Definitions of Sobolev classes on metric spaces

Franchi, Bruno; Hajłasz, Piotr; Koskela, Pekka

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 $W^{1,p}$ qu’on obtient à partir de la métrique de Carnot-Carathéodory associée à une famille de champs de vecteurs $\{X_1,\cdots ,X_m\}$ coïncide pour $p>1$ avec l’espace naturel des fonctions $u\in L^p$ telles que $X_{j}u\in L^p$ 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 $p=1$.

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