Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces

The problem of the boundedness of the fractional maximal operator _Mα$M_{\alpha }$, 0<α<n$0<\alpha <n$, in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted _Lp$L_p$-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.