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

Let P(ℝⁿ) be the class of all exponents p for which the Hardy- Littlewood maximal operator M is bounded on L^p(·) (ℝⁿ). A recent result by T. Kopaliani provides a characterization of P in terms of the Muckenhoupttype 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 (_p(·), _p(·)) property of M in terms of A for radially decreasing _p. Finally, we construct an example showing that _p ε P(ℝⁿ) does not imply _p(·) - α ε P(ℝⁿ) for all α < _p- - 1. Similarly, _p ε P(ℝ_n) does not imply α_p(·) ε P(ℝⁿ) for all a > 1/_p-.