Boundedness of the anisotropic fractional maximal operator in anisotropic local Morrey-type spaces

In this paper we study the boundedness of the anisotropic fractional maximal operator in anisotropic local Morrey-type spaces. We reduce this problem to the problem of boundedness of the supremal operator in weighted $L_p$-spaces on the cone of non-negative non-decreasing functions. This makes it possible to derive sharp sufficient conditions for boundedness for all admissible values of the numerical parameters, which, for a certain range of the numerical parameters, coincide with the necessary ones.