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Let $(X,d,\mu)$ be a doubling metric measure space. We consider the behaviourof the fractional maximal function $M^\alpha$ for $0\leq \alpha<Q$, where $Q$is the doubling dimension, acting on functions of bounded mean oscillation(BMO) and vanishing mean oscillation ($VMO$). For $\alpha>0$, we additionallyassume that the space is bounded. We show that $M^\alpha$ is bounded from $BMO$to $BLO$, a subclass of $BMO$, and maps $VMO$ to itself when $\mu$ has theannular decay property. We also show by means of examples that the action of$M^\alpha$ is not continuous on these function spaces.

Fractional maximal functions and mean oscillation on bounded doubling metric measure spaces