  The existence of local unique mild solutions to the Navier-Stokes equations
in the whole space with an initial tempered distribution datum in critical
homogeneous or inhomogeneous Sobolev spaces is shown. Especially, the case when
the integral-exponent is less than 2 is investigated. The global existence is
also obtained for the initial datum in critical homogeneous Sobolev spaces with
a norm small enough in suitable critical Besov spaces. The key lemma is to
establish the bilinear estimates in these spaces, due to the point-wise decay
of the kernel of the heat semigroup.
