Musielak−Orlicz−Sobolev spaces on arbitrary metrique space
Commentationes Mathematicae, Tome 56 (2016) no. 2.

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In this article we define Musielak−Orlicz−Sobolev spaces on arbitrary metric spaces with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a pointwise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds, and that the Lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each Musielak−Orlicz−Sobolev function has a quasi-continuous representative.
Mots-clés : Metric measure space, Musielak−Orlicz−Sobolev spaces, capacity 
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     author = {Akdim Youssef and Noureddine Aissaoui and My Cherif Hassib},
     title = {Musielak\ensuremath{-}Orlicz\ensuremath{-}Sobolev spaces on arbitrary metrique space},
     journal = {Commentationes Mathematicae},
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     year = {2016},
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Akdim Youssef; Noureddine Aissaoui; My Cherif Hassib. Musielak−Orlicz−Sobolev spaces on arbitrary metrique space. Commentationes Mathematicae, Tome 56 (2016) no. 2. https://geodesic-test.mathdoc.fr/item/COMA_2016__56_2_292379/