  In this paper, we consider the norm inequalities for sublinear operators with
rough kernel generated by fractional integrals and commutators on generalized
local Morrey spaces and on generalized vanishing local Morrey spaces including
their weak versions under generic size conditions which are satisfied by most
of the operators in harmonic analysis, respectively. As an example to the
conditions of these theorems are satisfied, we can consider the Marcinkiewicz
operator.
