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

Abstract:

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 $\alpha >1/p_-$.