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Hardy-Littlewood maximal operator on $$L^{p(x)}(\mathbb{R}^n)$$. (English) Zbl 1059.42016
Let $$L^p(\mathbb{R}^n)$$ be a generalized Lebesgue space corresponding to a measurable function $$p: \mathbb{R}^n \to [1,\infty)$$ and $$M$$ be the Hardy-Littlewood maximal operator corresponding to the collection of all cubes in $$\mathbb{R}^n$$. The author proves the following
Theorem. If a continuous function $$p$$ on $$\mathbb{R}^n$$ satisfies the following conditions: $1<\inf_{x\in \mathbb{R}^n} p(x) \leq \sup_{x\in \mathbb{R}^n} p(x)<\infty,\tag{1}$ (2) there is a $$c>0$$ such that $| p(x)-p(y)| \leq \dfrac{c}{\log\frac{1}{| x-y| }}\text{ for any }x,y\in \mathbb{R}^n\text{ with }0<| x-y| <\frac{1}{2},$ (3) there are numbers $$p_\infty>1$$ and $$c>0$$ such that $\int_{\{x:| p(x)-p_\infty| >0\}} | p(x)-p_\infty| c^{1/| p(x)-p_\infty| }\,dx<\infty,$ then $$M$$ is bounded on $$L^p(\mathbb{R}^n)$$.
The theorem sharpens the previous result of L. Deining [“Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $$L^{p(x)}$$ and $$W^{k,p(x)}$$”, University of Freiburg, preprint (2002)] where condition (3) is replaced by the requirement that $$p$$ be constant outside of a fixed ball. The author also gives an example showing that the theorem does not remain true if the condition 3) is replaced by the following weaker condition: there is a number $$p_\infty>1$$ such that $\lim_{r\to \infty} | \{x\in \mathbb{R}^n: | x| >r, \;| p(x)-p_\infty| \geq \delta\}| =0\text{ for each }\delta>0,$ where $$| \cdot|$$ denotes the Lebesgue measure.

##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
##### Keywords:
maximal operator; Lebesgue space; variable exponent
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