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On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. (English) Zbl 1069.47056

This is a survey of recent developments related to Lebesgue spaces \(L^{p(\cdot)} ( \Omega), \;\Omega \subseteq \mathbb R^n,\) and Sobolev spaces \(W^{m,p(\cdot)}( \Omega)\) with variable order \(p(x)\). The following topics are considered: denseness of \(C_0^\infty\)-functions, Hardy type operators, maximal operators, singular integral operators, potential operators, compactness of some integral operators, Fredholm theory for singular integral equations. The connection with variational problems and PDE is discussed.

MSC:

47G10 Integral operators
47S99 Other (nonclassical) types of operator theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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