Propagation of singularities

Abstract

We present a proof of Bony’s propagation of singularities result for solutions to nonlinear PDE. As mentioned in the Introduction, we emphasize how C^r regularity of solutions rather than H^n/2+r regularity yields propagation of higher order microlocal regularity, giving in that sense a slightly more precise result than usual. Our proof also differs from most in using S ^m_{1, δ} calculus, with δ < 1. This simplifies the linear analysis to some degree, but because of this, in another sense our result is slightly weaker than that obtained using B^r S ^m_1,1 calculus by Bony and Meyer; see also Hörmander’s treatment [H4] using $\tilde{S}_{{1,1}}^m$calculus. Material developed in §3.4 could be used to supplement the arguments of §6.1, yielding this more precise result. In common with other approaches, our argument is modeled on Hörmander's classic analysis of the linear case.