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Zbl 0954.46022
Hajłasz, Piotr; Koskela, Pekka
Sobolev met Poincaré.
(English)
[J] Mem. Am. Math. Soc. 688, 101 p. (2000). ISSN 0065-9266

Let $X$ be a metric space with the metric $d$ and the Borel measure $\mu$. Let $p>0$. Then the Sobolev space $M^1_p(X,\mu)$ is the collection of all $u\in L_p(X)$ for which there is a function $0\le g\in L_p(X)$ such that $$|u(x)- u(y)|\le d(x,y)\ (g(x)+ g(y))\quad\text{a.e.}$$ This is one way to extend the classical Sobolev space $W^1_p(\Omega)$ from $\bbfR^n$ to metric spaces ($\Omega$ is a domain in $\bbfR^n$). Alternatively one may ask whether for given $u$ there is function $g$ such that $$\not\mkern-7mu\int_B|u- u_B|d\mu\le Cr\Biggl( \not\mkern-7mu\int_{\sigma B} g^pd\mu\Biggr)^{{1\over p}},\ u_B\text{ mean value},$$ (Poincaré inequality) or $$\Biggl( \not\mkern-7mu\int_B|u- u_B|^q d\mu\Biggr)^{{1\over q}}\le Cr\Biggl( \not\mkern-7mu\int_{\sigma B} g^p d\mu\Biggr)^{{1\over p}}$$ (Sobolev-Poincaré inequality), $q> p\ge 1$. Here $B$ and $\sigma B$ are concentric balls of radius $r$ and $\sigma r$ with $\sigma\ge 1$.\par The main aim of this paper is the study of these types of Sobolev spaces on metric spaces (often homogeneous spaces when $\mu$ has the doubling condition) and their interrelations. Furthermore, various extensions and applications are given: Trudinger inequalities, Rellich-Kondrachov assertions, Sobolev classes on John domains, Poincaré inequalities on Riemannian and topological manifolds, Carnot-Carathéodory spaces, applications to PDE's.
[Hans Triebel (Jena)]
MSC 2000:
*46E35 Sobolev spaces and generalizations

Keywords: Poincaré inequality; Sobolev-Poincaré inequality; Borel measure; Sobolev space; homogeneous spaces; doubling condition; Trudinger inequalities; Rellich-Kondrachov assertions; Sobolev classes; John domains; Riemannian and topological manifolds; Carnot-Carathéodory spaces

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