Decay estimates of a tangential derivative to the light cone for the wave equation and their application

We consider wave equations in three space dimensions, and obtain new weighted $L^{\mathrm{\infty }}$-$L^{\mathrm{\infty }}$ estimates for a tangential derivative to the light cone. As an application, we give a new proof of the global existence theorem, which was originally proved by Klainerman and Christodoulou, for systems of nonlinear wave equations under the null condition. Our new proof has the advantage of using neither the scaling nor the pseudo-rotation operators.