Optimal estimates for the uncoupling of differential equations

Our aim in this note is to give optimal conditions on the spectral gap for the existence of an uncoupling of a differential equation of the formż = Cz + H(=) into a system ofuncoupled equations of the form (x, y) = (Ax, By) + (F(x, Φ(x)), G(ψ(y),y)), whereC=A×B is a bounded linear operator on a Banach spaceZ=X×Y satisfying a spectral gap condition, andH=(F,G) is a Lipschitz function withH(0) = 0. We also give optimal conditions for the regularity of the manifoldsgraph Φ andgraph ψ, and optimal conditions for the regularity of the leaves of two foliations of the phase space associated to the uncoupling. Sharp estimates for the Lipschitz constant ofΦ andψ and for the Hölder exponent of the uncoupling homeomorphism and its inverse are also given.