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New characterizations of Hajłasz-Sobolev spaces on metric spaces. (English) Zbl 1092.46026

The author introduces on spaces of homogeneous type \((X, \rho, \mu)\), consisting of a set \(X\), a quasi-metric \(\rho\) and a doubling Borel measure \(\mu\), fractional Sobolev spaces \(W^s_p (X)\) where \(1 \leq p \leq \infty\) and \(s>0\). The paper deals with equivalent characterisations of these spaces in terms of local means and maximal functions.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B35 Function spaces arising in harmonic analysis
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