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Definitions of Sobolev classes on metric spaces. (English) Zbl 0938.46037
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 $$1\leq p<\infty$$. Then $$u\in M^1_p(S,d,\mu)$$ if $$u\in L_p(S)$$ and $|u(x)- u(y)|\leq d(x,y)(g(x)+ g(y))\quad\text{for some }0\leq g\in L_p(S).$ Or: $$u\in P^1_p(S,d,\mu)$$ if for some $$0\leq g\in L_p(S)$$ and some $$C>0$$, $$\lambda\geq 1$$, $\not\mkern-7mu\int^\infty_B|u- u_B|d\mu\leq Cr\Biggl(\not\mkern-7mu\int_{\lambda B} g^p d\mu\Biggr)^{1/p},$ 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.
Reviewer: H.Triebel (Jena)

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