epsilon-energies for weakly hyperbolic operators

Energy estimates are a fundamental tool to obtain many results for linear and nonlinear hyperbolic equations: well-posedness, dispersive estimates, regularity of the solutions, ... Approximated energies (or epsilon-energies) were introduced by Colombini-De Giorgi-Spagnolo and Colombini-Jannelli-Spagnolo in order to treat non regular or degenerate hyperbolic operators of second order. There are (at least) three methods to extend the notion of epsilon-energies to higher order equations. We prove here that all these methods are essentially equivalent.