Characterizations of Sobolev spaces with variable exponent via averages on balls. (English) Zbl 1490.46033
This work is based on the approach used in the paper of [R. Alabern et al., Math. Ann. 354, No. 2, 589–626 (2012; Zbl 1267.46048)] where Sobolev spaces \(W^{\alpha,p}(\mathbb R^n)\) with \(\alpha\in[1,\infty)\) and the constant exponent \(p\in(1,\infty)\) were characterized. In the present paper, the author derives analogous characterization of the Sobolev spaces \(W^{\alpha,p(\cdot)}(\mathbb R^n)\) where \(\alpha\in[1,\infty)\) and \(p(\cdot)\) is a variable exponent which belongs to the globally log-Hölder continuous class \(C^{\log}(\mathbb R^n)\). Notice that this method can be used to define Sobolev spaces with variable exponent on metric spaces.
Reviewer: Petr Gurka (Praha)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B35 | Function spaces arising in harmonic analysis |
Citations:
Zbl 1267.46048References:
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