Weighted Hardy operators in the local generalized variable exponent weighted Morrey spaces and applications to differential operators

We consider local generalized weighted Morrey spaces M{x0}p(⋅),ω,φ(Rn) with variable exponent p(x), φ is a weight and a general function ω(r) defining the Morrey-type norm. Under the assumption φ is a variable exponent Muckenhoupt weight, we prove the M{x0}p(⋅),ω,φ(Rn) boundedness of the weighted Hardy operators and their commutators. Furthermore, we give the applications of our results to the differential equations.