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Maximal and fractional operators in weighted \(L^{p(x)}\) spaces. (English) Zbl 1099.42021

The article addresses the classical questions of the theory of singular operators on the following generalization of \(L^p\)-spaces: \(L^{p(x)}(\Omega)\) is a Banach space of functions with the finite norm \[ \| f\| _{p(x)}=\inf\{\lambda > 0: \int_{\Omega} \biggl| \frac{f(x)}{\lambda}\biggr| ^{p(x)}\,dx\leq 1\}. \] (Notice that those spaces are not translation invariant.)
The authors show that corresponding maximal, potential, Hardy and Hankel-type operators on these spaces are bounded, provided that the exponent \(p(x)\) is (for the most) uniformly bounded away from 1 and infinity, and is a bit better than continuous, \(\Omega\) has no casps at the point of the singularity, and the parameters of the operators satisfy corresponding restrictions.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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References:

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