Spacetime integral bounds for the energy-critical nonlinear wave equation

In this paper we prove a global spacetime bound for the quintic, nonlinear wave equation in three dimensions. This bound depends on the L t ∞ L x 2 L_{t}^{\infty } L_{x}^{2} and L t ∞ H ˙ 2 L_{t}^{\infty } \dot {H}^{2} norms of the solution to the quintic problem. The main motivation for this paper is the use of an interaction Morawetz estimate for the nonlinear wave equation.