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Orlicz-Sobolev spaces on metric measure spaces. (English) Zbl 1068.46022
Annales Academiæ Scientiarum Fennicæ. Mathematica. Dissertationes 135. Helsinki: Suomalainen Tiedeakatemia; Jyväskylä: Univ. of Jyväskylä, Dept. of Mathematics and Statistics (Thesis). 86 p. (2004).
Let \((X,d,\mu)\) be a metric space, where \(X\) is a set, \(d\) a metric and \(\mu\) a Borel measure. A function \(g \geq 0\) on \(X\) is an upper gradient of the real function \(u\) if \[ | u(x) - u(y)| \leq \int_\gamma g ds, \quad x\in X, \quad y \in X,\tag{*} \] for all curves in \(X\) connecting \(x\) and \(y\). Let \(\Psi\) be a Young function. Then the Orlicz-Sobolev space \(N^{1, \Psi} (X)\) is the collection of all \(u \in L^\Psi (X)\) such that there is an upper gradient \(g \in L^\Psi (X)\) with (\(*\)). The main aim of this PhD-thesis is the detailed study of these Orlicz-Sobolev spaces.

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)