Riesz potentials with radially symmetric densities are examined from the standpoint of Mellin multipliers. Various results are deduced from the underlying multipliers, including a decomposition of the potential into a product of Erdélyi-Kober fractional integrals. Distributional versions of these results are also produced and shown to be valid under less severe restrictions on the parameters than those required in a weighted Lp setting.

The Fréchet space fractional power theory developed in Schiavone1 is applied to the retarded potential associated with the three (space)–dimensional wave equation. It is shown that the powers obtained coincide with the corresponding Riesz fractional integral. Standard properties of the fractional integral are deduced from the fractional power theory. An indication of how the techniques developed here may be extended to higher dimensional fractional integrals is given.

A bilateral Laplace multiplier theory, based on Rooney's class , is developed for certain operators defined on the Fréchet spaces Dp,μ. The theory is applied to Riesz fractional integrals associated with the one-dimensional wave operator.

A theory of fractional powers of operators on an arbitrary Fréchet space is discussed. As special cases, multivariable fractional integrals and derivatives defined on certain spaces of test functions and generalised functions are obtained. In particular, properties of the two-dimensional Riesz fractional integral are determined and used to solve a distributional initial value problem involving the wave operator.