  We prove space-time decay estimates of suitable weak solutions to the
Navier-Stokes Cauchy problem, corresponding to a given asymptotic behavior of
the initial data of the same order of decay. We use two main tools. The first
is a result obtained by the authors in the paper "A remark on the partial
regularity of a suitable weak solution to the Navier-Stokes Cauchy problem"
(submitted), on the behavior of the solution in a neighborhood of $t=0$ in the
$L^\infty_{loc}$-norm, which enables us to furnish a representation formula for
a suitable weak solution. The second is the asymptotic behavior of the
$L^2(\R^3\setminus B_R)$ norm of $u(t)$ for $R\to\infty$. Following a Leray's
point of view, roughly speaking our result proves that a possible space-time
turbulence does not perturb the asymptotic spatial behavior of the initial data
of a suitable weak solution.
