There have been recent attempts to develop the theory of Sobolev spaces 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 .
Dans cet article nous comparons les différentes définitions qui ont été données de l’espace de Sobolev associé à un espace métrique qui n’admet aucune structure différentielle. Nous prouvons en particulier que l’espace de Sobolev qu’on obtient à partir de la métrique de Carnot-Carathéodory associée à une famille de champs de vecteurs coïncide pour avec l’espace naturel des fonctions telles que pour lorsque toute fonction lipschitzienne satisfait une inégalité de Poincaré intrinsèque, convenable.
@article{AIF_1999__49_6_1903_0, author = {Franchi, Bruno and Haj{\l}asz, Piotr and Koskela, Pekka}, title = {Definitions of {Sobolev} classes on metric spaces}, journal = {Annales de l'Institut Fourier}, pages = {1903--1924}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {49}, number = {6}, year = {1999}, doi = {10.5802/aif.1742}, zbl = {0938.46037}, mrnumber = {2001a:46033}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1742/} }
TY - JOUR AU - Franchi, Bruno AU - Hajłasz, Piotr AU - Koskela, Pekka TI - Definitions of Sobolev classes on metric spaces JO - Annales de l'Institut Fourier PY - 1999 SP - 1903 EP - 1924 VL - 49 IS - 6 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1742/ DO - 10.5802/aif.1742 LA - en ID - AIF_1999__49_6_1903_0 ER -
%0 Journal Article %A Franchi, Bruno %A Hajłasz, Piotr %A Koskela, Pekka %T Definitions of Sobolev classes on metric spaces %J Annales de l'Institut Fourier %D 1999 %P 1903-1924 %V 49 %N 6 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1742/ %R 10.5802/aif.1742 %G en %F AIF_1999__49_6_1903_0
Franchi, Bruno; Hajłasz, Piotr; Koskela, Pekka. Definitions of Sobolev classes on metric spaces. Annales de l'Institut Fourier, Volume 49 (1999) no. 6, pp. 1903-1924. doi : 10.5802/aif.1742. https://aif.centre-mersenne.org/articles/10.5802/aif.1742/
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