# Article

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Keywords:
variable Lebesgue space; maximal operator; $\gamma$-rectangle; Besicovitch's covering theorem; weak-type inequality; strong-type inequality
Summary:
The classical Hardy-Littlewood maximal operator is bounded not only on the classical Lebesgue spaces $L_{p}(\mathbb {R}^d)$ (in the case $p >1$), but (in the case when $1/p(\cdot )$ is log-Hölder continuous and $p_{-} = \inf \{ p(x) \colon x \in \mathbb R^d \} > 1$) on the variable Lebesgue spaces $L_{p(\cdot )}(\mathbb {R}^d)$, too. Furthermore, the classical Hardy-Littlewood maximal operator is of weak-type $(1,1)$. In the present note we generalize Besicovitch's covering theorem for the so-called $\gamma$-rectangles. We introduce a general maximal operator $M_{s}^{\gamma ,\delta }$ and with the help of generalized $\Phi$-functions, the strong- and weak-type inequalities will be proved for this maximal operator. Namely, if the exponent function $1/p(\cdot )$ is log-Hölder continuous and $p_{-} > s$, where $1 \leq s \leq \infty$ is arbitrary (or $p_{-} \geq s$), then the maximal operator $M_{s}^{\gamma ,\delta }$ is bounded on the space $L_{p(\cdot )}(\mathbb {R}^d)$ (or the maximal operator is of weak-type $(p(\cdot ),p(\cdot ))$).
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