Anisotropic Fractional Maximal Operator in Anisotropic Generalized Morrey Spaces

In this paper it is proved that anisotropic fractional maximal operator $M_{a,sigma}$, $0 le a < |sigma|$ is bounded on anisotropic generalized Morrey spaces $M_{p,varphi,sigma}$, where $|sigma|=sum_{i=1}^n sigma_i$ is the homogeneous dimension of $Rn$. We find the conditions on the pair $(varphi_1,varphi_2)$ which ensure the Spanne-Guliyev type boundedness of the operator $M_{a,sigma}$ from anisotropic generalized Morrey space $M_{p,varphi_1,sigma}$ to $M_{q,varphi_2,sigma}$, $1 As applications, we establish the boundedness of some Sch"{o}dinger type operators on anisotropic generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse H"{o}lder class.