Sobolev met Poincaré. (English) Zbl 0954.46022

Hajłasz, Piotr; Koskela, Pekka

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.

Reviewer: Hans Triebel (Jena)

MSC:

46E35

Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

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