Certain operators of fractional integration arising in connection with singular differential operators, Hankel transforms, and dual integral equations involve integration of fractional order with respect to $r^2$ and multiplication of functions by fractional powers of the independent variable. Such operations are not meaningful for distributions. In this paper a class of generalized functions is introduced on which such operations can meaningfully be performed. The operations are defined as adjoints of corresponding operations on a suitably selected space of testing functions. Relations to spherically symmetric n-dimensional distributions and to the singular differential operator \[ \frac{{d^2 }}{{dr^2 }} + \frac{{2\nu + 1}}{r} + \frac{d}{{dr}} \] are discussed.