## The maximal function on variable $$L^p$$ spaces.(English)Zbl 1037.42023

Let $$\Omega$$ be an open subset of $$\mathbf R^n$$. The authors study the Hardy-Littlewood maximal function on the variable Lebesgue space $$L^{p(x)} (\Omega )$$, the Banach space of measurable functions on $$\Omega$$ such that $\int_{\Omega}\Big | \frac{f(x)}{\lambda}\Big | ^{p(x)}dx<\infty$ for some $$\lambda>0$$, with norm $\| f\| _{p(x),\Omega}=\inf\Big \{\lambda>0:\int_{\Omega}\Big | \frac{f(x)}{\lambda}\Big | ^{p(x)}dx\geq 1 \Big \}.$ They prove that the maximal function is bounded on $$L^{p(x)}(\Omega )$$ when $$p\,$$ satisfies the following three conditions:
1. $$1<\inf\{ p(y):y\in\Omega\} \leq\sup\{ p(y):y\in\Omega\} <\infty$$,
2. $$| p(x)-p(y)| \leq\frac{C}{-\log| x-y| }, \text{ for }\, x,y\in\Omega \,\text{and}\,| x-y| <\frac{1}{2}$$,
3. $$| p(x)-p(y)| \leq\frac{C}{log(e+| x| )}, \text{ for }\, x,y\in\Omega\, \text{and}\,| y| \geq| x| .$$
In addition, the authors prove a weak-type inequality under the sole condition that the function $$\frac{1}{p}$$ satisfies a reverse Hölder’s condition: there is a constant $$C>0$$ so that $\frac{1}{p(x)}\leq\frac{C}{| B| }\int_{B}\frac{1}{p(y)}dy$ for almost every $$x\in B$$, where $$B$$ is an arbitrary ball in $$\mathbf{R}^n$$.

### MSC:

 42B25 Maximal functions, Littlewood-Paley theory 42B35 Function spaces arising in harmonic analysis
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