Fractional operators on weighted Morrey spaces
  A necessary condition and a sufficient condition for one weight norm inequalities on Morrey spaces to hold are given for the fractional maximal operator and the fractional integral operator. We clarify the difference between the behavior of the fractional maximal operator and the one of the fractional integral operator which is originated from the structure of Morrey spaces. Both the necessary condition and the sufficient condition are also verified for the power weights. 
Shohei Nakamura, Yoshihiro Sawano and Hitoshi Tanaka
math.FA