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

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 W1,p qu’on obtient à partir de la métrique de Carnot-Carathéodory associée à une famille de champs de vecteurs {X1,⋯,Xm} coïncide pour p>1 avec l’espace naturel des fonctions u∈Lp telles que Xju∈Lp 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 W1,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},
     pages = {1903--1924},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
     number = {6},
     year = {1999},
     doi = {10.5802/aif.1742},
     mrnumber = {2001a:46033},
     zbl = {0938.46037},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1742/}
}
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Franchi, Bruno; Hajłasz, Piotr; Koskela, Pekka. Definitions of Sobolev classes on metric spaces. Annales de l'Institut Fourier, Tome 49 (1999) no. 6, pp. 1903-1924. doi : 10.5802/aif.1742. https://www.numdam.org/articles/10.5802/aif.1742/

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