Geodesic
Musielak-Orlicz-Sobolev spaces on metric measure spaces
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 435-474.

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Our aim in this paper is to study Musielak-Orlicz-Sobolev spaces on metric measure spaces. We consider a Hajłasz-type condition and a Newtonian condition. We prove that Lipschitz continuous functions are dense, as well as other basic properties. We study the relationship between these spaces, and discuss the Lebesgue point theorem in these spaces. We also deal with the boundedness of the Hardy-Littlewood maximal operator on Musielak-Orlicz spaces. As an application of the boundedness of the Hardy-Littlewood maximal operator, we establish a generalization of Sobolev's inequality for Sobolev functions in Musielak-Orlicz-Hajłasz-Sobolev spaces.
DOI : 10.1007/s10587-015-0187-0
Classification : 31B15, 31E05, 46E35
Keywords: Sobolev space; metric measure space; Sobolev's inequality; Hajłasz-Sobolev space; Newton-Sobolev space; Musielak-Orlicz space; capacity; variable exponent 
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Ohno, Takao; Shimomura, Tetsu. Musielak-Orlicz-Sobolev spaces on metric measure spaces. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 435-474. doi : 10.1007/s10587-015-0187-0. http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0187-0/

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