## Sobolev met Poincaré.(English)Zbl 0954.46022

Mem. Am. Math. Soc. 688, 101 p. (2000).
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 $$0\leq g\in L_p(X)$$ such that $|u(x)- u(y)|\leq d(x,y)\;(g(x)+ g(y))\quad\text{a.e.}$ This is one way to extend the classical Sobolev space $$W^1_p(\Omega)$$ from $$\mathbb{R}^n$$ to metric spaces ($$\Omega$$ is a domain in $$\mathbb{R}^n$$). Alternatively one may ask whether for given $$u$$ there is function $$g$$ such that $\not\mkern-7mu\int_B|u- u_B|d\mu\leq Cr\Biggl( \not\mkern-7mu\int_{\sigma B} g^pd\mu\Biggr)^{{1\over p}},\;u_B\text{ mean value},$ (Poincaré inequality) or $\Biggl( \not\mkern-7mu\int_B|u- u_B|^q d\mu\Biggr)^{{1\over q}}\leq Cr\Biggl( \not\mkern-7mu\int_{\sigma B} g^p d\mu\Biggr)^{{1\over p}}$ (Sobolev-Poincaré inequality), $$q> p\geq 1$$. Here $$B$$ and $$\sigma B$$ are concentric balls of radius $$r$$ and $$\sigma r$$ with $$\sigma\geq 1$$.
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.

### MSC:

 4.6e+36 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: