Distributional fractional powers of the Laplacean. Riesz potentials

For different reasons it is very useful to have at one's disposal a duality formula for the fractional powers of the Laplacean, namely, ${\displaystyle ({(-\Delta )}^{\alpha }u,\varphi )=(u,{(-\Delta )}^{\alpha }\varphi )}$, α ∈ ℂ, for ϕ belonging to a suitable function space and u to its topological dual. Unfortunately, this formula makes no sense in the classical spaces of distributions. For this reason we introduce a new space of distributions where the above formula can be established. Finally, we apply this distributional point of view on the fractional powers of the Laplacean to obtain some properties of the Riesz potentials in a wide class of spaces which contains the ${\displaystyle L^p}$-spaces.