On some questions related to the maximal operator on variable spaces - See more at: http://www.ams.org/journals/tran/2010-362-08/S0002-9947-10-05066-X/#sthash.OUNzVZOm.dpuf

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_-$. - See more at: http://www.ams.org/journals/tran/2010-362-08/S0002-9947-10-05066-X/#sthash.OUNzVZOm.dpuf