Fractional maximal functions on variable Hölder spaces over metric measure spaces

We study centered fractional maximal functions on homogeneous metric spaces with $\delta$-annular decay. For a variable Hölder exponent $\phi$ and a fractional order $0<\beta\leq\delta$, the target exponent is $\min\{\phi(\cdot)+\beta,\delta\}$. As the fractional maximal operator of a bounded function need not be finite, we first treat its restricted version and prove a full norm estimate. For the unrestricted operator, we separate the oscillation estimate from the boundedness of the fractional maximal function and then obtain a full norm estimate under an additional $L^q$ assumption and a uniform lower bound on the volume at large radius, with no continuity condition imposed on the exponent. We also prove continuity into strictly weaker variable Hölder spaces and give Euclidean and convex-domain consequences.