A necessary condition for the boundedness of the maximal operator on Lp(·) over reverse doubling spaces of homogeneous type

Let (X,d,μ) be a space of homogeneous type and p(·):X→[1,∞] be a variable exponent. We show that if the measure μ is Borel-semiregular and reverse doubling, then the condition ess\,infx∈ Xp(x)>1 is necessary for the boundedness of the Hardy-Littlewood maximal operator M on the variable Lebesgue space Lp(·)(X,d,μ).