Spaces of quasi-periodic functions and oscillations in differential equations

We build spaces of q.p. (quasi-periodic) functions and we establish some of their properties. They are motivated by the Percival approach to q.p. solu tions of Hamiltonian systems. The periodic solutions of an adequate partial differential equation are related to the q.p. solutions of an ordinary dif ferential equation. We use this approach to obtain some regularization theo rems of weak q.p. solutions of differential equations. For a large class of differential equations, the first theorem gives a result of density: a par ticular form of perturbated equations have strong solutions. The second the orem gives a condition which ensures that any essentially bounded weak solu tion is a strong one.