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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)

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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