Birkhoff normal forms for Hamiltonian PDEs in their energy space

We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a sufficiently strong energy estimate, we prove that its low super-actions are almost preserved for very long times. Roughly speaking, it means that only modes with the same linear frequency will be able to exchange energy in a reasonable time. Contrary to the previous existing results, we do not require the solutions to be particularly regular. They only have to live in the energy space. We apply our result to nonlinear Klein-Gordon equations in dimension d = 1 and nonlinear Schrödinger equations in dimension d ≤ 2 .