Let be a Radon measure on and be a ball with center and radius . The fractional maximal operator is defined as
When satisfies the growth condition , J. García-Cuerva and A. E. Gatto [Stud. Math. 162, No. 3, 245–261 (2004; Zbl 1045.42006)] defined the following fractional operator
and obtained boundedness of where .
Without assuming the growth condition on , the author considers some potential-like operator which satisfies the following:
is defined as follows: Let for with .
The author also considers some vector-valued inequalities of Fefferman-Stein type, uncentered maximal functions, and the boundedness on Morrey spaces.