On some questions related to the maximal operator on variable $L^p$ spaces

Let $\mathcal{P}(\mathbb{R}^n)$ be the class of all exponents $p$ for which the Hardy-Littlewood maximal operator $M$ is bounded on $L^{p(\cdot)}({\mathbb{R}}^n)$. A recent result by T. Kopaliani provides a characterization of $\mathcal{P}$ in terms of the Muckenhoupt-type condition $A$ under some restrictions on the behavior of $p$ at infinity. We give a different proof of a slightly extended version of this result. Then we characterize a weak type $\big(p(\cdot),p(\cdot)\big)$ property of $M$ in terms of $A$ for radially decreasing $p$. Finally, we construct an example showing that $p\in\mathcal{P}(\mathbb{R}^n)$ does not imply $p(\cdot)-\alpha\in \mathcal{P}(\mathbb{R}^n)$ for all $\alpha< p_--1$. Similarly, $p\in\mathcal{P}(\mathbb{R}^n)$ does not imply $\alpha p(\cdot)\in \mathcal{P}(\mathbb{R}^n)$ for all $\alpha>1/p_-$.