Fractional maximal functions in weighted Banach function spaces (Q1396771)

The fractional maximal operator \(M_\gamma \) is given by \(M_\gamma (fd\sigma)(x)=\sup (\mu B)^{\gamma -1} \int _{B}|f(y)|d\sigma \), where \(\gamma \in [0,1)\), and the supremum is taken over all balls of positive measure containing the point \(x\). The functions are defined on a homogeneous type space, that is, a topological space with measure in which compactly supported functions are dense in \(L^1\). In the main theorem, a necessary and sufficient condition on a pair of weighted Banach function spaces \((Y,Z)\) is given in order that the operator \(M_\gamma \) is bounded from \(Y\) to \(Z\).