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Zbl 0938.46037
Franchi, Bruno; Hajłasz, Piotr; Koskela, Pekka
Definitions of Sobolev classes on metric spaces.
(English)
[J] Ann. Inst. Fourier 49, No.6, 1903-1924 (1999). ISSN 0373-0956; ISSN 1777-5310/e

Let $(S,d,\mu)$ be a set $S$ equipped with a metric $d$ and a locally finite Borel measure $\mu$ satisfying the doubling condition. There are several ways to introduce Sobolev spaces: Let $1\le p<\infty$. Then $u\in M^1_p(S,d,\mu)$ if $u\in L_p(S)$ and $$|u(x)- u(y)|\le d(x,y)(g(x)+ g(y))\quad\text{for some }0\le g\in L_p(S).$$ Or: $u\in P^1_p(S,d,\mu)$ if for some $0\le g\in L_p(S)$ and some $C>0$, $\lambda\ge 1$, $$\not\mkern-7mu\int^\infty_B|u- u_B|d\mu\le Cr\Biggl(\not\mkern-7mu\int_{\lambda B} g^p d\mu\Biggr)^{1/p},$$ where $B$ is a ball of radius $r$, $u_B$ is the average, and $\not\mkern-7mu\int$ the average value of the integral (Poincaré inequality). The authors study the relations of these two possibilities. They apply their results to Sobolev spaces defined via vector fields of first-order differential operators.
[H.Triebel (Jena)]
MSC 2000:
*46E35 Sobolev spaces and generalizations

Keywords: Sobolev spaces; metric spaces; doubling measures; Carnot-Carathéodory spaces; Hörmander's rank condition; Poincaré inequality; doubling condition

Cited in: Zbl 1026.49029

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