Uniform decay estimates and the Lorentz invariance of the classical wave equation

An acceptable modification of some Sobolev spaces is found in order to obtain a useful decay rate in a nonlinear wave equation. Since the decay is dependent on invariance of the D'Alembertian of the associated Minkowski space-time, a family of first-order operators can be defined which yield generalized Sobolev norms. Two Lemmas are then definable for the smooth, compact decay to zero at infinity.