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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.

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.)
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