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

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