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 Simultaneously vanishing higher derived limits

2021 ◽  
Vol 9 ◽  
Author(s):  
Jeffrey Bergfalk ◽  
Chris Lambie-Hanson
Keyword(s):  
Euclidean Space ◽  
Metric Spaces ◽  
Inverse System ◽  
Finite Support ◽  
Partial Functions ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Strong Homology ◽  
Finite Support Iteration ◽  
Weakly Compact Cardinal

Abstract In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for every $n>0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a weakly compact cardinal, it is indeed consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for all $n>0$ . We show this via a finite-support iteration of Hechler forcings which is of weakly compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to $\lim ^n\textbf {A}=0$ will hold for each $n>0$ . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions $\mathbb {N}^2\to \mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:\mathbb {N}\to \mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .

scholarly journals Quasi-metric tree inT0-quasi-metric spaces

2013 ◽  
Vol 160 (13) ◽  
pp. 1794-1801 ◽  
Author(s):  
Olivier Olela Otafudu
Keyword(s):  
Metric Spaces ◽  
Metric Tree

scholarly journals Best proximity point results in geodesic metric spaces

2012 ◽  
Vol 2012 (1) ◽  
pp. 234 ◽  
Author(s):  
Maryam A Alghamdi ◽  
Mohammed A Alghamdi ◽  
Naseer Shahzad
Keyword(s):  
Metric Spaces ◽  
Best Proximity Point ◽  
Geodesic Metric

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 Dichotomy of global capacity density in metric measure spaces

2018 ◽  
Vol 11 (4) ◽  
pp. 387-404 ◽  
Author(s):  
Hiroaki Aikawa ◽  
Anders Björn ◽  
Jana Björn ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Spaces ◽  
Exact Formula ◽  
Doubling Measure ◽  
Metric Measure Spaces ◽  
Geodesic Metric ◽  
Measure Spaces ◽  
Variational Capacity ◽  
John Domains ◽  
Superlevel Sets ◽  
Sobolev Capacity

AbstractThe variational capacity {\operatorname{cap}_{p}} in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every {E\subset{\mathbb{R}}^{n}},\inf_{x\in{\mathbb{R}}^{n}}\frac{\operatorname{cap}_{p}(E\cap B(x,r),B(x,2r))}% {\operatorname{cap}_{p}(B(x,r),B(x,2r))}is either zero or tends to 1 as {r\to\infty}. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in {{\mathbb{R}}^{n}}. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

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 ★.

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