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Zbl 1068.46022
Tuominen, Heli
Orlicz-Sobolev spaces on metric measure spaces. (English)
[B D] Annales Academi\ae\ Scientiarum Fennic\ae. 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 $g \ge 0$ on $X$ is an upper gradient of the real function $u$ if $$\vert u(x) - u(y)\vert \le \int_\gamma g ds, \quad x\in X, \quad y \in X,\tag * $$ 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.
[Hans Triebel (Jena)]

MSC 2000:: *46E35 Sobolev spaces and generalizations
46E30 Spaces of measurable functions

Keywords: Orlicz-Sobolev spaces; metric spaces

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