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

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