We prove some existence, uniqueness and regularity results for the solutions to the Stokes problem in R n , n ≥ 2 in weighted Sobolev spaces W m,p α . This framework enables us to characterise for which data the problem has solutions with prescribed decay or growth at infinity. Moreover, we obtain an explicit representation as well as an asymptotic expansion of the solution for non-smooth decaying data. We also establish the density of smooth solenoidal vector fields in the subspace of v ∈ W 1,p α such that div v = 0.This paper is devoted to the Stokes problem that arises in the modelling of steadystate viscous fluid flows in R n , n ≥ 2. The velocity field u in the fluid and the pressure field π are governed by the equations:This problem has been widely studied when it is set in bounded domains with some additional boundary condition. In this case, the classical Sobolev spaces provide a suitable functional framework as Cattabriga 6 shows in the original paper dealing with this subject. However, in unbounded domains the latter framework may not be suitable. It is for instance the case when the problem is set in an exterior domain or in the whole space like in this paper. Hence, it is necessary to introduce some other functional spaces to study problem (S). *

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