Application of the realization of homogeneous Sobolev spaces to Navier-Stokes

Molecule spaces have been introduced by Furioli and Terraneo [Funkcial. Ekvac., 45 (2002), pp. 141-160] to study some local behavior of solutions to the Navier-Stokes equations. In this paper we give a new characterization of these spaces and simplify Furioli and Terraneo's result. Our analysis also provides a persistence result for Navier-Stokes in a subspace of L-2(R-3, (1 + \ x\(2))(alpha)dx), alpha < 5/2, which fills a gap between previously known results in the weighted-L-2 setting and those on the pointwise decay of the velocity field at infinity. Our main tool is the realization of homogeneous Sobolev spaces introduced by Bourdaud.