Some remarks on the Hardy-Littlewood maximal function on variable L ^p spaces

We show that any pointwise multiplier for BMO(ℝⁿ) generates a function p from the class P(ℝⁿ) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L ^p space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to P(ℝⁿ).