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Maximal functions on Musielak–Orlicz spaces and generalized Lebesgue spaces. (English) Zbl 1096.46013
It is proved that the uniform boundedness of averaging operators corresponding to families of disjoint cubes is equivalent to the boundedness of the Hardy–Littlewood maximal operator $$M$$ on generalized Lebesgue space $$L^{p(\cdot)}(\mathbb R^d)$$ with $$1< \operatorname{ess}\inf p\leq \operatorname{ess}\sup p< \infty$$. It is also proved that the boundedness of $$M$$ is equivalent to the following conditions: (a) $$M_q$$ is bounded on $$L^{p(\cdot)}(\mathbb R^d)$$ for some $$q> 1$$, where $$M_qf=(M(| f| ^q))^{1/q}$$; (b) $$M$$ is bounded on $$L^{\frac{p(\cdot)}{q}}(\mathbb R^d)$$ for some $$q> 1$$; (c) $$M$$ is bounded on $$L^{p'(\cdot)}(\mathbb R^d)$$, where $$\frac{1}{p}+\frac{1}{p'}=1$$. Applications concerning Calderon–Zygmund operators and the Korn inequality are also given.

MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B25 Maximal functions, Littlewood-Paley theory 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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