ABSTRACT
B. S. Rubin discussed the multidimensional fractional integrals which are related to the Riesz potential. In particular he showed that by making use of expansions in terms of spherical harmonics, we can relate these operators to the 1-dimensional Erdélyi-Kober operators, which were studied by A. C. McBride. Mapping properties of the n-dimensional operators are then inherited from the corresponding properties of the 1-dimensional operators, which are well known. This chapter examines operators on spaces that consist of smooth functions and are n-dimensional analogues of the test function spaces introduced by McBride and W. J. Spratt. These spaces provide an appropriate setting for the study of such operators.
