Abstract.

In this paper, we consider the generalized fractional maximal function and use it to introduce the space of generalized fractional maximal functions and the various cones of monotone functions generated by generalized fractional maximal functions MΦf. We introduced three function classes. We give equivalent descriptions of such cones when the function Φ belongs to some function classes. The conditions for their mutual covering are given. Then, these cones are used to construct a criterion for embedding the space of generalized fractional maximal functions into the re-arrangement invariant spaces (RIS). The optimal RIS for such embedding is also described.

Keywords:

re-arrangement function, invariant spaces, maximal function, function spaces, cones, mutual covering, embedding.