicitly the optimal synthesis. This constitutes the largest class of Carnot groups for which the curvature exponent coincides with the geodesic dimension. We stress that generalized H-type Carnot groups have step 2, include all corank 1 groups and, in general, admit abnormal minimizing curves. As a corollary, we prove the absolute continuity of the Wasserstein geodesics for the quadratic cost on all generalized H-type Carnot groups.

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