# zbMATH — the first resource for mathematics

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.

##### MSC:
 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
##### Keywords:
Orlicz-Sobolev spaces; metric spaces