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Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. (English) Zbl 0974.46038
The author studies a possible definition of Sobolev spaces in abstract metric spaces, and answers in the affirmative the question whether this definition yields a Banach space. The author also explores the relationship between this definition and the Sobolev spaces defined by Hajłasz. For specialized metric spaces, the Sobolev embedding theorems are established. Different versions of capacities are also explored, and these various definitions are compared. The main tool used in this paper is the concept of moduli of path families.
Reviewer: Yang Dachun (Jena)

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
43A85 Harmonic analysis on homogeneous spaces
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