Definitions of Sobolev classes on metric spaces
Annales de l'Institut Fourier, Volume 49 (1999) no. 6, p. 1903-1924

There have been recent attempts to develop the theory of Sobolev spaces W 1,p 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 p=1.

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 W 1,p qu’on obtient à partir de la métrique de Carnot-Carathéodory associée à une famille de champs de vecteurs {X 1 ,,X m } coïncide pour p>1 avec l’espace naturel des fonctions uL p telles que X j uL p pour j=1,...,m 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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {6},
     year = {1999},
     pages = {1903-1924},
     doi = {10.5802/aif.1742},
     zbl = {0938.46037},
     mrnumber = {2001a:46033},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_1999__49_6_1903_0/

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