[Caractérisation des graphes intrinsèques co-horizontaux uniformément différentiables dans les groupes de Carnot]
Dans les groupes de Carnot, nous étudions les graphes intrinsèques des fonctions avec codomaine horizontal. Ces graphes sont réguliers quand la fonction est uniformément intrinsèquement différentiable. Notre premier résultat est une caractérisation de la différentiabilité intrinsèque en termes de régularité Hölder des projections sur le graphe des champs de vecteurs invariants à gauche.
Nous améliorons le résultat dans les groupes de Carnot de rang 2 pour les fonctions avec codomaine unidimensionnel : dans ce cas, la régularité horizontale suffit pour obtenir le résultat. Nous remarquons que cette amélioration n’est pas vraie dans le groupe de Carnot de rang 3 le plus simple. Enfin on montre une formule de l’aire pour les fonctions uniformément intrinsèquement différentiables avec codomaine unidimensionnel et on donne une expression explicite de l’élément de surface en fonctions des dérivées intrinsèques de la fonction.
In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differentiability by means of Hölder properties along the projections of left-invariant vector fields on the graph.
We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by only requiring horizontal regularity. We remark that such a refinement is not possible already in the easiest step-3 group.
As a by-product of independent interest, in every Carnot group we prove an area-formula for uniformly intrinsically differentiable real-valued maps. We also explicitly write the area element in terms of the intrinsic derivatives of the map.
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Keywords: Carnot groups, intrinsically $C^1$ surfaces, co-horizontal surfaces, area formula, intrinsically differentiable functions, little Hölder functions, broad solutions
Mot clés : Groupes de Carnot, Surfaces intrinsèquement $C^1$, Surfaces co-horizontales, fonctions uniformément différentiables, fonctions petit-Hölder, broad solutions
Antonelli, Gioacchino 1 ; Di Donato, Daniela 2 ; Don, Sebastiano 3 ; Le Donne, Enrico 4
CC-BY-ND 4.0
@article{AIF_2024__74_6_2523_0,
author = {Antonelli, Gioacchino and Di Donato, Daniela and Don, Sebastiano and Le Donne, Enrico},
title = {Characterizations of uniformly differentiable co-horizontal intrinsic graphs in {Carnot} groups},
journal = {Annales de l'Institut Fourier},
pages = {2523--2621},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {74},
number = {6},
year = {2024},
doi = {10.5802/aif.3660},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3660/}
}
TY - JOUR AU - Antonelli, Gioacchino AU - Di Donato, Daniela AU - Don, Sebastiano AU - Le Donne, Enrico TI - Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups JO - Annales de l'Institut Fourier PY - 2024 SP - 2523 EP - 2621 VL - 74 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3660/ DO - 10.5802/aif.3660 LA - en ID - AIF_2024__74_6_2523_0 ER -
%0 Journal Article %A Antonelli, Gioacchino %A Di Donato, Daniela %A Don, Sebastiano %A Le Donne, Enrico %T Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups %J Annales de l'Institut Fourier %D 2024 %P 2523-2621 %V 74 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3660/ %R 10.5802/aif.3660 %G en %F AIF_2024__74_6_2523_0
Antonelli, Gioacchino; Di Donato, Daniela; Don, Sebastiano; Le Donne, Enrico. Characterizations of uniformly differentiable co-horizontal intrinsic graphs in Carnot groups. Annales de l'Institut Fourier, Tome 74 (2024) no. 6, pp. 2523-2621. doi : 10.5802/aif.3660. https://aif.centre-mersenne.org/articles/10.5802/aif.3660/
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