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Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in R 3 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(N) . 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 thatdoi:10.1088/0951-7715/21/6/005 fatcat:wr6vtwiksnao5mvzx4wq5tfh6m