Maximal operator on variable Lebesgue spaces for almost monotone radial exponent

We study general Lebesgue spaces with variable exponent p. It is known that the classes L and N of functions p are such that the Hardy–Littlewood maximal operator is bounded on them provided p∈L∩P. The class L governs local properties of p and N governs the behavior of p at infinity.In this paper we focus on the properties of p near infinity. We extend the class N to a collection D of functions p such that the Hardy–Littlewood maximal operator is bounded on the corresponding variable Lebesgue spaces provided p∈L∩D and the class D is essentially larger than N.Moreover, the condition p∈D is quite easily verifiable in the practice.