0000000000846233

AUTHOR

Gioacchino Antonelli

showing 3 related works from this author

Polynomial and horizontally polynomial functions on Lie groups

2022

We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $\mathfrak g$ of left-invariant vector fields on a Lie group $\mathbb G$ and we assume that $S$ Lie generates $\mathfrak g$. We say that a function $f:\mathbb G\to \mathbb R$ (or more generally a distribution on $\mathbb G$) is $S$-polynomial if for all $X\in S$ there exists $k\in \mathbb N$ such that the iterated derivative $X^k f$ is zero in the sense of distributions. First, we show that all $S$-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent $k$ in the previous defini…

Mathematics - Differential GeometryLeibman Polynomialnilpotent Lie groupsApplied Mathematicspolynomithorizontally affine functionsryhmäteoriaMetric Geometry (math.MG)polynomial mapsGroup Theory (math.GR)harmoninen analyysiFunctional Analysis (math.FA)Mathematics - Functional AnalysisdifferentiaaligeometriaMathematics - Metric GeometryDifferential Geometry (math.DG)precisely monotone setsFOS: Mathematicspolynomial on groupsMathematics - Group TheoryAnnali di Matematica Pura ed Applicata (1923 -)
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Lipschitz Carnot-Carathéodory Structures and their Limits

2022

AbstractIn this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not…

differentiaaligeometriaNumerical AnalysissäätöteoriaControl and OptimizationAlgebra and Number Theorysub-Riemannian geometryMitchell’s theoremControl and Systems Engineeringsub-Finsler geometryLipschitz vector fieldsmittateoria
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Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces

2020

This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a -hypersurface without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure . As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with…

codimension-one rectifiabilitysmooth hypersurface1ryhmäteoriaIntrinsic Lipschitz graphIntrinsic rectifiable setsubmanifoldsdifferentiaaligeometriaIntrinsic Cintrinsic Lipschitz graphCarnot groupsSmooth hypersurfaceMathematics::Metric Geometryintrinsic rectifiable setmittateoriaCodimension-one rectifiabilityCarnot groups; Codimension-one rectifiability; Intrinsic C; 1; submanifolds; Intrinsic Lipschitz graph; Intrinsic rectifiable set; Smooth hypersurface
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