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Some remarks on the Hardy-Littlewood maximal function on variable $L^p$ spaces. (English) Zbl 1092.42009
Let $p: \Bbb R^n\to[1,\infty)$ be a measurable function. Denote by $L^{p(\cdot)}(\Bbb R^n)$ the Banach space of measurable functions $f$ on $\Bbb R^n$ such that for some $\lambda>0$, $$\int_{\Bbb R^n}\vert f(x)/\lambda\vert ^{p(x)}\,dx<\infty,$$ with norm $$\Vert f\Vert _{L^{p(\cdot)}}=\inf\left\{\lambda>0; \int_{\Bbb R^n}\vert f(x)/\lambda\vert ^{p(x)}\,dx\le1\right\}.$$ The author discusses on the class $\Cal P(\Bbb R^n)$ of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable $L^{p(\cdot)}$ space. He gives some sufficient conditions for $p\in \Cal P(\Bbb R^n)$, using mean oscillations. Let $$\Cal C(p)=\Vert p\Vert _\infty+\sup_{Q:\text{cube}}\vert Q\vert ^{1/n}\vert Q\vert ^{-1}\int_Q \left\vert f(x)-\vert Q\vert ^{-1}\int_Qf(y)dy\right\vert dx.$$ This was introduced by {\it E. Nakai} and {\it K. Yabuta} to characterize the pointwise multipliers for $BMO(\Bbb R^n)$ [J. Math. Soc. Japan, 37, 207--218 (1985; Zbl 0546.42019)]. The author’s results are the following: (1) There is a positive $\mu_n$ sufficiently small such that $2-p(x)\in \Cal P(\Bbb R^n)$ for any non-negative measurable function $p$ with $\Cal C(p)\le \mu_n$, and $2+p(x)\in \Cal P(\Bbb R^n)$ for any non-negative measurable function $p$ with $\text{ess. inf}\,p>0$ and $\Cal C(p)\le \mu_n$. (2) For any non-negative measurable function $p$ with $\Cal C(p)<\infty$, there exists $\alpha>0$ such that $\alpha-p(x)\in \Cal P(\Bbb R^n)$. Assuming additionally $\text{ess. inf }p>0$, $\alpha+p(x)\in \Cal P(\Bbb R^n)$. These improve known results by {\it L. Diening} [Math. Inequal. Appl. 7, No. 2, 245--253 (2004; Zbl 1071.42014)], {\it D. Cruz-Uribe, A. Fiorenza} and {\it C. J. Neugebauer} [Ann. Acad. Sci. Fenn., Math. 28, No. 1, 223--238 (2003; Zbl 1037.42023); Ann. Acad. Sci. Fenn., Math. 29, No. 1, 247--249 (2004; Zbl 1064.42500)], and {\it A. Nekvinda} [Math. Inequal. Appl. 7, No. 2, 255--265 (2004; Zbl 1059.42016)]. As a consequence, he gives a discontinuous $p\in \Cal P(\Bbb R^n)$, giving a positive answer to a conjecture by Diening.

MSC:
42B25Maximal functions, Littlewood-Paley theory
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References:
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