Maximal operator in Hölder spaces

We study the maximal operator on the variable exponent Hölder spaces in the setting of metric measure spaces. The boundedness is proven for metric measure spaces satisfying an annular decay property. Let us stress that there are no assumptions on the regularity of the variable exponent and the variable exponent can touch values $0$ and $1$. Furthermore, the continuity of the maximal operator between Hölder spaces is investigated. Those results are new even in the Euclidean setting.