The maximal operator on variable Lebesgue spaces: an ${\mathcal{A}}_{\mathrm{\infty }}$-characterization

In this paper we obtain a new boundedness criterion for the maximal operator $M$ on variable exponent spaces $L^{p(\cdot )}$. It is formulated in terms of the variable exponent analogue of the well known weighted $A_{\mathrm{\infty }}$ condition.