The propagation theorem proved in Chapter 8 is a general-purpose tool for analyzing the regularity of solutions of general linear PDEs \(P u=f\) when \(P\in \Psi ^m(M)\) has a real homogeneous principal symbol, assuming one has information on u somewhere to begin with. In particular, one cannot, in general, control u globally only in terms of f (cf. the a priori control assumption of microlocal regularity of u encoded by the term \(E u\) in the estimate ( 8.21 )). What is needed for global control of u is the existence of a subset of phase space where one can get unconditional control of u. There are two main settings in which this happens.