On an estimate for the wave equation and applications to nonlinear problems

Abstract

We prove estimates for solutions of the Cauchy problem for the inhomogeneous wave equation on $\mathbb R^{1+n}$ in a class of Banach spaces whose norms depend only on the size of the space--time Fourier transform. The estimates are local in time, and this allows one, essentially, to replace the symbol of the wave operator, which vanishes on the light cone in Fourier space, with an inhomogeneous symbol, which can be inverted. Our result improves earlier estimates of this type proved by Klainerman--Machedon [4, 5]. As a corollary, one obtains a rather general result concerning local well-posedness of nonlinear wave equations, which was used extensively in the recent article [8].