  Onsager conjectured that weak solutions of the Euler equations for
incompressible fluids in 3D conserve energy only if they have a certain minimal
smoothness, (of order of 1/3 fractional derivatives) and that they dissipate
energy if they are rougher. In this paper we prove that energy is conserved for
velocities in the function space $B^{1/3}_{3,c(\NN)}$. We show that this space
is sharp in a natural sense. We phrase the energy spectrum in terms of the
Littlewood-Paley decomposition and show that the energy flux is controlled by
local interactions. This locality is shown to hold also for the helicity flux;
moreover, every weak solution of the Euler equations that belongs to
$B^{2/3}_{3,c(\NN)}$ conserves helicity. In contrast, in two dimensions, the
strong locality of the enstrophy holds only in the ultraviolet range.
