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

Let $(X,d,\mu)$ be a doubling metric measure space. We consider the behaviour of 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 additionally assume 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 the annular decay property. We also show by means of examples that the action of $M^\alpha$ is not continuous on these function spaces.