Definitions of Sobolev classes on metric spaces
Franchi, Bruno ; Hajłasz, Piotr ; Koskela, Pekka
Annales de l'Institut Fourier, Tome 49 (1999), p. 1903-1924 / Harvested from Numdam
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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 u∈L p telles que X j u∈L p pour j=1,...,m lorsque toute fonction lipschitzienne satisfait une inégalité de Poincaré intrinsèque, convenable.

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.

@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},
     volume = {49},
     year = {1999},
     pages = {1903-1924},
     doi = {10.5802/aif.1742},
     mrnumber = {2001a:46033},
     zbl = {0938.46037},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 49 (1999) pp. 1903-1924. doi : 10.5802/aif.1742. http://gdmltest.u-ga.fr/item/AIF_1999__49_6_1903_0/

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