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For different reasons it is very useful to have at one’s disposal a duality formula for the fractional powers of the Laplacean, namely, $({(-\Delta )}^{\alpha }u,\varphi )=(u,{(-\Delta )}^{\alpha }\varphi )$, 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...