  We prove an extension of the Stein-Weiss weighted estimates for fractional
integrals, in the context of Lp spaces with different integrability properties
in the radial and the angular direction. In this way, the classical estimates
can be unified with their improved radial versions. A number of consequences
are obtained: in particular we deduce precised versions of weighted Sobolev
embeddings, Caffarelli-Kohn-Nirenberg estimates, and Strichartz estimates for
the wave equation, which extend the radial improvements to the case of
arbitrary functions. Then we apply this technology in order to give new a
priori assumptions on weak solutions of the Navier-Stokes equation so as to be
able to conclude that they are smooth. The regularity criteria are given in
terms of mixed radial-angular weighted Lebesgue space norms.

Analysis of PDEs